IIT-JAM PHYSICS
ELECTRICITY & MAGNETISM
ELECTROSTATICS
COLOMB FORCE & FIELD
1. In terms of mass (M), length (L), time (T) and charge (C) dimensionof permittivity is:
(a) 𝑀𝐿3 𝑇 −2 𝐶 −2
(b) 𝑀 −1 𝐿−3 𝑇 2 𝐶 2
(c) 𝑀 −1 𝐿−2 𝑇 2 𝐶 2
(d) 𝑀𝐿2 𝑇 −2 𝐶 −2
2. A charge 𝑞is placed at the center of the line joining two equal charges 𝑄. The system of these charges
will be in equilibrium if 𝑞 is equal to
[B.H.U.-2010]
𝑄
(a)
2
𝑄
(b) − 2
(c)
𝑄
4
𝑄
(d) − 4
3. The ratio of repulsive Coulomb force and the attractive gravitational force between two protons is
(a) 1.24 × 1036
(b) 2.31 × 1039
(c) 3.21 × 1026
(d) 1.91 × 1032
[B.H.U.-2010]
4. Three positively charged particles lie on a straight line at positions 0, 𝑥 and 10 as indicated in the figure
below. Their charges are 𝑄, 2𝑄 and 4𝑄 cm respectively. [TIFR 2016]
If the charges at 𝑥 = 0 and 𝑥 = 10 are fixed and the charge at 𝑥 is movable, the system will be in
equilibrium when 𝑥 =
(a) 8
(b) 2
(c) 20/3
(d) 10/3
5. Two points charges +𝑞1 and +𝑞2 are fixed with a finite a distance ‘d’ between them. It is desired to put
a third charge 𝑞1 in between these two charges on the line joining them so that the charges 𝑞1 is in
equilibrium. That is:
[JAM 2005]
(a) Possible only if 𝑞3 is positive
(b) Possible only if 𝑞3 is negative
(c) Possible irrespective of the sign of 𝑞3 .
(d) Not possible at all.
6. The ratio 𝐹𝑐 /𝐹𝐺 of the electrostatic Coulomb force 𝐹𝐶 to the gravitational force 𝐹𝐺 between the proton
and the electron in the first Bohr orbit (or radius 𝑟𝐵 ) of a hydrogen atom is closest to the following value.
(a) 2 × 1039
(b) 2 × 1049
(c) 2 × 1042 𝑟𝐵
(d) −2 × 1035 /𝑟𝐵
7. Three point charges are placed at the corners of an equilateral triangle of side ‘a’ as shown in the figure.
Magnitude of force on charge at A is:
(a)
(b)
(c)
(d)
√3𝐾𝑞2
𝑎2
𝐾𝑎 2
𝑎2
2𝐾𝑞2
𝑎2
𝐾𝑞2
2𝑎 2
8.In the previous question if a point charge 𝑞 is placed at the centroid of the triangle force on it will be:
(a)
(b)
(c)
(d)
2𝐾𝑎 2
𝑎2
3𝐾𝑞2
𝑎2
6𝐾𝑞2
𝑎2
3√3𝐾𝑞2
𝑎2
9. Consider two points charges 𝑞 and 𝜆𝑞 located at the points, 𝑥 = 𝑎 and 𝑥 = 𝜇𝑎, respectively. Assuming
that the sum of the two charges is constant, what is the value of 𝜆 for which the magnitude of the
electrostatic force is maximum?
[JEST 2015]
(a) 𝜇
(b) 1
1
(c) 𝜇
(d) 1 + 𝜇
10. Five negative electric charges of magnitude 𝑒 are placed symmetrically on a circle of radius 𝑅. The
magnitude of the electric field at the center of the circle is
[B.H.U-2012]
𝑒
(a)
2
4𝜋𝜀0𝑅
(b) 0
(c)
𝑒
4𝜋𝜀0𝑅 2
5𝑒
(d) 4𝜋𝜀
0𝑅
sin
2𝜋
5
2
11. When five equal point charges are placed at the corners of regular hexagon, electric field at the centre
is 𝐸0 . If another charge is removed such that charges from two adjacent corners are missing then electric
field at centre will be
(a) 2𝐸0
𝐸
(b) 0
2
(c) √3𝐸0
(d) 2√3𝐸0
12. Two free point charges +4Q and +Q are placed at a distance 𝑟. A third charge 𝑞 is so placed that all the
three are in equilibrium.
1
(a) 𝑞 is placed at a distance 3 𝑟 from 4Q
1
(b) 𝑞 is placed at a distance 3 𝑟 from Q
(c) 𝑞 =
4𝑄
9
(d) 𝑞 = −
4𝑄
9
13. Two equal charges are 2𝑑′ distance apart. How far form the mid point should another charge be
placed on perpendicular bisector so that it experiences maximum force?
(a) 𝑑
(b) 𝑑/2
(c) 𝑑/√2
(d) √2𝑑
14. Four charges 𝑄1 , 𝑄2 , 𝑄3 and 𝑄4 of same magnitude are fixed along the 𝑥-axis at 𝑥 = −2𝑎, −𝑎, +𝑎 and
+2𝑎 respectively. A positive charge 𝑞 is placed on the positive axis at a distance 𝑏 > 0. Four options of the
signs of these charges are given in List-I. The direction of the forces on the charges 𝑞 is given in List-II.
Match List-I with List-II and select the correct answer using the code given below the lists.
List-I
P. 𝑄1 , 𝑄2 , 𝑄3 , 𝑄4 all positive
Q. 𝑄1 , 𝑄2 positive 𝑄3 , 𝑄4 negative
R. 𝑄1 , 𝑄4 positive, 𝑄2 , 𝑄4 negative
S. 𝑄1 , 𝑄3 positive, 𝑄2 , 𝑄4 negative
List-II
1. +𝑥
2. −𝑥
3. +𝑦
4. −𝑦
(a) P-3, Q-1, R-4, S-2
(b) P-4, Q-2, R-3, S-1
(c) P-3, Q-1, R-2, S-4
(d) P-4, Q-2, R-1, S-3
15. Three charges are located on the circumference of a circle of radius ‘R’ as shown in the figure below.
The two charges Q subtend an angle 90° at the centre of a circle. The charge ‘q’ is symmetrically placed
with respect to the charges Q. If the electric field at the centre of the circle is zero, what is the magnitude
of Q?
(a) 𝑞/√2
(b) √2𝑞
(c) 2𝑞
(d) 4𝑞
16. A few electric field lines for a system of two charges 𝑄1 and 𝑄2 fixed at two different points on the xaxis are shown in the figure. These lines suggedst that
(a) |𝑄1 | > |𝑄2 |
(b) |𝑄1 | < |𝑄2 |
(c) at a finite distance to the left of 𝑄1 the electric field is zero
(d) at a definite distance to the right of 𝑄2 the electric field is zero
17. Four equal point charges 𝑞 are placed at the corners of a square of side ‘a’. Magnitude of force on each
charge is
(a)
3𝐾𝑞2
𝑎2
(b) (√2 + 1)
𝐾𝑞2
𝑎2
2
𝐾𝑞
(c) (√2 + 1⁄2) 2
𝑎
𝐾𝑞2
(
(d) √2 + 2) 𝑎2
18. In the previous question what charge must be placed at centre so that force on all charges becomes
zero.
(a)
(b)
−(√2+1)𝑞
2
−2(√2+1)𝑞
2
−(2√2+1)𝑞
(c)
4
(d) −4𝑞
19. Two particles each of mass 𝑚 are moving in a circle of radius ‘a’ under the effect of their electrostatic
force of attraction. If magnitude of charge on each particle is 𝑞, their speed is:
𝐾𝑞2
(a) √4𝑚𝑎
𝐾𝑞2
(b) √2𝑚𝑎
(c) √
𝐾𝑞2
𝑚𝑎
(d) √
2𝐾𝑞2
𝑚𝑎
20. Two point charges attract each other with a force of 2 × 10−3 𝑁 charge on one is twice the other. When
distance between them is increased by 10cm, force becomes 5 × 10−4 𝑁 value of their charges is
100
−200
(a) 3 𝑛𝑐, 3 𝑛𝑐
(b)
(c)
(d)
50
𝑛𝑐,
3
200
3
10
3
−100
𝑛𝑐,
𝑛𝑐,
𝑛𝑐
3
−400
20
3
3
𝑛𝑐
𝑛𝑐
21. Two charges 𝑞 and −2𝑞 are placed 𝑑 distance apart on a smooth table. At what distance from 𝑞 should
a third charge be placed so that force on all three charges becomes zero. What is value of third charge.
2
(a) 𝑑(√2 − 1), 2(√2 + 1) 𝑞
2
(b) 𝑑(√2 + 1), 2(√2 + 1) 𝑞
2
(c) 𝑑(√2 + 1), −2(√2 + 1) 𝑞
2
(d) 𝑑(√2 − 1), −2 (√2 + 1) 𝑞
22. Four equal point charge 𝑞 are fixed at the corners of square side ‘a’. Another point charge 𝑞0 is placed
above the centre of square at a height 𝑎/√2 from centre. Force on 𝑞0 is
2𝐾𝑞𝑞
(a) 2 0
(b)
(c)
(d)
𝑎
4𝐾𝑞𝑞0
𝑎2
√2𝐾𝑞𝑞0
𝑎2
2√2𝐾𝑞𝑞0
𝑎2
23. Eight equal charges 𝑞 are placed at the corners of a cube of side ‘a’. Force on each charge due to the
other charges is
(a)
(b)
(c)
(d)
𝐾𝑞2
3
1
[√3 + √ + ]
2
3
𝑎2
𝐾𝑞2
𝑎2
𝐾𝑞2
3
1
[√3 + √ + ]
2
√3
3
1
[√2 + √ + ]
2
3
𝑎2
𝐾𝑞2
𝑎2
2
1
[√2 + √ + ]
3
√2
24. Four identical simple pendula consisting of light string of length 𝑙 and small bob of mass 𝑚 are
suspended from common point of suspension. The bobs of pendula are given equal charge due to which
each string makes 60° with downward vertical. Charge on each bob is
√3𝑚𝑔𝑙2
(a) √(2√2+1)𝐾
3√3𝑚𝑔𝑙2
(b) √(2√2+1)𝐾
3𝑚𝑔𝑙2
(c) √(√2+1)𝐾
(d) √
𝑚𝑔𝑙2
(√2+1)𝐾
25. An 𝛼 particle passes rapidly through the exact centre of a hydrogen molecule, moving on a line
perpendicular to the internuclear axis. The distance between the nuclei is 𝑏. Where on its path does the 𝛼
particle experience the greatest force? Assume that the nuclei do not move much during the passage of
the 𝛼 particle. Also neglect the electric field of the electrons in the molecule.
𝑏
(a) 2
𝑏
(b) 2√2
(c)
𝑏
√2
(d) none
26. A charge distribution has the charge density given by 𝜌 = 𝑄{𝛿(𝑥 − 𝑥0 ) − 𝛿(𝑥 + 𝑥0 )}. For this charge
distribution the electric field at (2𝑥0 , 0,0)
2𝑄𝑥̂
(a) 9𝜋∈ 𝑥 2
0 0
𝑄𝑥̂
(b) 4𝜋∈
2
0 𝑥0
𝑄𝑥̂
(c) 4𝜋∈
(d)
2
0 𝑥0
𝑄𝑥̂
16 𝜋∈0 𝑥02
27. Ten identical charges of 500 𝜇𝑐 are placed on the periphery of a circle of radius 2𝑚. Spacing between
any two charge is same. A charge of −20𝜇𝑐 is placed on the axis, 2m from the centre of the circle,
magnitude of force on this charge is
(a) 65.2 N
(b) 72.3 N
(c) 69.5 N
(d) 79.5
28. A point charge 𝑞 = 50 𝜇𝑐 is located in the x-y plane at the point of the position vector 𝑟⃗0 = 2𝑖̂ + 3𝑗̂.
Electric force on a point charge 𝑞 ′ = 20 𝜇𝑐 placed at point 𝑟⃗ = 8𝑖̂ − 5𝑗̂ is
(a) 2 × 10−3 𝑁
(b) 4 × 10−3 𝑁
(c) 3 × 10−3 𝑁
(d) 9 × 10−3 𝑁
29. Four equal point charges are placed at the corners of a square. How many points are there in space
where electric field is zero.
(a) One
(b) Two
(c) Five
(d) Infinite
30. If 𝐸⃗⃗1 = 𝑥𝑦𝑖̂ + 2𝑦𝑧𝑗̂ + 3𝑥𝑧𝑘̂ and 𝐸⃗⃗2 = 𝑦 2 𝑖̂ + (2𝑥𝑦 + 𝑧 2 )𝑗̂ + 2𝑦𝑧𝑘̂ then
[H.C.U-2016]
(a) both are not static electric fields
(b) both are static electric fields
(c) only 𝐸⃗⃗1 is a possible static electric field
(d) only 𝐸⃗⃗2 is a possible static electric field
31. The electric field in the x-y plane is given by 𝐸⃗⃗ = 𝑖̂8𝑥 − 𝑗̂4𝑦. Then equation for the lines of force is
given by
[H.C.U-2015]
(a) 𝑥𝑦 2 = constant
(b) 𝑥 2 𝑦 = constant
(c) 𝑥 2 + 𝑦 = constant
(d) 𝑥𝑦 = constant
32. In the laboratory, four point charges +𝑄, −𝑄, +𝑄, −𝑄 are placed at the four ends of a horizontal
square of side a, as shown in the figure below. The number of neutral points (where the electric field
vanishes) is
(a) ∞
(b) 4
(c) 1
(d) zero
[TIFR 2010]
33. A thin spherical oil drop carrying a net charge 𝑞 is balanced in still air with a vertical uniform electric
81
field of strength × 𝑉𝑚−1 . When the field is switched off the drop is observed to fall with terminal
7
velocity 2 × 10−3 𝑚𝑠 −1 . Given, 𝑔 = 9.8 𝑚𝑠 −2 viscosity of the air = 1.8 × 10−5 𝑁𝑠 𝑚−2 and the density of oil
= 900 kg 𝑚−3 , the magnitude of q is
(a) 1.6 × 10−19 𝐶
(b) 3.2 × 10−19 𝐶
(c) 4.8 × 10−19 𝐶
(d) 8.0 × 10−19 𝐶
34. Four point charges, each of +q, are rigidly fixed at the four corners of a square planar soap film of side
‘a’. The surface tension of the soap film is 𝛾. The system of charges and planar film are in equilibrium, and
𝑞2
1/𝑁
𝑎 = 𝑘[ ]
𝛾
, where ′𝑘 ′ is a constant. Then 𝑁 is _________
35. An electric field in a region is given by 𝐸⃗⃗ (𝑥, 𝑦, 𝑧) = 𝑎𝑥𝑖̂ + 𝑐𝑧𝑗̂ + 6𝑏𝑦𝑘̂. For which values of a, b, c does
this represent an electrostatic field? [JEST 2012]
(a) 13, 1., 12
(b) 17, 6, 1
(c) 13, 1, 6,
(d) 45, 6, 1
36.Which of the following cannot describe a conservative force field? [IISc. 2007]
(a) (𝑥 2 − 𝑦 2 )𝑖̂ − 2𝑥𝑦𝑗̂2𝑥𝑦 𝑘̂
(b) 𝑥𝑖̂ + 𝑦𝑗̂ − 2𝑧𝑘̂
(c) sin(𝑥) 𝑖̂ − cos(𝑦) 𝑗̂
(d) 𝑥𝑖̂ − 𝑦𝑗̂ + 𝑧𝑘̂
37. Two spherical, nonconducting and very thin shells of uniformly distributed positive charge 𝑄 and
radius 𝑑 are located a distance 10𝑑 from each other. A positive point charge 𝑞 is placed inside one of the
shells at a distance 𝑑/2 from the center, on the connecting the centers of the two shells, as shown in the
figure below. What is the net force on the charge 𝑞?
(a)
𝑞𝑄
(b) 361 𝜋𝜀
𝑞𝑄
(c) 441𝜋𝜀
(d)
to the left
361 𝜋𝜀0𝑑 2
𝑞𝑄
0𝑑
0𝑑
2
2
𝑞𝑄
441𝜋𝜀0𝑑 2
to the right
to the left
to the right
[B.H.U-2012]
38. Which of the following force fields, given below in spherical polar coordinates, is not a conservative
force field? [IISc. 2008]
𝑒̂
(a) 𝑟𝑟
(b)
𝑒̂ 𝜙
𝑟
sin 𝜃
(c) 𝑟 𝑒̂𝜃
(d) cos 𝜃 𝑒̂𝑥 − sin 𝜃 𝑒̂𝜃
39. If 𝐸⃗⃗1 = 𝑥𝑦𝑖̂ + 2𝑦𝑧𝑗̂ + 3𝑥𝑧𝑘̂ and 𝐸⃗⃗2 = 𝑦 2 𝑖̂ + (2𝑥𝑦 + 𝑧 2 )𝑗̂ + 2𝑦𝑧𝑘̂ then [JEST 2013]
(a) Both are impossible electrostatic fields
(b) Both are possible electrostatic fields
(c) Only 𝐸⃗⃗1 is a possible electrostatic field
(d) Only 𝐸⃗⃗2 is a possible electrostatic field
40. A circular loop of radius 𝑅, carries a uniform line charge density 𝜆. The electric field, calculated at a
distance 𝑧 directly above the center of the loop, is maximum if 𝑧 is equal to, [JEST 2015]
𝑅
(a)
(b)
√3
𝑅
√2
𝑅
(c) 2
(d) 2𝑅
41. Two positive point charges of magnitude Q each are separated by a distance “2d”. A test charge ′𝑞0 ′ is
located in a plane which is normal to the line joining these charges and midway between them. The locus
of the points in this plane for which the force on the test charge has a maximum value is
(a) a circle of radius 𝑟 = 𝑎/√2
(b) a circle of radius 𝑟 = 𝑎
(c) a straight line
(d) an ellipse
42. Three positive charges of equal to value 𝑞 are placed at the vertices of an equilateral triangle. The
resulting lines of force should be sketched as in
43.Figure shows three small spheres that have charges of equal magnitudes, (whose sign are shown) and
are at rest on a frictionless surface, 𝑦 and 𝑧 are fixed in place and are at equal distance from sphere 𝑥.
Which of the four paths shown will 𝑥 take?
(a) A
(b) B
(c) C
(d) D
44. Two identical point charges are placed at a separation of 𝑙. P is a point on the joining the charges, at a
distance 𝑥 from any one charge. The field at P is E. E is plotted against 𝑥 for values of 𝑥 from close to zero
to slightly less than 𝑙. Which of the following best represents the resulting curve?
45. The electric potential in the 𝑥𝑦-plane in a certain region of space is give by 𝑉(𝑥, 𝑦) = 𝐶(6𝑥 2 𝑦 − 2𝑦 3 ),
where 𝑥 and 𝑦 are in meters and 𝑉 is in volts and C is a constant. What is the magnitude of the ycomponent of the electric field at the point (-1, 2)? [H.C.U.-2011]
(a) 0 V/m
(b) 18 C V/m
(c) 24C V/m
(d) 30C V/m
46. Two connected charges of +𝑞 and −𝑞 respectively are at fixed distance AB apart in a non-uniform
electric field whose lines of force are shown in the figure. The resultant effect on the two charges is
(a) a torque vector in the plane of the paper and no resultant force
(b) a resultant force in the plane of the paper and no torque
(c) a torque vector normal to the plane of the paper and no resultant force
(d) a torque vector normal to the plane of the paper and a resultant force in the plane of the paper
47. Two positive and two negative charges are kept in 𝑥-𝑦 plane in free space as shown in the figure. The
magnitude of electric field due to the system of charges at a point 𝑃(0, 𝑦) will be (𝑦 ≫ 𝑑)
√5𝑞𝑑
(a) 4𝜋𝜀
3
0𝑦
2𝑑𝑞
(b) 4𝜋𝜀
(c)
(d)
0𝑦
3
𝑑𝑞
4𝜋𝜀0𝑦3
3𝑑𝑞
4𝜋𝜀0 𝑦3
48. Five charged particles, each having a charge 𝑞, are placed at the five vertices of a regular hexagon. The
sides of the hexagon have equal length ‘a’. The magnitude of electric field at the centre of the hexagon is
𝑞
(a) 4𝜋∈
0𝑎
2𝑞
(b) 4𝜋∈
2
(c) 4𝜋∈
2
0𝑎
𝑞
0𝑎
(d) none of these
49. A block of mass m having charge 𝑞 is hung by a spring of spring constant 𝑘 in a vertical electrostatic
field E. The string extension in equilibrium will be
(a)
(b)
(c)
(d)
𝑚𝑔
𝑘
𝑞𝐸
𝑘
𝑚𝑔+𝑞𝐸
𝑘
𝑚𝑔−𝑞𝐸
𝑘
50. A wooden block SHM on a frictionless surface with frequency, 𝑣0. The block carries a charge +Q on its
surface. If now a uniform electric field 𝐸⃗⃗ is switched on shown, then the SHM of the block will be
(a) of the same frequency and with shifted mean position
(b) of the same frequency and with same mean position
(c) of changed frequency and with shifted mean position
(d) of changed frequency and with the same mean position
51. A mass M is attached to a solid support with an ideal spring of spring constant k. The natural
frequency of the system is oo. If the mass now carries a charge +Q and an external electric field E along
the axis of the springis turned on, the frequency for small oscillations [IISc. 2007]
(a) becomes zero
(b) increases
(c) decreases
(d) remains unchanged
52. Six charges, three positive and three negative of equal magnitude are to be placed at the vertice ofa
regular hexagon such that the electric field at O is double the electric field at O is double the electricfield
when only one positive charge of same magnitude is placed at R. Which of the following arrangmentsof
charges is possible for P, Q, R, S, T and U respectively?
(a) +, −, +, −, −, +
(b) +, −, +, −, +, −
(c) +, +, −, +, −, −
(d) −, +, +, −, +, −
53. An electron of mass m and charge 𝑒, initially at rest, gets accelerated by a constant electric field 𝐸. The
rate of change of de-Broglie wavelength of this electron with time, (ignoring relativistic effects) is given
by
ℎ
(a) − 2
(b) −
𝑒𝐸𝑡
𝑒ℎ𝑡
𝐸
𝑚ℎ
(c) − 𝑒𝐸𝑡 2
ℎ
(4) − 𝑒𝐸
54. A cube of side a has point charges +𝑄 located at each of its vertices except for the origin where the
charge is −𝑄 as shown below. [IISc. 2010]
The electric field at the center is:
−𝑄
(a) 3√3𝜋𝜀 𝑎2 (𝑥̂ + 𝑦̂ + 𝑧̂ )
𝑄
0
(b) 3√3𝜋𝜀
2
(𝑥̂ + 𝑦̂ + 𝑧̂ )
2
(𝑥̂ + 𝑦̂ + 𝑧̂ )
2
(𝑥̂ + 𝑦̂ + 𝑧̂ )
0𝑎
−2𝑄
(c) 3√3𝜋𝜀
0𝑎
2𝑄
(d) 3√3𝜋𝜀
0𝑎
55. Four charges are placed at the corners of a square as shown in figure. Electric field at the centre of
square is
√2𝑞
(a) 𝜋∈
(b)
(c)
0𝑎
2
𝑞
√2𝜋∈0 𝑎 2
2√2𝑞
𝜋∈𝑎 2
𝑞
(d) 2√2𝜋∈
0𝑎
2
56. A particle of charge q is moving in a circular path of radius R with constant speed 𝑣(≪ 𝑐). Magnitude
of rate of change of electric field at centre is
𝑞𝑣
(a)
2
(b)
4𝜋∈0𝑅
𝑞𝑣
2𝜋∈0 𝑅 3
(c) zero
𝑞𝑣
(d) 8𝜋∈ 𝑅3
0
57. A charge Q is uniformly distributed on a circular ring of radius R. A particle of mass m and charge q
placed on its axis near centre executes S.H.M. Time period of oscillation of the particle is T. If radius ofthe
ring andcharge are doubled then time period of oscillation becomes
(a) 2T
(b) 4T
(c) √2T
(d) 2√2T
58. In the figure shown, four point charges are placed at the corners of a square of side ‘a'. Electric field at
the centre of square is
(a)
(b)
(c)
(d)
4𝑘𝑞
𝑎2
2√2𝐾𝑞
𝑎2
4√2𝐾𝑞
𝑎2
8𝐾𝑞
𝑎2
59. A charge q is uniformly distributed over a uniform rod of length L and a point charge 𝑞 ′ is placed at a
distance ′𝑑′ from its one end (distance 𝑑 is measured parallel to the length of the rod). Electric force
between the rod and point charge is
(a)
(b)
𝐾𝑞𝑞′
𝑑2
𝐾𝑞𝑞′
2
(𝑑+ 𝐿⁄2)
𝐾𝑞𝑞′
(c) 𝑑(𝑑+𝐿)
𝐾𝑞𝑞′
(d) (𝑑+𝐿)2
60. A segment of a circular wire of radius 𝑅, extending from 𝜃 = 0 to 𝜋⁄2, carries a constant linear charge
density, 𝜆. The electric field at origin ‘O’ is: [JAM 2012]
𝜆
(a) 4𝜋𝜀 𝑅 (−𝑥̂ − 𝑦̂)
0
1
𝜆
(b) 4𝜋𝜀 𝑅 (−
0
𝜆
1
√2
𝑥̂ −
1
1
√2
𝑦̂)
(c) 4𝜋𝜀 (− 2 𝑥̂ − 2 𝑦̂)
(d) 0
0
61. For an infinitely long wire with uniform line-charge density, 𝜆, along the z-axis, the electric field at a
point (𝑎, 𝑏, 0) away from the length origin is (𝑒̂𝑥 , 𝑒̂𝑦 , and 𝑒̂𝑧 are unit vectors in Cartesian- coordinate
system) [JAM 2016]
𝜆
(𝑒̂𝑥 + 𝑒̂𝑦 )
(a)
2
2
2𝜋𝜀0√𝑎 +𝑏
𝜆
(b) 2𝜋𝜀
(c)
(d)
2
2
0 (𝑎 +𝑏 )
𝜆
2𝜋𝜀0√𝑎 2+𝑏2
𝜆
2𝜋𝜀0 √𝑎 2+𝑏2
(𝑎𝑒̂𝑥 + 𝑏𝑒̂𝑦 )
𝑒̂𝑥
𝑒̂𝑧
62. A very circular wire of radius a carries electric charge of uniform linear density 𝜌. On its axis, the
magnitude of the electric field attains its maximum value at a perpendicular distance d form the plane of
the wire (see the figure below). The value of d is [JNU 2011]
(a) 0
(b) 0.32 a
(c) 0.50 a
(d) 0.71 a
63. The electric fields of a long straight wire varies with distance r (perpendicular to the wire) as
(a) 1/𝑟
(b) ln 𝑟
(c) 1/𝑟 2
(d) 1/𝑟 3
64. A charge 𝑞 is uniformly distributed on a thin ring of radius R. A point charge q is placed on the axis of
the ring. Maximum value of force on the point charge is
𝐾𝑞2
(a) 3√3𝑅2
(b)
(c)
(d)
2𝐾𝑞2
3√3𝑅 2
√2𝐾𝑞2
3𝑅 2
2𝐾𝑞2
𝑅2
65. In the figure shown a long line charge of linear charge density 𝜆 is lying along the axis of a uniformly
charged ring of same linear charge density 𝜆. One end of the line charge is at the centre of the ring.
Electric force between the two is
(a)
(b)
(c)
(d)
𝜆2
2∈0
𝜆2
4∈0
𝜆2
∈0
2𝜆2
∈0
66. In the figure shown a long line charge of charge density 𝜆 is connected to a semi-circle of radius 𝑅.
Electric field at the centre of circle is,
(a)
(b)
(c)
(d)
𝜆𝐾
𝑅
2𝜆𝐾
𝑅
√2𝜆𝐾
𝑅
2√2𝜆𝐾
𝑅
67. Two long line charges having charge densities 𝜆 and – 𝜆are joined at one end at right angle, as shown
in figure. Electric field at point P is
(a)
(b)
(c)
(d)
𝐾𝜆
𝑅
2𝐾𝜆
𝑅
√2𝐾𝜆
𝑅
𝐾𝜆
√2𝑅
68. Two uniform line charges of 𝜆 = 4𝑛 C/m each are parallel to the z-axis at (0, 4)m and (0, -4)m.
Magnitude of electric field at points (±4, 0,0 ) is
(a) 9 V/m
(b) 18 V/m
(c) 4.5 V/m
(d) 9√2 V/m
69. The charge per unit length of a circular wire of radius 𝑎 in the 𝑥𝑦-plane, with its center at the origin, is
𝜆 = 𝜆0 cos 𝜃, where 𝜆0 is a constant and the angle 𝜃 is measured from the positive 𝑥-axis. The electric field
at the center of the circle is
𝜆
(a) 𝐸⃗⃗ = − 4𝜀 0𝑎 𝑖̂
0
𝜆
(b) 𝐸⃗⃗ = 4𝜀 0𝑎 𝑖̂
0
𝜆
(c) 𝐸⃗⃗ = − 4𝜀 0𝑎 𝑗̂
0
𝜆
(d) 𝐸⃗⃗ = 4𝜋𝜀0 𝑎 𝑘̂
0
70. A large flat metal surface has a uniform charge density +𝜎. An electron of mass m and charge 𝑒 leaves
the surface at point A with speed 𝑢, and returns to it at point B. Disregard gravity. The maximum value of
AB is
(a)
(b)
𝑢 2 𝑚𝜀0
𝜎𝑒
𝑢 2 𝑒𝜀0
𝑚𝜎
𝑢 2𝑒
(c) 𝜀
(d)
0 𝜎𝑚
𝑢 2 𝜎𝑒
𝜀0 𝑚
71. An electron revolves around a long wire of uniform linear charge density. Which of the following
statements is/are correct.
(a) speed of the particle is independent of distance from wire
(b) acceleration of particle is independent of distance from wire
(c) acceleration of particle is inversely proportional to distance from wire
(d) work done on electron due to electric force is zero
72. In the figure shown electric field at the centre of semi-circle is
𝜆
(a) 2𝜋∈ 𝑎 𝑖̂
0
(b)
𝜆
4𝜋∈0 𝑎
𝜆
𝑖̂
(c) 2𝜋∈ 𝑎 𝑗̂
0
(d)
𝜆
2𝜋∈ 0𝑎
(−𝑗̂)
73. A long line charge having uniform charge density 𝜆 lies along the straight line 𝑦 = 𝑚𝑥 + 𝑐. Electric
field at the origin is
𝐾𝜆
(a)
(b)
(c)
(d)
𝑐
2𝐾𝜆
𝑐
2𝐾𝜆
𝑐
𝐾𝜆
𝑐
√1 + 𝑚2
√1 + 𝑚2
74. A flat circular disc of radius R carries a uniform surface charge density 𝜎. The potential at a distance 𝑧𝜎
(√𝑅2 + 𝑧 2 − 𝑧). The electric field at z is [H.C.U-2016]
perpendicularly above the centre of this disk is
2∈ 0
𝜎
𝑧
(a) 2∈ (1 −
0
𝜎
(b) 2∈ (
0
(c)
𝜎
4∈0
𝜎
√𝑧 2+𝑅 2
𝑧
2√𝑧 2+𝑅 2
𝑧
(1 −
(d) 4∈ (1 −
0
)
− 1)
√𝑧 2+𝑅 2
𝑧
)
2√𝑧 2+𝑅 2
)
75. A thin rod of length L and uniform charge 𝑞 is placed near a long line charge of charge density 𝜆 as
shown in the figure. Force on the thin rod is
(a)
(b)
(c)
(d)
2𝐾𝜆𝑞
𝐿
2𝐾𝜆𝑞
𝐿
𝐾𝜆𝑞
𝐿
log 𝑑
log (
log (
𝐿
2𝐾𝜆𝑞
𝐿
𝑑+𝐿
𝑑
𝑑+𝐿
)
𝑑
𝑑+𝐿
log (
𝐿
)
)
76. An electron revolves around a long line charge of density 𝜆 under the effect of electrostatic force as
shown in figure. Kinetic energy of electron is
𝑒𝜆
(a) 2𝜋∈
𝑒𝜆
0
(b) 4𝜋∈
0
𝑒𝜆
(c) 𝜋∈
0
2𝑒𝜆
(d) 𝜋∈
0
77. A block of mass m containing a net positive charge q is placed on a smooth horizontal table
whichterminates in a vertical wall as shown in figure. The distance of the block from the wall is d. A
horizontalelectric field E towards right is switched on. Assuming elastic collisions (if any) the time period
of theresulting oscillatory motion is ____________ second.
(Take 𝑚 = 10−3 𝑘𝑔, 𝑑 = 2𝑚, 𝑞 = 10−3 𝐶, 𝐸 = 1𝑉/𝑚)
78. Acharge Q is uniformly distributed on a circular ring of radius R. A particle of mass m and charge q
placed on its axis near centre executes S.H.M. Time period of oscillation of the particle is T. If particle of
same mass and charge q oscillates parallel to plane of ring about centre, then time period will be
(a) √2𝑇
(b) 4𝑇
(c) 2T
(d) 2√2𝑇
79. In the figure shown tension in the string at equilibrium position is
𝑞𝜆
(a) √(𝑚𝑔)2 + (2𝜋∈ 𝑙 )
2
0
2
𝑞𝜆
(b) √(𝑚𝑔)2 + (2𝜋∈
0 𝑙 sin 𝜃0
)
𝑞𝜆
(c) 𝑚𝑔 + 2𝜋∈
0𝑙
(d) √(𝑚𝑔)2 + (2𝜋∈
𝑞𝜆
0 𝑙 cos 𝜃
)
2
80. A circular ring of radius ‘a’ and uniform charge density 𝜆is bent along diameter to make to halves
perpendicular to each other, electric field at the centre of bent ring is
2𝐾𝜆
(a)
(b)
(c)
(d)
𝑎
√2𝐾𝜆
𝑎
2√2𝐾𝜆
𝑎
4√2𝐾𝜆
𝜋𝑎 2
81. A charge 𝑞 is uniformly distributed over the periphery of a circular thread of radius ‘a’. Another point
charge 𝑞 is placed at the centre of the ring. Tension in the thread due to electrostatic repulsion between
charges on the periphery and that at the centre is
(a)
𝐾𝑞2
2𝜋𝑎 2
𝐾𝑞2
(b) 𝜋𝑎2
(c)
(d)
𝐾𝑞2
4𝜋𝑎 2
2𝐾𝑞2
𝜋𝑎 2
82. Which of the following electric fields could exist in a region of space that contains no charges? (In
these expressions, 𝐴 is a constant, and I, j and k are unit vectors pointing in the 𝑥, 𝑦 and 𝑧 directions,
respectively). [B.H.U-2012]
(a) 𝐴(2𝑥𝑦𝑖 − 𝑥𝑧 𝑘)
(b) 𝐴(−𝑥𝑦 𝑗 + 𝑥𝑧 𝑘)
(c) 𝐴(𝑥𝑧 𝑖 + 𝑥𝑧 𝑖)
(d) 𝐴 𝑥𝑦𝑧 (𝑖 + 𝑗)
83. Two infinitely wires are connected to a semi-circular wire of radius 𝑅. If there is uniform charge
density 𝜆 everywhere on the wires. Electric field at the centre of semi-circle is
(a)
𝜆
𝜋∈0 𝑅
𝜆
(b) 2𝜋∈
0𝑅
(c) zero
2𝜆
(d) 𝜋∈ 𝑅
0
84. In the figure shown in 𝐸𝐴 and 𝐸𝐵 be magnitude of electric field at A and B due to the hemisphere of
radius R and uniform surface charge density 𝜎 then which of the following is correct.
𝜎
(a) 𝐸𝐴 − 𝐸𝐵 = 4∈
0
𝜎
(b) 𝐸𝐴 + 𝐸𝐵 = 4∈
0
𝜎
(c) 𝐸𝐴 − 𝐸𝐵 = − 4∈
(d) 𝐸𝐴 = 𝐸𝐵
0
85. A long thin sheet of charge of uniform charge density 𝜎 lies in x-y plane as shown in figure. Width of
the sheet is ‘2a’. Electric field at point (𝑑, 0,0) where 𝑑 > 𝑎 is
𝑑+𝑎
(a) 2𝐾𝜎 log (𝑑−𝑎 )
𝑑
(b) 2𝐾𝜎 log (𝑎 )
𝑑+𝑎
(c) 𝐾𝜎 log (𝑑−𝑎 )
𝑑
(d) 𝐾𝜎 log 𝑎
86. In prevoius question electric field at point (0,0, 𝑑) is
𝑎
(a) 4 𝐾 𝜎 tan−1 𝑑
𝑑
(b) 4𝐾 𝜎 tan−1 𝑎
𝑎
(c) 2𝐾𝜎 tan−1 𝑑
(d) 2𝐾𝜎 tan−1
𝑑
𝑎
87. A semi infinite line charge of linear charge density 𝜆 has the shape as shown in the figure. Portion ABC
forms three-fourth of a circle of radius R while the straight portion from C to infinity is parallel to BOA.
The field at the centre of circle (O) is
(a)
𝜆
2√2𝜀0𝑅
𝜆
(b) 2𝜋𝜀
0𝑅
(c) zero
(d) None of these
88. From an infinite non-conducting sheet having uniform surface charge density 𝜎a circular portion of
radius 𝑅 is removed from its middle region. Electric field at a distance √3𝑅 on axis of circular portion is
𝜎
(a)
(b)
2∈0
𝜎
√3∈ 0
√3𝜎
(c) 2∈
0
√3𝜎
(d) 4∈
0
89. Charge 𝑞 is uniformly distributed over a thin half ring of radius R. The electric field at the centre of the
ring is
𝑞
(a) 4𝜋∈ 𝑅2
0
𝑞
(b) 2𝜋2 ∈
0𝑅
𝑞2
2
(c) 2𝜋∈
2
0𝑅
𝑞2
(d) 4𝜋∈
0𝑅
2
𝑟
𝑅
𝑅
𝛽
90. A sphere of radius R has volume charge density 𝜌 = 𝜌0 (1 − ). Electric field is maximum at 𝑟 = ,
value of 𝛽 is ________
91. A circular ring of radius 𝑟 of fine wire carries a uniformly distributed positive charge 𝑞. The electric
field intensity at the centre of the ring, due to the charge on a portion of the ring subtending an angle 𝜃 at
the centre is
𝑞
𝜃
(a) 2 2 sin ( )
4𝜋 𝜀0𝑟
𝑞
(b) 4𝜋𝜀
0𝑟
𝑞
(c) 4𝜋𝜀
0
2
𝜃
( )
2 sin
2
𝜃
sin2 ( 2 )
𝑟2
𝑞
(d) 4𝜋2𝜀
𝜃
0𝑟
2
sin2 ( 2 )
92. The given diagram shows two infinite lines of charges having equal (in magnitude) linear charge
density but with opposite sign. The electric field at any point on x axis (for 𝑥 > 0) is along the unit vector
(a) cos 𝜃 𝑖̂ + sin 𝜃 𝑗̂
(b) 𝑖̂
(c) 𝑗̂
(d) − sin 𝜃 𝑖̂ + sin 𝜃 𝑗̂
93. A large sheet carries uniform surface charge density 𝜎. A rod of length 2𝑙 has a linear charge density 𝜆
on one half and – 𝜆 on the other half. The rod is hinged at mid point O and makes 𝜃 with the normal to the
sheet. The torque experienced by the rod is
(a)
𝜎𝜆𝑙2
2𝜀0
𝜎𝜆𝑙
cos 𝜃
(b) 2𝜀 cos2 𝜃
0
(c)
𝜎𝜆𝑙2
2𝜀0
𝜎𝜆𝑙
sin 𝜃
(d) 2𝜀 sin2 𝜃
0
94. A small charged particle of mass 𝑚 and charge 𝑞 is suspended by an insulated thread in front of a very
large sheet of charge density 𝜎. The angle made by the thread with the vertical in equilibrium is
(a) tan−1 (
𝜎𝑞
)
2𝜀0𝑚𝑔
𝜎𝑞
−1
(b) tan (𝜀 𝑚𝑔)
0
2𝜎𝑞
(c) tan−1 (𝜀 𝑚𝑔)
0
(d) zero
95. A particle of mass m and charge q is constrained to move along a straight line joining two other equal
charges 𝑞 fixed at 𝑥 = ±𝑎. The time period of small oscillation is:
2𝜋𝑎
(a) 𝑇 = 𝑞 √𝜀0 𝑎𝑚
(b) 𝑇 =
4𝜋𝑎
𝑎
𝑞
√𝜋𝜀0 𝑎𝑚
(c) 𝑇 = 𝑞 √𝜋𝜀0 𝑎𝑚
(d) 𝑇 =
2𝜋𝑎
𝑞
√𝜋𝜀0 𝑎𝑚
96. Four equal charges of +𝑄 each are kept at the vertices of a square of side 𝑅. A particle of mass m and
charge +𝑄, oscillating parallel to plane about centre of square has time period 𝑇1 . If a particle of same
mass and charge – 𝑄 oscillates perpendicular to plane then its time period is 𝑇2 . The value of 𝑇1 /𝑇2 is
(a) 1
(b) 2
(c) √2
1
(d)
√2
97. A positively charged thin metal ring of radius R is fixed in the 𝑥𝑦 plane with its centre at the origin O. A
negatively charged particle P is released from rest at the point (0,0, 𝑧0 ) where 𝑧0 > 0. Then the motion of
P is
(a) periodic for all values of 𝑧0 satisfying 0 < 𝑧0 < ∞
(b) simple harmonic for all values of 𝑧0 satisfying 0 < 𝑧0 ≤ 𝑅
(c) approximately simple harmonic provided 𝑧0 ≪ 𝑅
(d) such that P croses 0 and continuous to move along the negative z-axis towards 𝑧 = −∞
98. Six equal charges 𝑞 are fixed at the corners of a hexagon of side ‘a’. A particle of mass 𝑚 and charge
−𝑞0 is released from a height 𝑧 ≪ 𝑎 above the centre of hexagon. Angular frequency of oscillation of the
particle about centre of hexagon is
(a) √
3𝐾𝑞𝑞0
(b) √
(c) √
𝑚𝑎 3
6𝐾𝑞𝑞0
𝑚𝑎 3
2𝐾𝑞𝑞0
𝑚𝑎 3
𝐾𝑞𝑞0
(d) √ 𝑚𝑎3
99. Two equal negative charges −𝑞 are fixed at the points (0, 𝑎) and (0, −𝑎) on the y-axis. A positive
charge
Q is released from rest at the point (2𝑎, 0) on the x-axis. The charge Q will
(a) execute simple harmonic motion about the origin
(b) move to the origin and remain at rest
(c) move to infinity
(d) execute oscillatory but not simple harmonic motion
100. An electron revolves around a long wire of uniform linear charge density. Which of the following
statementsis incorrect.
(a) speed of the particle is independent of distance from wire
(b) acceleration of particle is independent of distance from wire
(c) acceleration of particle is inversely proportional to distance from wire
(d) work done on electron due to electric force is zero
ELECTRIC FLUX & GAUSS LAW
1
1. An electron field 𝐸⃗⃗ (𝑟⃗) = 𝑟 (𝛼𝑟̂ + 𝛽 𝑠𝑖𝑛 𝑠𝑖𝑛𝜃 𝑐𝑜𝑠 𝑐𝑜𝑠𝜙𝜙⃗⃗) exists in space. What will be the total charge
enclosed in a sphere of unit radius at the origin ?
(a) 4𝜋𝜀0 𝛼
(b) 4𝜋𝜀0 (𝛼 + 𝛽)
(c) 4𝜋𝜀0 (𝛼 − 𝛽)
(d) 4𝜋𝜀0 𝛽
2. A charge q is at the center of two concentric spheres. The outward electric flux through the inner
sphere is 𝜙 while that through the outer sphere is 2𝜙 . The amount of charge contained in the region
between the two spheres is [JAM 2015]
(a) 2𝑞
(b) 𝑞
(c) – 𝑞
(d) – 2𝑞
3. A point charge q sits at a corner of a cube of side a, as shown in the figure on the right. The flux of the
electric field vector through the shaded side is [TIFR 2013]
𝑞
(a) 8𝜀
0
𝑞
(b) 16𝜀
𝑞
(c) 24𝜀
(d)
𝑞
0
0
6𝜀0
4. The flux of electric field through a circle of radius R placed in the x-y plane with its centre at the origin
due to a point charge 𝑄 placed at (0, 0, d) is
[JUN 2013]
𝑄
(a) 2𝜀 [1 −
𝑄
(b) 2𝜀
𝑑
1
(𝑑 2+𝑅 2 )2
𝑑3
0
]
3/2
0 (𝑑 2 +𝑅 2 )
𝑄 𝑑
(c)4𝜀
0𝑅
𝑄 𝑅2
(d) 4𝜀
2
0𝑑
5. Consider a charge Q at the origin of 3-dimensional coordinate system. The flux of the electric field
through the curved surface of a cone that has a height ‘h’ and a circular base of radius R (as shown in the
figure) is
𝑄
(a) ∈
0
𝑄
(b) 2 ∈
0
ℎ𝑄
(c) 𝑅∈
0
𝑄𝑅
(d) 2ℎ ∈
0
6. Suppose the electric field in some region is found to be 𝐸⃗⃗ = 𝑘𝑟 2 𝑟⃗ (where k is some constant). The total
charge contained in the sphere of radius R centred at origin is:
(a) 5 ∈0 𝑘𝑟 2
(b) 4 ∈0 𝑘𝑟 2
(c) 4 ∈0 𝑘𝑅5
16
(d) 5 𝜋 ∈0 𝑘𝑅5
7. A Gaussian surface encloses no net charge. Only one of the following is true for a point on it
(a) Electric field must be zero
(b) Electric potential must be zero [B.H.U-2016]
(c) Electric field and potential are zero
(d) Electric flux is zero
8. A uniformly charged and infinitely long line having a linear charge density ‘ ’ is placed at a normal
distance y from a point O. Consider a imaginary sphere of radius R with O as centre and R > y. Electric flux
through the surface of the sphere is
(a) zero
2𝜆𝑅
(b)
𝜀0
(c)
(d)
2𝜆√𝑅 2−𝑦2
𝜀0
𝜆√𝑅 2+𝑦2
𝜀0
9. A hemispherical surface of radius R is in a constant electric field 𝐸⃗⃗ directed perpendicular to the base,
as shown. The flux through the curved surface S of the hemisphere is
(a) 0
(b) 𝜋𝑅2 𝐸
(c) 2𝜋𝑅2 𝐸
(d) 4𝜋𝑅2 𝐸
𝑟2
𝑟
2𝑟
10. The total charge within a sphere of radius r in a charge cloud is given by :𝑞 𝛼2 (𝑒 −𝑎 − 𝑒 𝑎 ). The electric
field at the surface of the above sphere is given by [H.C.U]
𝑟
𝑟
𝑞
(𝑒 −𝑎 − 𝑒 𝑎 )
(a)
2
4𝜋∈0𝑎
𝑞
(b) ∈
0𝑎
(c) ∈
𝑞
2
0𝑎
𝑞
(d) 4𝜋∈
0
𝑟
2𝑟
𝑟
2𝑟
(𝑒 −𝑎 − 𝑒 − 𝛼 )
2
(𝑒 −𝑎 − 𝑒 − 𝛼 )
𝑟
𝑎2
2𝑟
(𝑒 −𝑎 − 𝑒 − 𝑎 )
11. A sphere of radius R and a uniform total charge 𝑄 is centered at origin, total electric flux through a
Gaussian sphere of radius R centered at (0,0, d) is
(a) zero if 𝑑/2𝑅
𝑄
(b) ∈ 𝑖𝑓𝑑 = 0
0
𝑄
(c) 2∈ 𝑖𝑓𝑑 = 𝑅
0
𝑄
(d) > 2∈ 𝑖𝑓𝑑 < 𝑅
0
12. If 𝜌 is the charge density in a closed volume having a surface 𝑆, 𝐸⃗⃗ is the electric field and 𝑄 is the total
charge then the value of ∯𝑠 𝐸⃗⃗. 𝑛⃗⃗𝑑𝑆 is
(a) 𝜌
𝜌
(b) 𝜀
0
𝑄
(c) 𝜀
0
(d) zero
13. A sphere of radius 𝑅 lies in a region where electric field 𝐸⃗⃗ = 𝑥𝑖̂ + 𝑦𝑗̂ + 𝑧𝑘̂, total electric flux through
the sphere is
(a) 4𝜋𝑅3
4
(b) 3 𝜋𝑅3
(c) 8𝜋𝑅3
(d) zero
14. For a sphere of radius 𝑅 and charge density 𝜌 = 𝜌𝜌 value of
1
𝜌(2𝜌)
𝜌
2
𝜌( )
is
(a) 16
(b)
1
4
1
(c) 8
(d) 1
15. Two long line charge has density 𝜌. If they lines 𝜌 = 𝜌 and 𝜌 = −𝜌 then electric field at (0, 0 𝜌) is
√2𝜌
(a) 𝜌∈
0𝜌
𝜌
(b) 𝜌∈
0𝜌
𝜌
(c) 2𝜌∈
(d)
0𝜌
𝜌
√2𝜌∈ 0𝜌
16. Two charge each equal to 𝜌 are fixed at (– 𝜌, 0, 0)and (𝜌, 0, 0). A particle of charge –𝜌 and mass 𝜌
moves in circle or radius ‘𝑎’ in 𝑦 − 𝑧 plane with its center at origin. Speed of particle is
(a)
(b)
√𝑄2
2√2𝜋∈0 𝑎𝑚
√𝑄2
4√2𝜋∈0 𝑎𝑚
√𝑄2
(c) 4𝜋∈
(d)
0 𝑎𝑚
√𝑄2
2√2𝜋∈ 0 𝑎𝑚
17. A charge 𝑄 is uniformely distributed on surface of a sphere of radius 𝑅 force exerted byone half on the
other half is
(a)
𝑄2
8𝜋∈0 𝑎𝑚
𝑄2
(b) 16𝜋∈
0 𝑎𝑚
𝑄2
(c) 32𝜋∈
0 𝑎𝑚
𝑄2
(d) 64𝜋∈
0 𝑎𝑚
18. The total outward electric flux going from a cube with 0 ≤ 𝑥 ≤ 1, 0 ≤ 𝑦 ≤ 1, 0 < +𝑥 ≤ 1 metres
containing a volume charge dentisty𝜌 = 16𝑥𝑦𝑧, will be [B.H.U-2010]
(a) 16
(b) 8
(c) 4
(d) 2
19. A disk of radius 𝛼/4 having uniformly distributed charge 6𝐶 is placed in the 𝑥𝑦-plane with its center
−𝑎
at ( , 0, 0 ). A rod of length a carrying a uniformly distributed charge 8𝐶 is placed on the 𝑥-axis from
2
𝑎
𝑥 = 4 to 𝑥 =
5𝑎
4
𝑎 𝑎
. Two point charge- 7𝐶 and 3𝐶 are placed at ( 4 , 4 , 0)and (
𝑎
a
−3𝑎 3𝑎
4
,
4
, 0), respectively. Consider
a cubical surface formed by six surface = ± 2 , 𝑦 = ± 2 . The electric flux through this cubical surface is
(a)
(b)
(c)
(d)
−2𝐶
𝜀0
2𝐶
𝜀0
10𝐶
𝜀0
12𝐶
𝜀0
20. Charges 𝑞 and – 𝑞 placed at (-1, 0, 0) and (1, 0, 0) respectively. Electric flux through plane 𝑥 = 0 is
(a) 0
𝑞
(b) ∈
0
𝑞
(c) 2∈
0
(d)
2𝑞
∈0
21. A Charge 𝑞 sits at one corner of a cube as shown in figure. The flux electrostatic field 𝐸⃗⃗ through the
total shaded area is [H.C.U.-2013]
(a) 𝑞/(48 ∈0 )
1 𝑟−𝛼
(b)
2 𝑟̂
(c)
∈0 𝑟
1 𝑏−𝑎
∈ 0 𝑟2
1 𝑟−𝑎
𝑟⃗
(d) ∈
2
0 𝑟
22. A hollow spherical shell with inner radius ‘a’ and outer radius ‘b’ carries a charge density 𝜌(𝑟) = 1/𝑟 2
in the region 𝑎 ≤ 𝑟 ≤ 𝑏. The electric field in the region 𝑎 < 𝑟 < 𝑏 is [H.C.U.-2014]
1
(a) 4𝜋∈ 𝑟2 𝑟̂
0
(b)
1 𝑟−𝑎
∈0 𝑟2
1 𝑏−𝑎
(c) ∈
2
0 𝑟
1 𝑟−𝑎
(d) ∈
2
0 𝑟
𝑟̂
𝑟̂
𝑟̂
23. An electric field 𝐸⃗⃗ = 𝛼𝑥 3 𝑖̂ exists in a region. Total electric flux through a sphere of radius 𝑅 centered at
origin is
3
(a) 5 𝜋𝛼𝑅5
4
(b) 5 𝜋𝛼𝑅5
(c)
(d)
𝜋𝛼𝑅 5
5
2𝜋𝛼𝑅 5
5
24. Consider electric field 𝐸⃗⃗ = 𝐸0 𝑥̂, where 𝐸0 is a constant. The flux through the shaded area (as shown in
the figure) due to this field is
(a) 2𝐸0 𝑎
(b) √2𝐸0 𝑎2
(c) 𝐸0 𝑎2
(d)
𝐸0 𝑎 2
√2
25. Let the electric field in a certain region of sapce be given 𝐸⃗⃗ = (𝑟⃗) = 𝐶𝑟⃗/∈0 𝑎3 where 𝑎 has dimension
of length and 𝐶 is a constant. The charge density 𝜌(𝑟⃗) is given by
(a) 𝜌 = 0
(b) 𝜌 = 3𝐶/𝑎3
(c) 𝜌 = 𝐶 ∈0 /𝑎
(d) 𝜌 = 𝐶/∈0
26. A charge 𝑞 sits at the back corner of a cube as shown in the figure. The electric flux through the shaded
surface is [H.C.U.2013]
𝑞
(a) 6𝜀
0
𝑞
(b) 12𝜀
0
(c) 𝑞/𝜀0
𝑞
(d) 24𝜀
0
𝑄
27. If the electric field due to a point charge 𝑄 is expressed as 𝐸⃗⃗ = 4𝜋∈
is [H.C.U.-2012]
3𝑄
(a) 4𝜋∈ 𝑟2
𝑟̂
2
0𝑟
, then the divergence of this field
0
(b)
2𝑄
4𝜋∈0 𝑟
(c) 0
𝑄
(d) 4𝜋∈
0𝑟
28. A cylindrical of radius R with the right side cut at an angle 30° as shown in the figure below is kept in a
uniform electric field 𝐸⃗⃗ , with its axis along the direction of electric field.
The total electric flux linked with the entire cylinder is [H.C.U.-2011]
(a) 0
(b) – (𝜋𝑅2 𝐸√3)/2
(c) – 𝜋𝑅2 𝐸/2
(d) 𝜋𝑅2 𝐸
29. A region contains a volume charge density𝜌 = 𝛼𝑟 3 . If electric flux through a sphere of radius 𝑅
centred at origin is 𝜙0 . Then electric flux through concentric sphere of radius 2𝑅 is
(a) 𝜙0
(b) 8𝜙
(c) 32𝜙
(d) 64𝜙
30. Consider the charge configuration and a spherical Gaussian surface as shown in the figure. When
calculating the flux of the electric field over the spherical surface, the electric field will be due to
(a) 𝑞2
(b) only the positive charges
(c) all the charges
(d) +𝑞1 𝑎𝑛𝑑 − 𝑞1
31. An infinite, uniformly charged sheet with surface charge density 𝜎 cuts through a spherical Gaussian
surface of radius 𝑅at a distance 𝑥 from its center, as shown in the figure below. The electric flux 𝛷
through the Gaussian surface is [B.H.U-2012]
(a)
(b)
(c)
(d)
𝜋𝑅 2 𝜎
𝜀0
2𝜋𝑅 2 𝜎
𝜀0
𝜋(𝑅−𝑥)2 𝜎
𝜀0
𝜋(𝑅 2 −𝑥 2 )𝜎
𝜀0
32. A cubical region of side ‘a’ has its centre at the origin It encloses three fixed point charges,
𝑎
−𝑞𝑎𝑡 (0, − 04 ) , +3𝑞𝑎𝑡(0, 0, 0)𝑎𝑛𝑑 − 𝑞𝑎𝑡 (0, +𝑎/4,0) . Choose the correct options (s)
𝑎
(a) the net electric flux crossing the plane 2 𝑥 = + 2 𝑎 is equal to the net electric flux crossing the plane 2
𝑎
𝑥=−
2
(b) the net electric flux crossing the plane 2
𝑎
𝑎
𝑦 = + 2is more than the net electric flux crossing the plane 2𝑦 = − 2
(c) the net electric flux crossing entire region is
𝑞
𝜀0
(d) the net electric flux crossing the plane 2
𝑎
𝑎
𝑧 = + 2is equal to the net electric flux crossing the plane 2𝑥 = + 2
33. Charge is uniformly distributed in a space. The net flux passing through the surface of an imaginary
cube of side “a” in the space is . The net flux passing through the surface of an imaginary sphere of radius
“a” in the space will be
(a) 𝜙
3
(b) 4𝜋 𝜙
(c)
(d)
2𝜋
3
4𝜋
3
𝜙
𝜙
𝑘
34. A hollow spherical shell carries charge density 𝜌 = 2 in the region 𝑎 ≤ 𝑟 < 𝑏. The electric field in the
𝑟
region 𝑎 < 𝑟 < 𝑏 is [B.H. U-2012]
𝑘
𝑎−𝑏
(a) 𝐸 = 𝜀 ( 𝑟2 ) 𝑟̂
0
(b) 𝐸 =
(c) 𝐸 =
𝑘
𝜀0
𝑘
𝜀0
𝑘
(
𝑟−𝑎
𝑟2
𝑎
) 𝑟̂
( 2 ) 𝑟̂
𝑟
𝑏
(d) 𝐸 = 𝜀 (𝑟2 ) 𝑟̂
0
35. Electric field inside a charged sphere is uniform. Total charge contained inside the sphere is
proportional to
(a) radius square
(b) radius cube
(c) radius
(d) four power of radius
36. Suppose the electric field in some region is found to be E
total
(a) 5 ∈0 𝑘𝑟 2
(b) 4 ∈0 𝑘𝑟 2
(c) 4 ∈0 𝑘𝑅5
16
(d) 5 ∈0 𝑘𝑅5
kr2r (where k is some constant). The
37. If the electric flux through a closed surface S is equal to zero, then [B.H.U-2012]
(a) the volume charge density is zero everywhere inside the surface S
(b) the electric field at every point on S is zero
(c) the electric field is zero at every point inside S
(d) the net charge enclosed by S is zero
38. An electric field 𝐸⃗⃗ = 𝛼𝑖̂ + 𝛽𝑗̂ = 𝛾𝑘̂ exists in space, where 𝛼, 𝛽, 𝛾 are constant. Magnitude of electric flux
per unit area through a plane surface 𝑎𝑥 + 𝑏𝑦 + 𝑐𝑧 = 𝑑 is (𝑎, 𝑏, 𝑐 and 𝑑 are constant)
𝑎𝛼+𝑏𝛽+𝑐𝛾
(a) 2 2 2
(b)
(c)
√𝑎 +𝑏 +𝑐
𝑎𝛼+𝑏𝛽+𝑐𝛾
𝑎+𝑏+𝑐
√𝑎 2 𝛼2 +𝑏2 𝛽 2 +𝑐 2 𝛾
√𝑎 2 +𝑏2 +𝑐 2
(d) 𝛼 + 𝛽 + 𝛾
39. Consider a hollow charged shell of inner radius 𝑅 and outer radius 2𝑅. The volume charge density is
𝛼
𝜌(𝑟) = 𝑟2 where 𝛼 is constant. If 𝜙(𝑟) represents electric flux through a concentric sphere of radius ′𝑟′
then correct graph 𝜙 for𝜙(𝑟) versus ′𝑟′ is
40. An electric field 𝐸⃗⃗ = 𝛼𝑖̂ + 𝛽𝑗̂ + 𝛾𝑘̂ exists in space. Electric flux through the curved surface of
hemisphere shown in the figure is
(a) (𝛼 + 𝛽)𝜋𝑅2
(b) – (𝛼 + 𝛽)𝜋𝑅2
(c) 𝛾𝜋𝑅2
(d) – 𝛾𝜋𝑅2
41. If total electric flux through a closed surface is zero. Which of the following statements is correct.
(a) Electric field at every point on the surface must be zero
(b) Electric field at every point on the surface must be parallel to the surface
(c) There is no charge enclosed inside the surface
(d) Sum of total charge enclosed inside the surface must be zero.
42. An electric field 𝐸⃗⃗ = 𝛼𝑟 3 𝑟̂ exists in space. Total electric flux through a sphere of radius 𝑅 centered at
origin is
(a) 4𝜋𝛼𝑅3
(b) 4𝜋𝛼𝑅5
(c) 𝛼𝑅3
(d) 𝛼𝑅5
43. An electric field 𝐸⃗⃗ = 𝛽𝑧 3 𝑘̂ exists in space. Total electric flux through a sphere of radius 𝑅 centered at
origin is
(a) 4𝜋𝛽𝑅3
(b) 4𝜋𝛽𝑅5
(c) 𝛽𝑅3
(d) 𝛽𝑅5
44. An electric field 𝐸⃗⃗ = 𝛼𝑟̂ + 𝑓(𝜃)𝜙̂ exists in space. Total flux through a sphere centered at origin will
(a) depend on 𝑓(𝜃)
(b) not dependent on radius of sphere
(c) depended both on 𝑓(𝜃) and radius
(d) depend on radius but not on 𝑓(𝜃)
45.A point charge 𝑞 is placed at the centre of a cylinder of radius R and length 2L. Electric flux through the
curved surface of cylinder is
𝑞
(a) 2∈
0
(b)
𝑞
(c) ∈
(d)
.
𝐿
∈0 𝑅
𝑞
𝐿
0 √𝐿2 +𝑅 2
𝑞
𝑅
∈0 √𝐿2 +𝑅 2
46. A point charge q is placed at (0, 0, 0). An imaginary spherical surface of radius 𝑅 is centered at
(0,0, −𝑎)where 𝑎 > 𝑅. If the sphere starts moving along +𝑧 direction with constant velocity 𝑣0 then
which of the following option for electric flux through the sphere versus time is correct?
47. 47. In the figure shown electric flux through closed surface 𝑆1 and 𝑆3 is zero whereas flux 𝑆2 is not
zero. Which of the following statement is true
(a) 𝑞1 = 0, 𝑞2 ≠ 0, |𝑞3 | ≠ |𝑞3 |
(b) |𝑞1 | = |𝑞2 |, 𝑞3 = 0
(c) 𝑞1 = 0, 𝑞2 ≠ 0, 𝑞3 = −𝑞2
(d) 𝑞2 ≠ 0, 𝑞1 = 𝑞2 = 0
48. A Point charge 𝑞 is placed at a distance ′𝑑′ from the center of a hemisphere as shown in figure. Electric
flux through curved surface of hemisphere is
(a)
𝑞
2∈0
𝑞
(b) 2∈ [1 +
0
(c)
𝑞
2∈0
𝑞
[1 −
(d) 2∈ [1 −
0
𝑑
√𝑑 2 +𝑅 2
𝑑
√𝑑 2 +𝑅 2
𝑅
]
]
√𝑑 2 +𝑅 2
]
49. In a spherical distribution the charges density varies as 𝜌(𝑟) = 𝑄𝜋𝑅12 for 𝑅1 < 𝑟 < 𝑅2 as shown in the
figure A point charge A lies at the centre of the sphere at 𝑟 = 0 Choose the appropriate curve for variation
of electric field E in the space as a function of 𝑟
50. A charge +Q is distribution uniformly within a solid sphere of radius𝑟1 .It is surrounded by concentric
hollow sphere of inner and outer radii 𝑟2 and 𝑟3 ,respectively, having charge -Q distributed over its volume.
The plot of electric.field intensity versus distance 𝑥 from the centre shows that
(a) electric field is maximum at a distance 𝑟2
(b) electric field is maximum at a distance 𝑟3
(c) electric field first increases, reaches a maximum
(d) none of the above
51. Charge 𝑞 and 2𝑞 are placed at (−1,0,0)𝑎𝑛𝑑(1,0, 0) respectively. Electric flux through plane 𝑥 = 0 is
(a) 0
𝑞
(b) ∈
0
𝑞
(c) 2∈
0
2𝑞
(d) ∈
0
52. A region contains a volume charge density 𝜌 = 𝛼𝑥. If electric flux through a sphere of radius 𝑅centred
at origin is 𝜙0 . Then electric flux through concentric sphere of radius 2𝑅 is
(a) 𝜙0
(b) 2𝜙0
(c) 32𝜙
(d) 64𝜙
53. An electric field 𝛼𝑥𝑖̂ + 𝛽𝑦𝑗̂ + 𝛾𝑧𝑘̂ exists in space where 𝛼, 𝛽, 𝛾 are constant . Electric flux through a
sphere of unit radius centered at (1, 1, 1) is 𝜙0 . Electric flux through a sphere of unit radius centered at (0,
0, 0) is
(a) 𝜙0
(b) 2𝜙0
𝜙
(c) 20
(d) zero
54. A space charge of density 𝜌 is uniformly distributed in an infinitely cylinder of radius 𝑅 Then, for any
point at distance 𝑟 from the axis (relative permittivity=1),
𝜌𝑟
(a) the electric field is 𝐸 =
for 0 < 𝑟 < 𝑅
(b) the electric field is 𝐸 =
2∈0
𝜌𝑅 3
2∈0 𝑟2
𝜌𝑟
for 𝑟 > 𝑅
(c) the electric field is 𝐸 = ∈ for 0 < 𝑟 < 𝑅
(d) the electric field is
𝜌𝑅 2
2∈0 𝑟
0
for 𝑟 > 𝑅
𝑞
55. The electric field due to a charge 𝑞 is given by 𝐸⃗⃗ = 𝑟2 𝑟̂ . The value of the surface integral ∮𝑠
depends only
(a) On the area of the surface
(b) On the radial distance 𝑟.
(c) On the charge
(d) On the shape of the surface
𝐸⃗⃗ . ⃗⃗⃗⃗⃗
𝑑𝑠
56. Consider a sphere 𝑆1 of radius 𝑅 which carries a uniform charge of density 𝜌. A smaller sphere 𝑆2 of
𝑅
𝑅
radius 𝑎 < 2 is cut out and removed from it . The centres of the sphere are separated by the vector 𝑏⃗⃗ = 𝑛̂ 2
as shown in the figure .
The electric field at a point 𝑃 inside 𝑆2 is
𝜌𝑅
(a) 3𝜀 𝑛̂
0
(b)
𝜌𝑅
3𝜀0 𝑎
𝜌𝑅
(𝑟⃗ − 𝑛̂𝑎)
(c) 6𝜀 𝑛̂
0
𝜌𝑅
(d) 3𝜀 𝑟⃗
0
57. A non-conducting solid sphere of radius R is uniformly charged. The magnitude of the electric field
due to the sphere at a distance r from its centre:
(a) increases as r increases for 𝑟 < 𝑅
(b) decreases as r increases for 0 < 𝑟 < ∞
(c) decreases as r increases for 𝑅 < 𝑟 < ∞
(d) is discontinuous at 𝑟 = 𝑅
58. Three infinitely long charge sheets are placed as shown in figure. The electric field at point 𝑃 is
2𝜎
(a) 𝜀 𝑘̂
0
2𝜎
(b) − 𝜀 𝑘̂
(c)
4𝜎
𝜀0
(d) −
0
𝑘̂
4𝜎
𝜀0
𝑘̂
59. For spherical symmetrical charge distribution, variation of electric potential with distance from centre
is given in diagram
Given that: 𝑉 =
𝑞
4𝜋𝜀0 𝑅0
for 𝑟 ≤ 𝑅0 and 𝑉 =
𝑞
4𝜋𝜀0 𝑟
for 𝑟 ≥ 𝑅0
Then which options(s) are correct:
(a) Total charge within 2𝑅0 is 𝑞
(b) 𝛻⃗⃗. 𝐸⃗⃗ = 0everywhere
(c) At 𝑟 = 𝑅0 electric field is discontinuous
(d) There will be no charge anywhere except at 𝑟 = 𝑅0
60. Two large nonconducting sheets one with a fixed uniform positive charge and another with a fixed
uniform negative charge are placed at a distance of 1 meter from each other. The magnitude of the surface
chargedensities are 𝜎+ 6.8𝜇𝐶/𝑚2 for the positively charged sheet and𝜎− = 4𝜇𝐶/𝑚2 for the negatively
charged sheet. What is the electric field in the region between the sheets? [JEST 2014]
(a) 6.30 × 105 𝑁/𝐶
(b) 3.84 × 105 𝑁/𝐶
(c) 1.40 × 105 𝑁/𝐶
(d) 1.16 × 105 𝑁/𝐶
61. A spherical portion has been removed from a solid sphere having a charge distributed uniformly in its
volume as
(a) zero everywhere
(b) non-zero and uniform
(c) non-uniform
(d) zero only at its centre
62. Consider a hollow charged shell of inner radius ‘a’ and outer radius ‘b’ The volume charge density is
𝑘
𝜌(𝑟) = 2 (where 𝑘 is a constant) in the (region 𝑎 < 1 < 𝑏) . The magnitude of the electric field produced
𝑟
at distance 𝑟 > 𝑎 is:
𝑘(𝑏−𝑎)
(a) 𝜀 𝑟2 𝑓𝑜𝑟 𝑟 > 𝑎
0
(b)
(c)
(d)
𝑘(𝑏−𝑎)
𝜀0 𝑟 2
𝑘(𝑟−𝑎)
𝜀0 𝑟 2
𝑘(𝑟−𝑎)
𝜀0 𝑟 2
𝑘𝑏
𝑓𝑜𝑟 𝑎 < 𝑟 < 𝑏 𝑎𝑛𝑑 𝜀
2
𝑓𝑜𝑟 𝑟 > 𝑏
2
𝑓𝑜𝑟 𝑟 > 𝑏
0𝑟
𝑘𝑏
𝑓𝑜𝑟 𝑎 < 𝑟 < 𝑏 𝑎𝑛𝑑 𝜀
𝑓𝑜𝑟 𝑎 < 𝑟 < 𝑏 𝑎𝑛𝑑
0𝑟
𝑘(𝑏−𝑎)
𝜀0 𝑎 2
𝑓𝑜𝑟 𝑟 > 𝑏
63. Charge is distributed uniformly throughout the volume of an infinitely long cylinder of radius 𝑅 The
electric field at a distance 𝑟 form the cylinder axis (for 𝑟 < 𝑅)is
𝜌𝑟
(a) 𝜀
0
𝜌𝑟
(b) 2𝜀
0
𝜌𝑟
(c) 2𝜀
0
𝜌𝑟
(d) 4𝜀
0
𝑎𝑟
64. A solid sphere of radius 𝑅 has a charge density, given by 𝜌(𝑟) = 𝜌0 (1 − 𝑅 ), where 𝑟 is the radial coordinate and 𝜌0 , 𝑎 and 𝑅 are positive constants. If the magnitude of the electric field at 𝑟 = 𝑅/2 is 1.25 times
that at 𝑟 = 𝑅, then value of 𝑎 is
(a) 2
(b) 1
1
(c) 2
1
(d) 4
65. Consider a set of two stationary point charges 𝑞1 and 𝑞2 as shown in the figure. Which of the following
statement is correct ?
(a) The electric field at 𝑃 is independent of 𝑞2
(b) The electric flux crossing the closed surface S is independent of 𝑞2
(c) The line integral of the electric field 𝐸⃗⃗ over the closed contour 𝐶 depends on 𝑞1 and 𝑞2.
(d) ⃗∇⃗. 𝐸⃗⃗ = 0 everywhere
66. A solid sphere of radius 𝑅 carries a uniform volume charge density 𝜌. The magnitude of electric field
inside the sphere at a distance 𝑟 from the centre is:
𝑟𝜌
(a) 3𝜀
0
𝑅𝜌
(b) 3𝜀
0
𝑅2𝜌
(c) 𝑟𝜀
0
𝑅3𝜌
(d) 𝑟2 𝜀
0
67. A sphere of radius 𝑅 carries a volume charge density 𝜌 = 𝜌0 𝑟, where 𝜌0 is a constant and 𝑟 is the
radial distance from the centre of the charged sphere. The intensity of the electric field is proportional to:
(a) 𝑟 2 for 𝑟 ≤ 𝑅 and 𝑟 2 for 𝑟 > 𝑅
(b) 𝑟 for 𝑟 ≤ 𝑅 and 1/𝑟 2 for 𝑟 ≥ 𝑅
(c) 1/𝑟 2 for both 𝑟 < 𝑅 and 𝑟 > 𝑅
(d) 1/𝑟 for 𝑟 ≤ 𝑅 and 1/𝑟 2 for 𝑟 > 𝑅
68. Consider an infinite horizontal surface with fixed surface charge density 𝜎, where 𝑛̂ is the upward
normal to the surface. If an electric field 𝐸⃗⃗𝑏 = 𝐸𝑛̂ is applied from below, the electric field 𝐸⃗⃗𝑎 in the above
the surface is:
𝜎
(a) 𝐸⃗⃗𝑎 = 𝐸⃗⃗𝑏 − 2𝜀 𝑛̂
0
𝜎
(b) 𝐸⃗⃗𝑎 = 𝐸⃗⃗𝑏 + 2𝜀 𝑛̂
0
𝜎
(c) 𝐸⃗⃗𝑎 = 𝐸⃗⃗𝑏 − 𝜀 𝑛̂
0
𝜎
(d) 𝐸⃗⃗𝑎 = 𝐸⃗⃗𝑏 + 𝜀 𝑛̂
0
𝑞
69. The electric field due to an unknown charge distribution given by 𝐸 = 𝑟2 exp(−4𝑟) 𝑟̂
The total integrated charge over all space ∫ 𝜌 𝑑𝑉 is given by [IISc.2009]
(a) 0
(b) 𝑞
𝑞
(c) 4
𝑞
(d) 𝜋
70. An infinitely long solid cylinder of radius 𝑅 has a uniform volume charge 𝜌. It has a spherical cavity of
𝑅
radius 2 with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric
field at the point 𝑃, which is at distance 2𝑅 from the axis of the cylinder, is given by the expression
The value of 𝑘 is _______
23𝜌𝑅
16𝑘𝜀0
.
71. Two non-conducting solid spheres of radii 𝑅 and 2𝑅, having uniform volume charge densities 𝜌1 and
𝜌2 respectively, touch each other. The net electric field at a distance 2𝑅 from the center of the smaller
sphere, along the line joining the centers of the spheres, is zero. The ratio 𝜌1 /𝜌2 can be
(a) −4
32
(b) − 25
32
(c) 25
(d) 4
72. Two non-conducting spheres of radii 𝑅1 and 𝑅2 and carrying uniform volume charge densities +𝜌 and
−𝜌, respectively, are placed such that they partially overlap, as shown in the figure.
At all points in the overlapping region,
(a) the electrostatic field is zero
(b) the electrostatic potential is constant
(c) the electrostatic field is constant in magnitude
(d) the electrostatic field has same direction
73. Let 𝐸1 (𝑟), 𝐸2 (𝑟) and 𝐸3 (𝑟) be the respective electric field at a distance 𝑟 from point charge , an
infinitely long wire with constant linear charge density 𝜆, and an infinite plane with uniform surface
charge density 𝜎. If 𝐸1 (𝑟0 ) at a given distance 𝑟0 , then
(a) 𝒬 = 4𝜎𝜋02
𝜆
(b) 𝑟0 = 2𝜋𝜎
𝑟
𝑟
𝑟
𝑟
(c) 𝐸1 ( 20 ) = 2𝐸2 ( 20 )
(d) 𝐸2 ( 20 ) = 4𝐸3 ( 20 )
𝑅
74. A sphere of radius 𝑅 has a uniform charge density 𝜌. A sphere of smaller radius 2 is cut from the
𝑅
original sphere, as shown in the figure below. The centre of the cut out sphere lies at 𝑧 = . After the
𝑅
𝜌𝑅
2
smaller sphere has been cut out, the magnitude of the electric field at 𝑧 = 2 is 𝑛∈ . The value of the integer
𝑛 is ______[JAM 2017]
0
75. Three infinite plane sheets carrying uniform charge densities – 𝜎, 2𝜎, 3𝜎 are placed parallel to the 𝑥𝑧palne at 𝑦 = 𝑎, 3𝑎, 4𝑎 respectively. The electric field at the point (0, 2𝑎, 0) is [JAM-2018]
4𝜎
(a) 𝜀 𝑗̂
0
3𝜎
(b) − 𝜀 𝑗̂
0
2𝜎
(c) − 𝜀 𝑗̂
(d)
𝜎
𝜀0
0
𝑗̂
𝑘𝑟 2 ; 𝑟 < 𝑅
76. Given a spherically symmetric charge 𝜌(𝑟) = {
, (𝑘 being a constant), the electric field for
0; 𝑟 > 𝑅
𝑟 < 𝑅 is (take the total charge as 𝒬)
(a)
(b)
𝒬𝑟3
4𝜋𝜀0 𝑅 5
3𝒬𝑟2
4𝜋𝜀0 𝑅 5
5𝒬𝑟3
(c) 8𝜋𝜀
5
0𝑅
𝒬𝑟3
(d) 4𝜋𝜀
0𝑟
5
𝑟̂
𝑟̂
𝑟̂
𝑟̂
ELECTRIC POTENTIAL
1. A charge +𝑞 is fixed at each of the points 𝑥 = 𝑥0 , 𝑥 = 3 𝑥0 , 𝑥 = 5 𝑥0 … . . ∞ on the x-axis and a charge −𝑞
is fixed at each of the points 𝑥 = 2𝑥0 , 𝑥 = 4 𝑥0 , 𝑥 = 6𝑥0 , … . , ∞. Here 𝑥0 is a positive constant.
𝑄
Take the electric potential at a point due to a charge Q at a distance 𝑟 from it to be 4𝜋∈ 𝑟. Then the
potential at the origin due to the above system of charge is
(a) 0
𝑞
(b) 8𝜋∈ 𝑥 ln 2
0
0 0
(c) ∞
𝑞 ln(2)
(d)
4𝜋∈0 𝑥0
2. Point charges each equal to 𝑄 are placed at 𝑥0 , 2𝑥0 , 3𝑥0 , 4𝑥0 , …. Magnitude of electric field at the origin is
𝜋𝑄
given to be 𝛼𝜀 𝑥 2. Value of 𝛼 is ___________
0 0
3. Electric potential in a particular region of space is 𝑉 = 5𝑥 − 3𝑥 2 𝑦 + 2𝑦𝑧 2 . The magnitude of electric
field at point 𝑃(𝑙𝑚, 0, −2𝑚) is
(a) 5 N/C
(b) 6 N/C
(c) 7.08 N/C
(d) 9.0 N/C
4. Electrical potential in an electric field is given by 𝑉 = 𝐾/𝑟 (K being constant). If the position vector 𝑟⃗ =
2𝑖̂ + 3𝑗̂ + 6𝑘̂ , then the electric field will be
(a)
(b)
(c)
(d)
̂)
(2𝑖̂ +3𝑗̂ +6𝑘
𝐾
243
̂)
(2𝑖̂+3𝑗̂ +6𝑘
343
̂)
(3𝑖̂ +2𝑗̂ +6𝑘
𝐾
𝐾
243
̂)
(6𝑖̂+ 2𝑗̂ +6𝑘
343
𝐾
5. Suppose that the electric field on the xy-plane is given by
1
𝐸⃗⃗ = 𝑃 [𝑥𝑦𝑖̂ + ( 𝑥 2 + 𝑦 2 ) 𝑗̂]
2
The magnitude of the difference in potential between the origin (0,0) and the point (1,2) is:
5
(a) 4 𝑃
(b)
11
3
7
𝑃
(c) 2 𝑃
(d) Not unique, it depends on the path joining the two points.
[JNU 2009]
6. In a uniform electric field potential difference between two points (1, 2, 1) and (2,4,1) is 10 volt. Which
of the following option is correct
(a) minimum magnitude of electric field must be 10 V/m
(b) minimum magnitude of electric field must be 2√5 V/m
(c) maximum magnitude of electric field must be 10 V/m
(d) maximum magnitude of electric field must be 5 V/m
7. Electric potential in a region is 𝑉(𝑥, 𝑦, 𝑧) = 2𝑥𝑦 volt, line integral of electric field from point (1, 1, 1) to
(2, 2, 2) is
(a) 6 volt
(b) – 6 volt
(c) 4 volt
(d) – 2 volt
8. Electric field at a point is zero. Electric potential at that point
(a) must be zero
(b) must not be zero
(c) may or not be zero
(d) none of these
9. Electric potential at a point is zero. Electric field at that point
(a) must be zero
(b) must not be zero
(c) may or may not be zero
(d) none of these
10. A uniform electric field pointing in positive 𝑥-direction exists in a region. Let A be the origin, B be the
point on the x-axis at 𝑥 = +1 cm and C be the point on the y-axis at 𝑦 = +1 cm. Then the potentials at the
points A, B and C satisfy:
(a) 𝑉𝐴 < 𝑉𝐵
(b) 𝑉𝐴 > 𝑉𝐵
(c) 𝑉𝐴 < 𝑉𝐶
(d) 𝑉𝐴 > 𝑉𝐶
11. The adjacent figure shows charged spherical shells A, B and C having charge densities 𝜎, −𝜎, 𝜎 and
radii 𝑎, 𝑏, 𝑐 respectively. If 𝑉𝐴 = 𝑉𝐶 , then equals to
(a) 𝑎 + 𝑐
(b) √𝑎2 + 𝑏 2
(c) √𝑎𝑐
(d) none of these
12. A conducting sphere A of radius 𝑎, with charge 𝑄, is placed concentrically inside a conducting shell B
of radius 𝑏. B is earthed. C is the common centre of A and B. Which is not correct?
𝑄
(a) The filed at a distance 𝑟 from C, where 𝑎 ≤ 𝑟 ≤ 𝑏, is 𝑘 𝑟2
𝑄
(b) The potential at a distance 𝑟 from C, where 𝑎 ≤ 𝑟 ≤ 𝑏, is 𝑘 𝑟
1
1
(c) The potential differences between A and B is 𝑘𝑄 (𝑎 − 𝑏 )
1
1
(d) The potential at a distance 𝑟 from C, where 𝑎 ≤ 𝑟 ≤ 𝑏, is 𝑘𝑄 (𝑟 − 𝑏)
13. A sphere of radius 𝑅 has volume charge density proportional to distance from centre Total charge
contained in the sphere is 𝑄. If electric potential at infinity is taken to be zero potential at its centre is
𝑄
(a) 2𝜋∈ 𝑅
0
𝑄
(b) 3𝜋∈
𝑄
0𝑅
(c) 4𝜋∈
0𝑅
3𝑄
(d) 8𝜋∈
0𝑅
14. The variation of potential with distance 𝑟 from a fixed point is shown in the figure. The electric field at
𝑟 = 3cm and 𝑟 = 5𝑐𝑚 are, respectively.
(a) 0, 2V/cm
(b) 2V/cm, - 2V/cm
(c) 0, -2V/ cm
(d) 2V/cm, 0
15.The long with cylindrical glass rod shown below has length 𝑙 and is insulted from its surroundings. The
rod has an excess charge 𝑄 uniformly distributed along its length. Assume the electric potential to be zero
at infinite distance from the rod. If 𝑘 is the constant in Coulomb’s law, the electric potential at a point P
𝑘𝑄
along the axis of the rod and a distance 𝑙 from one end is 𝑙 multiplied by [B.H.U-2012]
(a)
4
9
1
(b) 2
2
(c)
3
(d) ln 2
16. Two uniformly charged insulting solid spheres A and B, both of radius 𝑎, carry total charges +𝑄 and
−𝑄. respectively. The spheres are placed touching each other as shown in the figure.
If the potential at the center of the sphere 𝐴 is 𝑉𝐴 and that at the center of B is 𝑉𝐵 , then the difference 𝑉𝐴 −
𝑄
𝑉𝐵 is 4𝜋𝜀 𝑎 𝛽, value of 𝛽 is ________
0
17. There are two concentric metal shells of radii 𝑟 and 𝑟2 (> 𝑟1 ). If the outer shell has a c charge 𝑞 and the
inner shell is grounded, the charge on the inner shell is
(a) zero
𝑟
(b) − (𝑟1 ) 𝑞
2
(c) 𝑟1 𝑟2 𝑞
(d) ∞
18. At a certain distance from a point charge the field intensity is 500 V/m and the potential is – 300V.
The distance to the charge and the magnitude of the charge respectively are
(a) 0.6 m and 60 nC
(b) 0.4 m and 20 nC
(c) 0.6 m and 40 nC
(d) 0.6 m and 20 nC
19. A solid insulating sphere of radius R is given a charge Q. If at a point inside the sphere the potential is
1.5 times the potential at the surface, this point will be
(a) at a distance of 2R/3 from the centre
(b) at the centre
(c) at a distance of 2R/3 from the surface
(d) data insufficient
20. The concentric thin spherical shells of radii a, b and c (𝑎 < 𝑏 < 𝑐) carry uniform surface electric
charge of densities 𝜎, −𝜎, and 𝜎, respectively. The electric potential at the surface of the outermost shell
is [JNU 2011]
𝜎
(a) 𝜀 (𝑐 − 𝑏 + 𝑎)
0
𝜎
𝑐2
(b) 𝜀 ( 𝑎 −
0
𝜎
𝑏2
𝑎
𝑐2
(c) 𝜀 ( 𝑏 − 𝑏
0
𝜎
(d) 𝜀 (𝑐 −
0
𝑏2
𝑐
+ 𝑎)
𝑎2
𝑏
)
+
𝑎2
𝑐
)
21. Two concentric conducting spheres having radii 𝑎 and 𝑏 are charge to 𝑞1 and 𝑞2, respectively. The
potential difference between 1 and 2 will be
𝑞1
(a)
−
𝑞2
4𝜋𝜀0 𝑎
4𝜋𝜀0 𝑏
𝑞2
1
1
(b) 4𝜋𝜀 (𝑎 − 𝑏 )
𝑞
0
1
1
(c) 4𝜋𝜀1 (𝑎 − 𝑏)
0
(d) none of these
22. Two concentric spheres of radii 𝑅 and 2𝑅 has uniform charges Q and 2Q respectively. Potential
difference between two spheres is 𝑉0 . If charge on the inner sphere is doubled then potential difference
will become
(a) zero
(b) 2𝑉0
𝑉
(c) 20
(d)
3𝑉0
2
23. A sphere of radius 𝑅 has volume charge density proportional to distance from centre. Total charge
contained in the sphere is 𝑄. If electric potential at infinity is taken to be zero and potential at its centre is
𝛼𝑄
then value of 𝛼 is ______
3𝜋∈ 𝑅
0
24. Consider the charge distribution shown in the figure. Which of the following statement is correct.
(a) Electric potential at every point on z-axis is zero
(b) Electric field at every point on z-axis is in 𝑗̂ direction.
𝜆
(c) Electric field at centre of circle is 𝜋∈ 𝑎
0
(d) Electric dipole moment of system is zero
25. Which of the following statement is correct
(a) if electric field is zero electric potential may not be zero
(b) if electric potential is zero electric field may not be zero
(c) electric potential is extremum in a charge free region
(d) electric field is directed from higher potential to lower potential.
26. An arc subtends an angle 𝛼 at the center. If radius of circle is 𝑅 and the arc has uniform charge then
𝐸𝑅
value of 𝑉 at the center is (where 𝐸 = Electric field, 𝑉 = Potential)
(a)
sin 𝛼
𝛼
(b)
(c)
(d)
sin
𝛼
2
𝛼
𝛼
sin
2
2𝛼
𝛼
2 sin
2
𝛼
27. If 𝐸⃗⃗ = 2𝑖̂ + 3𝑗̂ + 4𝑘̂, value of potential difference between point 𝑎(1, −2,1) and 𝑏(2,1, −2) is
(a) −1
(b) −2
(c) 0
(d) 1.5
28. A Cube has a constant electric potential 𝑉 on its surface. If there no charges inside the cube, the
potential at the center of the cube is [B.H.U-2012]
(a) zero
𝑉
(b)
8
𝑉
(c)
2
(d) 𝑉
29. An electric field 𝐸⃗⃗ = 𝐴(3𝑥 2 𝑦 2 𝑧 2𝑖̂ + 2𝑥 3 𝑦𝑧 2 𝑗̂ + 2𝑥 3 𝑦 2 𝑧𝑘̂) exists in space. If electric potential at (2, 2, 2)
is taken to be zero then potential at point (1, 2, 2) is
(a) 128 A
(b) 64 A
(c) 32 A
(d) 112 A
30. Electric potential in a region is 𝑉(𝑟) = 𝐴𝑟 3 + 𝐵. Charge enclosed by a sphere of radius R is centred at
origin is 𝑄0. Charged enclosed by concentric sphere of radius 2𝑅 is
(a) 𝑄0
(b) 4𝑄0
(c) 16 𝑄0
(d) 32 𝑄0
𝑅
31. A sphere of radius R has uniformly distributed charge 𝑄. A spherical portion of radius 2 is cut from it
to form a cavity as shown in the figure. Electric potential at the centre of cavity is [Take 𝑉(𝑟 = ∞) = 0]
𝑄
(a) 4 𝜋∈
0𝑅
𝑄
(b) 𝜋∈
0𝑅
𝑄
(c) 2𝜋∈
0𝑅
𝑄
(d) 4𝜋∈
0𝑅
32. Two concentric conducting spheres of radii R and 3R are maintained at potential 2𝑉0 and 𝑉0
3𝑅
respectively. Potential at 𝑟 = 2 is
(a)
3𝑉0
2
(b) 2𝑉0
4𝑉
(c) 30
(d) 𝑉0
33. Consider an annular ring of inner and outer radii a and b. A charge Q is uniformly distributed on the
ring. Potential at the center of the ring is
𝑄
(a)
(𝑎+𝑏)
4𝜋∈0
𝑄
(b) 4𝜋∈
0 (𝑏−𝑎)
(c)
(d)
𝑄
2𝜋∈0 𝜋(𝑏+𝑎)
𝑄
2∈0 𝜋(𝑏−𝑎)
34. Adjacent figure depicts the electric field lines in a region of space. Then, we can conclude that
(a) potential at C is less than potential at B
(b) electric field intensity at B is greater than that at C
(c) electric field intensity at A is greater than that at C
(d) in moving a negative charge from B to C, the external agent has to do a negative work
35. A hollow conducting sphere of inner radius R and outer radius 2R is given a charge Q as shown in the
figure, then the
(a) potential at A and B is different
(b) potential at O and B is different
(c) potential at O and C is different
(d) potential at A, B, C and O is same
36. A charge Q is distributed over two concentric hollow spheres of radii R and 𝑟(𝑅 > 𝑟) such that their
surface densities are equal. The common potential at the centre is
(a) zero
𝑟+𝑅
(b) 2𝑄 ( 2 2)
𝑟 +𝑅
𝑟+𝑅
𝑄
(c) 4𝜋∈ (𝑟2 +𝑅2)
0
1
𝑄
𝑟+𝑅
(d) 2 (4𝜋∈ ) (𝑟2+𝑅2)
0
37. Consider a uniformly charged disc of radius ‘a’ and surface charge density 𝜎. Consider a point P on the
axis of the disc at a distance z from the disc. The potential at P is given by [H.C.U.-2015]
𝜎 1
(a) ∈ 𝑧 2
0
𝜎
(b) 2∈ [√𝑎2 + 𝑧 2 − 𝑧]
0
𝜎
(c) ∈ [√𝑎2 + 𝑧 2 − √𝑎2 − 𝑧 2 ]
0
(d)
𝜎
∈0
[√𝑎2 + 𝑧 2 − 𝑎]
38. Electric potential in a region is 𝑉(𝑟) = 𝐴𝑟 3 + 𝐵. The electric flux through a sphere of radius 𝑅 is
proportional to
(a) 𝑅 2
(b) 𝑅 3
(c) 𝑅 4
(d) 𝑅 5
39. A charge 𝑞 is distributed uniformly over the surface of a thin circular insulting disc of radius 𝑎. The
potential at the rim of the disc is given by [H.C.U.-2015]
𝑞
(a) 2
(b)
𝜋 𝑎∈0
𝑞
4𝜋 2 𝑎∈0
𝑞
(c) 𝜋𝑎2∈
0
(d)
𝑞
4𝜋 2 𝑎 2 ∈0
40. A charge Q is distributed over two concentric hollow spheres of radii R and 𝑟(𝑅 > 𝑟) such that their
surface densities are equal. The potential at the outer sphere is s
(a) zero
𝑟+𝑅
(b) 2𝑄 ( 2 2)
(c)
𝑟 +𝑅
𝑟+𝑅
𝑄
4𝜋∈0
𝑄
(
𝑟2 +𝑅 2
)
(d) 4𝜋∈
0𝑅
41. A loop of radius 𝑟 carries a uniformly distributed charge. Assuming the potential at infinity to be zero,
the ratio of the potential at a height ‘2r’ on the axis to that at a height 3𝑟 is: [IISc. 2008]
(a) 1/2
(b) 2
(c) √1/2
(d) √2
∆𝑉
42. A quantity X is given by ∈0 𝐿 ∆𝑡 where ∈0 is the permittivity of free space, L is a length, ∆𝑉 is a
potential difference and ∆𝑡 is a time interval. The dimensional formula for X is the same as that of
(a) resistance
(b) charge
(c) voltage
(d) current
43. Electric charges of +10𝜇𝐶, +5𝜇𝐶, −3𝜇𝐶 and +8 𝜇𝐶 are placed at the corners of a square of side √2 𝑚.
The potential at the centre of the square is[H.C.U.-2014]
(a) 1.8 V
(b) 1.8 × 105 𝑉
(c) 1.8 × 106 𝑉
(d) 1.8 × 104 𝑉
44. A sphere of radius R has uniform volume charge density 𝜌 and uniform surface charge density 𝜎. If
electric potential outside the sphere is constant then value of 𝜌𝑅/𝜎 is
(a) −3
(b) 3
(c) −1
(d) 1
45. If an electric potential difference of 10 volts is applied between two plates kept 2 cm apart, the electric
field between the plates is [H.C.U.-2014]
(a) 20 V/m
(b) 500 V/m
(c) 5 V/m
(d) 250 V/m
46. The potential on a charged soap bubble is 16 V. If the radius of the bubble is double then potential will
now be [H.C.U.-2014]
(a) 2V
(b) 4V
(c) 8V
(d) 16V
47. A long, hollow conducting cylinder is kept coaxially inside another long, hollow conducting cylinder of
larger radius. Both the cylinders are initially electrically neutral.
(a) a potential difference appears between the two cylinders when a charge density is given to the
innercylinder.
(b) a potential difference appears between the two cylinders when a charge density is given to the
outercylinder.
(c) no potential difference appears between the two cylinders when a uniform line charge is kept along
the axisof the cylinders.
(d) no potential difference appears between the two cylinders when same charge density is given to both
thecylinders.
48. Consider an infinite line charge with linear charge density𝜆. At a distance r from the line, the
electrostaticpotential has the form
𝜆𝑎
(a) 4𝜋𝜀 𝑟
𝜆
0
𝑟
(b) 4𝜋𝜀 exp (− 𝑎)
0
𝜆
(c) 4𝜋𝜀 ln(𝑟⁄𝑎)
0
(d)
𝜆
𝑟
4𝜋𝜀0 𝑎
49. Two concentric spheres of radii R and 2R has uniform charges 𝑄 and 2𝑄 respectively. Potential
difference between the two spheres is 𝑉0 . If charge on the inner sphere is doubled and outersphere is half
then potential difference will become
(a) zero
(b) 2𝑉0
𝑉
(c) 0
(d)
2
3 𝑉0
2
50. Maximum electric field on the dotted line is 𝐸0 . Maximum electric potential on the dotted line is
(assume reference for potential is taken at far point)
(a)
√3
𝐸𝑑
2 0
3√3
𝐸0 𝑑
2
(b)
(c) 𝐸0 𝑑
𝐸 𝑑
(d) 0
√2
51. A sphere S of radius R has volume charges density 𝜌(𝑟) = 𝛽𝑟 2 , where ‘r’ is distance from centre and 𝛽
is a constant. The electric potential on the surface of sphere is
(a)
(b)
𝛽𝑅 4
4𝜀0
𝛽𝑅 3
4𝜀0
𝛽𝑅 4
(c) 5𝜀
(d)
0
𝛽𝑅 3
5𝜀0
52. A shere of radius R has uniform volume charge density. The electric potential at a point 𝑟(𝑟 < 𝑅) is
(a) Due to the charge inside a sphere of radius r only
(b) Due to the entire charge of the sphere
(c) Due to charge in the spherical shell of inner and outer radii r and R, only
(d) Independent of 𝑟
53. Consider an infinity long cylinder of radius 𝑅, placed along the 𝑧-axis, which carries a static charge
density 𝜌(𝑟) = 𝑘𝑟, where 𝑟 is the perpendicular distance from the axis of the cylinder and 𝑘 is a constant.
The electrostatic potential 𝜙(𝑟) inside the cylinder is proportional to [TIFR 2015]
2
(a) − (
𝑟3
3 𝑅3
+ 1)
𝑟
(b) −2 ℓ𝑛 ( )
2
𝑟3
𝑅
(c) − 3 (𝑅3 − 1)
𝑅
(d) −2 ℓ𝑛 ( 𝑟 )
54. Consider a charged of radius R having uniform volume charge density. Electric field at point ∈0 < 𝑅
has contribution due to charge in the region
(a) 𝑟0 < 𝑟0 only
(b) 0 ≤ 𝑟 ≤ 𝑅
(c) 𝑟 > 𝑟0
(d) none of these
55. In previous question potential at 𝑟0 < 𝑅 has contribution due to charge in the region
(a) 𝑟 < 𝑟0 only
(b) 0 ≤ 𝑟 ≤ 𝑅
(c) 𝑟 > 𝑟0
(d) none of these
56. The electrostatic potential due to a uniformly charged circular disc is[IISc. 2010]
(a) the same at all points on the disc
(b) is larger at the centre than at the edge
(c) is larger at the edge than at the centre
(d) has a maximum volume at half the radius
57. For a spherical shell of radius R which carries a uniform surface charge,[H.C.U.-2010]
(a) both the field and the potential inside the shell are non-zero
(b) both the field and the potential inside the shell are zero
(c) the field inside the shell is non-zero while potential inside the shell is zero
(d) the field inside the shell is zero while potential inside the shell is non-zero
58. A soap bubble 10 cm is radius with a wall thickness of 3.3 × 10−6 cm is charged to a potential of 100 V.
The bubble burst and falls as a spherical drop. The potential of the drop is [H.C.U. -2010]
(a) 10 kV
(b) 1 kV
(c) 100 kV
(d) 10−2 𝑘𝑉
59. Consider a thin spherical shell of radius 𝑅 with its centre at the origin, carrying uniform positive
surface charge density. The variation of the magnitude of the electric field |𝐸⃗⃗ (𝑟)| and the electric potential
𝑉(𝑟) with the distance ′𝑟′ from the centre, is best represented by which graph?
60. One of electrostatic boundary conditions for the potential V for a surface carrying a surface charge
density 𝜎 can be written as
𝜎
(a) 𝑉𝑎𝑏𝑜𝑣𝑒 − 𝑉𝑏𝑒𝑙𝑜𝑤 = − ∈
0
𝜎
(b) 𝑉𝑎𝑏𝑜𝑣𝑒 − 𝑉𝑏𝑒𝑙𝑜𝑤 = − ∈
(c)
(d)
𝜕𝑉𝑎𝑏𝑜𝑣𝑒
𝜕𝑛
𝜕𝑉𝑎𝑏𝑜𝑣𝑒
𝜕𝑛
−
−
𝜕𝑉𝑏𝑒𝑙𝑜𝑤
𝜕𝑛
𝜕𝑉𝑏𝑒𝑙𝑜𝑤
𝜕𝑛
𝜎
0
=∈
0
𝜎
= −∈
0
61. Consider a uniform spherical charge distribution with a total charge 𝑄. The potential at the center of
the charge distribution is given by [B.H.U-2012]
(a) 0
𝑄
(b) (8𝜋𝜀 𝑎)
0
3𝑄
(c) (8𝜋𝜀
0 𝑎)
𝑄
(d) (4𝜋𝜀
2
0𝑎 )
62. A small spherical ball having charge q and mass m, is tied to a thin massless non-conducting string of
length /. The other end of the string is fixed to an infinitely extended thin non-conducting sheet with
uniform surfacecharge density s. Under equilibrium, the string makes an angle 45° with the sheet as
shown in the figure. Then s is given by [JAM 2019]
(g is the acceleration due to gravity and 𝜀0 is the permittivity of free space)
𝑚𝑔𝜀
(a) 𝑞 0
(b) √2
(c) 2
(d)
𝑚𝑔𝜀0
𝑞
𝑚𝑔𝜀0
𝑞
𝑚𝑔𝜀0
𝑞√2
63. The electrostatic potential 𝑉(𝑥, 𝑦) in free space in a region where the charge density 𝜌 is zero is given
by 𝑉(𝑥, 𝑦) = 4𝑒 2𝑥 + 𝑓(𝑥) − 3𝑦 2. Given that the x-compotent of the electric field, 𝐸𝑥 , and V are zero at the
origin 𝑓(𝑥) is:
(a) 3𝑥 2 − 4𝑒 2𝑥 + 8𝑥
(b) 3𝑥 2 − 4𝑒 2𝑥 + 16𝑥
(c) 4𝑒 2𝑥 − 8
(d) 3𝑥 2 − 4𝑒 2𝑥
64. The potential in a medium is given by 𝜙(𝑟) =
[H.C.U.-2015]
𝑞
(a) − ∈ 𝑒 −𝑟/𝜆
0
𝑞
𝑒 −𝑟/𝜆
4𝜋∈0
𝑟
. The charge density for 𝑟 ≠ 0 is given by
𝑞
(b) − 4𝜋𝑟𝜆2 𝑒 −𝑟/𝜆
𝑞
(c) − 4𝜋∈
𝑞
(d) −
0 𝑟𝜆
4𝜋∈0
2
𝑒 −𝑟/𝜆
𝑒 −2𝑟/𝜆
65. If the electrostatic potential in spherical polar coordinates is
𝜑(𝑟) = 𝜑0 𝑒 −𝑟/𝑟0
Where 𝜑0 and 𝑟0 are constants, then the charge density at a distance 𝑟 = 𝑟0 will be
𝜀 0 𝜑0
(a) 𝑒𝑟
2
0
(b)
𝜀𝜀0 𝜑0
2𝑟02
𝜀 0 𝜑0
(c) −
𝑒𝑟02
2𝜀𝜀0 𝜑0
(d) −
𝑟02
66. If the electrostatic potential were given by 𝜙 = 𝜙0 (𝑥 2 + 𝑦 2 + 𝑧 2 ), where 𝜙0 is constant, then the
charge density giving rise to the above potential would be:
(a) 0
(b) −6 𝜙0 𝜀0
(c) −2𝜙0 𝜀0
6𝜙
(d) − 𝜀 0
0
67. The one dimensional potential 𝑉(𝑥) for an electrostatic fields in a region of space which does not
contain any charge has the form (in the following a and b are constants)
(a) 𝑎 sin 𝑏𝑥
2
(b) 𝑎𝑒 −𝑥
1
(c) 𝑎𝑥 2
2
𝑛𝑎
(d) 2𝜋
68. The electrostatic potential is given by, Φ = 0 at 𝑥 = 0 and 𝑥 = 𝑎; Φ = 𝑉 0 ≤ 𝑥 ≤ 𝑎. Then Φ is given by
Φ = ∑∞
𝑛=1 𝐴𝑛 sin(𝛼 𝑥) where 𝛼 is:
𝑛𝜋
(a) 2
(b)
(c)
(d)
𝑛𝜋
𝑎
𝑛𝑎
𝜋
𝑛𝑎
2𝜋
𝑄
69. If 𝜙 = 4𝜋𝜀 𝑟, where Q is the charge and 𝑟 is the distance of the point from 𝑄, then ∇2 𝜙 is
(a) zero
𝑄
(b) 2𝜋𝜀 𝑟2
0
𝑄
(c) 2𝜋𝜀
𝑄
0𝑟
(d) 4𝜋𝜀
3
𝑟⃗
3
0𝑟
0
1
70. The electrostatic potential in a given region of space is 𝜙(𝑟) = ∈ [𝑥 2 + 𝑦 2 − 2𝑧 2 ]. The charge density
0
𝜌(𝑟) in the region is
−𝑥−𝑦+4𝑧
(a)
3
𝑟
𝑥+𝑦−2𝑧
(b)
𝑟
(c) zero
4
(d) 𝑟2
71. Solving Poisson’s equation ∇2 𝜑 = −𝜌0 /𝜀0 for the electrostatic potential 𝜑(𝑥⃗) in a region with a
constant charge density 𝜌0, two students find different answers, viz, [TIFR 2014]
1 𝜌 𝑥2
1 𝜌 𝑦2
𝜑1 (𝑥⃗) = − 0 and 𝜑2 (𝑥⃗) = − 0
2 𝜀0
2 𝜀0
The reason why these different solutions are both correct is because
(a) space is isotropic and hence 𝑥 and 𝑦 are physically equivalent.
(b) We can add solutions of Laplace’s equation to both 𝜑1 (𝑥⃗) and 𝜑2 (𝑥⃗)
(c) The electrostatic energy is infinite for a constant charge density.
(d) The boundary conditions are different in the two cases.
72. In a region of space, that does not contain any electric charge, the electrostatic potential satisfies
Laplace’s equation. Suppose that the potential in such a situation is given by the formula. [JNU 2009]
1
𝑉(𝑥, 𝑦, 𝑧) = 𝐴 (𝑏𝑥 2 + 𝑦 2 − 𝑧 2 )
2
Where A and b are constants, There the value of b
(a) can be arbitrary
(b) must be zero
(c) must be 1/2
(d) must be 1.
𝑒 −𝜆𝑟
73. The electrostatic potential due to a change distribution is given by 𝑉(𝑟) = 𝐴 𝑟 , where A and 𝜆are
constants. The total charge enclosed within a sphere of radius 1/𝜆, with its origin at 𝑟 = 0is given by,
[JEST 2015]
8𝜋𝜀 𝐴
(a) 𝑒 0
(b)
4𝜋𝜀0 𝐴
𝑒
𝜋𝜀 𝐴
(c) 𝑒0
(d) 0
74. Five sides of a hollow metallic cube are grounded and the sixth side is insulated from the rest and is
held at a potential Φ (see figure). [TIFR 2012]
The potential at the center O of the cube is
(a) 0
(b) Φ/6
(c) Φ/5
(d) 2Φ/3
ELECTRIC DIPOLE/MULTI POLE & ASYMPTOTIC VARIATION
1. The charge on one proton is 1.6 × 10−19 𝐶 and bond length of HCl molecule is 1.28 Å. Magnitude of its
electrical dipole moment is [B.H.U.-2010]
(a) 1.6 × 10−19
(b) 2.05 × 10−29
(c) 1.28 × 10−11
(d) 2.05 × 10−31
2. Consider two charges +𝑄 and −𝑄 placed at the points (−𝑎, 0) in a plane, as shown in the figure on the
right. If the origin is moved to the point (𝑋, 𝑌), the magnitude of the dipole moment of the given charge
distribution with respect to this origin will be [TIFR 2013]
(a) 𝑄√(𝑎 − 𝑋)2 + 𝑦 2 − 𝑄√(𝑎 + 𝑋)2 + 𝑦 2
(b) 2𝑄𝑎
(c) 𝑄(𝑎 − 𝑋) − 𝑄(−𝑎 + 𝑋)
(d) 2𝑄𝑎 √𝑋 2 + 𝑌 2
3. An electric dipole is constructed by fixing two circular charged rings, each of radius ‘a’, with an
insulting contact (see figure). One of these rings has total charge +𝑄. If the charge is distributed uniformly
along each ring, the dipole moment about the point of contact will be [TIFR 2014]
𝑄𝑎
(a) 𝜋 𝑧̂
(b) 4𝜋𝑄𝑎𝑧̂
(c) 2𝑄𝑎 𝑧̂
(d) Zero
4. A charge −𝑞 is distributed uniformly over a sphere, with a positive charge 𝑞 at its center in (i). also in
(ii), a charge −𝑞 is distributed uniformly over an ellipsoid with a positive charge 𝑞 at its center. With
respect to the origin of the coordinate system, which one of the following statements is correct?
(a) The dipole moment is zero in both (i) and (ii)
(b) The dipole moment is non-zero in (i) but zero in (ii)
(c) The dipole moment is zero in (i) but non-zero in (ii)
(d) The dipole moment is non-zero in both (i) and (ii)
5. A sphere of radius R has surface charge density 𝜎 = 𝜎0 sin 𝜃. Which of the following statements is/are
correct
(a) dipole moment is non-zero
(b) electric field inside sphere is non-zero
(c) potential at centre is zero
(d) asymptotically field varies as 1/𝑟 3
6. An electric dipole is placed at the origin and is directed along the 𝑥-axis. At a point 𝑃, far away from the
dipole, the electric field is parallel to the 𝑦-axis. OP makes an angle 𝜃 with the 𝑥-axis.
(a) tan 𝜃 = √3
(b) tan 𝜃 = √2
(c) 𝜃 = 45°
1
(d) tan 𝜃 =
√2
7. An electric dipole of dipole moment 𝑝0 𝑖̂ is placed at origin which of the following statements is NOT
correct?
(a) Electric flux through 𝑥𝑦 plane is zero
(b) Electric flux through 𝑦𝑧 plane is zero
(c) Electric flux through 𝑥𝑧 plane is zero
𝑝0 𝑥
(d) Electric potential at a point in space is 4 𝜋∈ (𝑥 2+𝑦
2 +𝑧 2 )3/2
0
8. Four electric charges of strength 𝑄, 𝑄, −𝑄 and −𝑄 are placed respectively at the points
(−𝑏, 0), (𝑏, 0), (0, −𝑎) and (0, 𝑎)on the xy-plane. At a distance d very far away from these charges
(𝑑 >> 𝑎, 𝑏), the electric field will appear to be approximately that of [JNU 2009]
(a) a point charge
(b) a dipole
(c) a quadrupole
(d) a mixture of a dipole and a qaudrupole
9. The potential due to an electric dipole P placed at the origin at the origin in known to be of the form
𝜙(𝑟) = 𝑃⃗⃗, 𝑟⃗/𝑟 3. The total flux of the electric field through a spherical surface of radius R with the dipole at
the centre is given by
[JNU 2009]
(a) zero
2𝜋 |𝑃|
(b) − 3 𝑅
(c) +
(d) +
2𝜋 |𝑃|
3 𝑅
4𝜋 |𝑃|
3 𝑅
10. Four point charges ±𝑞1 and ±𝑞2 are placed at the corners of a rectangle of sides a and b as shown in
the figure: [JNU 2011]
What is the magnitude of the dipole moment of the system? [JNU 2011]
(a) (𝑞1 + 𝑞2 )√𝑎2 + 𝑏 2
(b) (𝑞1 − 𝑞2 )(𝑎 − 𝑏)
(c) √(𝑞1 + 𝑞2 )2 𝑎2 + (𝑞1 − 𝑞2 )2 𝑏 2
(d) The dipole moment will on the choice of origin
11. Four point charges are placed in a plane at the following positions:
+𝑄 at (1,0), −𝑄 at (−1,0), +𝑄 at (0,1) and −𝑄 at (0, −1)
At large distances the electrostatic potential due to this charge distribution will be dominated by the
(a) monopole moment
(b) dipole moment
(c) quadrupole moment
(d) octopole moment
12. A circular disc of radius 𝑎 on the 𝑥𝑦 plane has a surface density, 𝜎 =
moment of this charge distribution is
(a)
(b)
𝜎0 𝜋𝑎 4
4
𝜎0 𝜋𝑎 3
(c) −
(d) −
𝑎
. The electric dipole
𝑥̂
𝑥̂
4
𝜎0 𝜋𝑎 3
4
𝜎0 𝜋𝑎 4
4
𝜎0 cos 𝜃
𝑥̂
𝑥̂
13. Three charges are kept on a straight line as shown in figure. With A and B denoting constants, the
potential at a far point (𝑟 >> 𝑎) on the axis is given by [H.C.U.-2013]
(a) 𝐴/𝑟
(b) 𝐴/𝑟 2
(c) 𝐴/𝑟 3
𝐴
𝐵
(d) 𝑟2 + 𝑟3 + ⋯
14. The electrostatic potential 𝜑(𝑟) of a distribution of point charges has the form 𝜑(𝑟𝑅) ∝ 𝑟 −3 at a
distance 𝑟 from the origin (0,0,0) where 𝑟 ≫ 𝑎. Which of the following distributions can give rise to this
potential? [TIFR 2015]
15. An insulting sphere of radius a carries a charge density 𝜌(𝑟⃗) = 𝜌0 (𝑎2 − 𝑟 2 ) cos 𝜃 ; 𝑟 < 𝑎. The leading
order term for the electric field at a distance 𝑑, for away from the charge distribution, is proportional to
(a) 𝑑 −1
(b) 𝑑−2
(c) 𝑑 −3
(d) 𝑑 −4
16. Electric charges 𝑞, 𝑞 and −2𝑞 are placed at the corners of an equilateral triangle ABC of side L. The
magnitude of electric dipole moment of the system is
(a) 𝑞𝐿
(b) 2𝑞𝐿
(c) (√3)𝑞𝐿
(d) 4𝑄𝐼
17. If a dipole 𝑝⃗ situated at the origin (0,0,0) is pointing in the +𝑧-direction, then the force on a point
charge 𝑞 at (𝑎, 0,0) is [H.C.U.-2012]
2𝑝𝑞
(a) 𝐹 = 4 𝜋∈ 𝑎3 𝑧̂
0
(b) 𝐹 =
−2𝑝𝑞
4𝜋∈0 𝑎 3
𝑝𝑞
(c) 𝐹 = − 4𝜋∈
(d) zero
𝑧̂
0𝑎
3
𝑧̂
18.An electric dipole moment 𝑝0 𝑖̂ is placed in an external electric field 𝐸⃗⃗ = 𝐴(𝑥 3 − 𝑥 2 )𝑖̂. Force on dipole is
𝑥
zero at 𝑥1 (≠ 0) and minimum at 𝑥2 , then value of 𝑥1 is _______
2
19. An electric dipole of dipole moment 𝑝0 𝑖̂ is placed at origin which of the following statements is/are
correct?
(a) Electric flux through 𝑥𝑦 plane is zero
(b) Electric flux through 𝑦𝑧 plane is zero
(c) Electric flux through 𝑥𝑧 plane is zero
𝑝0 𝑥
(d) Electric potential at a point is space is 4𝜋∈ (𝑥 2+𝑦
2 +𝑧 2 )3/2
0
20. Consider an electric dipole of moment 𝑝. Electric field at any point (𝑟, 𝜃) due to the dipole is given as
𝑘𝑝
𝜕𝐸⃗⃗
𝐸⃗⃗ = 3 (2 cos 𝜃 𝑟̂ + sin 𝜃 𝜃̂). Value of is
𝑟
𝑘𝑝
(a) 𝑟3 [−2 cos 𝜃 𝑟̂ + sin 𝜃 𝜃̂]
3𝑘𝑝
[− sin 𝜃 𝑟̂ + cos 𝜃 𝜃̂]
(b)
(c)
𝜕𝜃
𝑟3
𝑘𝑝
𝑟3
𝑘𝑝
[−3 sin 𝜃 𝑟̂ + 2 cos 𝜃̂]
(d) 𝑟3 [−3 sin 𝜃 𝑟̂ + 2 cos 𝜃 𝜃̂]
21. Charges 𝑄, 𝑄 and −2𝑄 are placed on the vertices of an equilateral triangle ABC of sides of length a, as
shown in the figure
The dipole moment of this configuration of charges, irrespective of the choice of origin, is
(a) +2𝑎𝑄𝑖̂
(b) +√3𝑎𝑄𝑗̂
(c) −√3𝑎𝑄𝑗̂
(d) 0
22. A ring of radius 𝑅 has charge density +𝜆 on one half and – 𝜆 on the other half. Its dipole moment is
(a) 4𝜆𝑅2
(b) 2𝜆𝑅2
(c) 3𝜆𝑅2
(d) 𝜆𝑅2
23. A particle of charge e and mass m is located at the midpoint of the line joining two mixed collinear
dipoleswith unit charges as shown in the figure. (The particle is constrained to move only along the line
joining thedipoles). Assuming that the length of the dipoles is much shorter than their separation, the
natural frequency of oscillation of the particle is
6𝑒 2 𝑅 2
(a) √𝜋𝜀
0 𝑚𝑑
5
6𝑒 2 𝑅
(b) √𝜋𝜀
0 𝑚𝑑
4
6𝑒 2 𝑑 2
(c) √𝜋𝜀
0 𝑚𝑅
5
6𝑒 2 𝑑
(d) √𝜋𝜀
0 𝑚𝑅
4
24. Four charges are placed at the four corners of a square of side a as shown in the figure. The electric
dipole moment of this configuration is:
(a) 𝑝⃗ = 𝑞𝑎𝑖̂ + 𝑞𝑎𝑗̂
(b) 𝑝⃗ = −𝑞𝑎𝑖̂ + 𝑞𝑎𝑗̂
(c) 𝑝⃗ = −𝑞𝑎𝑖̂ − 𝑞𝑎𝑗̂
(d) 𝑝⃗ = 𝑞𝑎𝑖̂ − 𝑞𝑎𝑗̂
25. Four charges are placed at the corners of a square as shown in figure. Which of the following
statements is/are correct.
√2𝑞
(a) Electric field at the centre of square is 𝜋𝜀
0𝑎
2
(b) Dipole moment of system is 2𝑞𝑎
1
(c) Field varies as 𝑟3 for 𝑟 >> 𝑎
(d) There are infinite number of neutral points
26. There point charges 𝑞, 𝑞 and −2𝑞 are located at (0, −𝑎, 𝑎), (0,0, −𝑎), respectively. The net dipole
moment of this charge distribution is:
(a) 4𝑞𝑎𝑘̂
(b) 2𝑞𝑎𝑘̂
(c) −4𝑞𝑎𝑖̂
(d) −2𝑞𝑎𝑗̂
27. The dipole moment of the following configuration about the centre:
(a) 4 𝑞𝑎 (𝑖̂ + 𝑗̂)
(b) 2𝑞 𝑎 (𝑖̂ − 𝑗̂)
(c) 0
(d) 8 𝑞𝑎2 𝑘̂
28. Two electric dipoles of dipole moment 𝑝1 and 𝑝2 are placed at origin such that one is along 𝑥-axis and
𝑝
the other along 𝑦-axis respectively. If potential at point (𝑟, 𝜃) is zero than | 1 | is
(a) 1
(b) cot 𝜃
(c) tan 𝜃
(d) sin 𝜃 . cos 𝜃
𝑝2
29. If the intensity of electric field at a distance 𝑥 from the centre in the axial position of a small electric
dipole is equal to intensity at distance 𝑦 in equatorial position, then
(a) 𝑥 = 𝑦
(b) 𝑥 = 𝑦/2
(c) 𝑦 = 𝑥/22/3
(d) 𝑦 = 𝑥/21/3
30. Electric force between two dipoles varies with distance as
1
(a) 2
𝑟
1
(b) 𝑟3
1
(c) 𝑟4
1
(d) 𝑟
31. An electric dipol is placed at origin along 𝑥-direction. The direction of electric field at (3,4,0) is
2𝑖̂− 36𝑗̂
(a) 10√13
(b)
(c)
(d)
2𝑖̂+36𝑗̂
10√13
36𝑖̂− 2𝑗̂
10√13
36𝑖̂+2𝑗̂
10√13
32. An electric point charge +𝑞 is placed at the point (1,1) of the xy-plane in which two grounded semiinfinite conducting plates along the positive x and y-axes meet see figure). The electric potential in the
positive quadrant at a large distance r goes as
(a) 𝑉(𝑟)~𝑟 −1
(b) 𝑉(𝑟)~𝑟 −2
(c) 𝑉(𝑟)~𝑟 −3
(d) 𝑉(𝑟)~𝑟 −4
33. Two point charges 𝑞 and −2𝑞 lie at 𝑥 = −𝑎 and 𝑥 = +𝑎 respectively. which of the following
statements is/are correct
(a) There is only one neutral point on 𝑥-𝑦 plane
𝐸(𝑟)
(b) At large distance 𝑟 ≫ 𝑎 𝐸(2𝑟) = 4
(c) Equipotential lines on 𝑥-𝑦 plane are circles
3𝑞
(d) Electric flux through y-z plane is 2∈
0
34. Four charges (two +𝑞 and two −𝑞) are kept fixed at the four vertices of a square of side ‘a’ as shown
At the point P which is at a distance R from the centre (𝑅 ≫ 𝑎), the potential is proportional to
1
(a)
𝑅
1
(b) 𝑅2
1
(c) 𝑅3
1
(d) 𝑅4
35. Consider the charge distribution shown in the figure. Which of the following statement is NOT correct.
(a) Electric potential at every point on 𝑥-axis is zero
(b) Electric field at every point on 𝑥-axis is in −𝑗̂ direction.
(c) Asymptotically field varies as 1/𝑟 4
𝜆
(d) Electric field at centre of circle is 2𝜋∈ 𝑎
0
36. The electrostatic lines of force due to a system of four point charges is sketched below.
At a large distance 𝑟, the leading asymptotic behaviour of the electrostatic potential is proportional to
(a) 𝑟
(b) 𝑟 −1
(c) 𝑟 −2
(d) 𝑟 −3
37. A spherically symmetric charge distribution 𝜌(𝑟) has zero net charge. For an arbitrarily chosen
coordinate system, which of the following statements is true? [IISc. 2009]
(a) Dipole moment is the first non-zero moment
(b) Quadrupole moment is always non-zero
(c) All moments are zero
(d) The monopole and the dipole moments always zero
38. A sphere has uniform volume charge density, if both radius and charged of the sphere are doubled,
then potential at the centre of the sphere becomes 𝛽 times, value of 𝛽 is ______
CONDUCTORS & IMAGE PROBLEMS
1. A solid spherical conductor encloses 3 cavities, a cross-section of which are as shown in the figure. A
net charge +q resides on the outer surface of the conductor. Cavities A and C contain point charges +𝑞
and – 𝑞, respectively. [TIFR 2015]
The net charge on the surface of these cavities are
(a) 𝐴 = −𝑞, 𝐵 = 𝑞, 𝐶 = 0
(b) 𝐴 = −𝑞, 𝐵 = 0, 𝐶 = +𝑞
(c) 𝐴 = 𝑞, 𝐵 = 0, 𝐶 = −𝑞
(d) 𝐴 = −𝑞, 𝐵 = 0, 𝐶 + 𝑞
The net charges on the surface of these cavities are
(a) 𝐴 = −𝑞, 𝐵 = 𝑞, 𝐶 = 0
(b) 𝐴 = −𝑞, 𝐵 + 0, 𝐶 = −𝑞
(c) 𝐴 = +𝑞, 𝐵 = 0, 𝐶 = −𝑞
(d) 𝐴 = −𝑞, 𝐵 = 0, 𝐶 = +𝑞
2. An elliptical cavity is carved within a perfect condcutor. A positive charge q is placed at the centre of
the cavity. The point A and B are on the cavity surface as shown in the figure. Then
(a) electric field near A in the cavity = electric field near B in the cavity q B
(b) charge density at A = charge density at B
(c) potential at A = potential at B
𝑞
(d) total electric field flux through the surface of the cavity is ∈
0
3. A metallic shell has a point charge ‘q’ kept inside its cavity. Which one of the following diagrams
correctly represents the electric lines of forces?
4. Three concentric metallic spherical shells of radii 𝑅, 2𝑅, 3𝑅 are given charges 𝒬1 , 𝒬2 , 𝒬3 respectively.
It is found that the surface charge densities on the outer surfaces of the shells are equal. Then, the ratio of
the charges given to the shells, 𝒬1 , 𝒬2 , 𝒬3 is
(a) 1 : 2 : 3
(b) 1 : 3 : 5
(c) 1 : 4 : 9
(d) 1 : 8 : 18
5. A solid metal sphere with a spherical cavity as shown below has a total charge +𝑄. [IISc. 2007]
O is the centre of the sphere, and P and R are two points equidistant form it. If P E and R E represent the
magnitude of the electric field at P and R respectively, which of the following statements is correct ?
(a) 𝐸𝑃 = 𝐸𝑅
(b) 𝐸𝑃 = 0 and 𝐸𝑅 = 0
(c) 𝐸𝑃 > 𝐸𝑅
(d) 𝐸𝑃 < 𝐸𝑅
6. A thin metallic shell contains charge 𝑄 on it. A point charge q is placed at the centre of the shell and
another charge 𝑞1 is placed outside it as shown in the adjacent figure. All charges are positive. The force
on the charge at the centre is
(a) towards left
(b) towards right
(c) upward
(d) zero
Common data for Q. 7 and Q. 8.
Consider two concentric conducting spherical shells with inner and outer radii a, b and c, d as shown in
the figure. Both the shells are given 𝑄 amount of positive charges
7. The electric field different regions are:
𝑄
𝑄
(a) 𝐸⃗⃗ = 0 𝑓𝑜𝑟 𝑟 < 𝑎; 𝐸⃗⃗ 4𝜋𝜀 𝑟2 𝑟̂ 𝑓𝑜𝑟 𝑎 < 𝑟 < 𝑏 ; 𝐸⃗⃗ = 0 𝑓𝑜𝑟 𝑏 < 𝑟 < 𝑐; 𝐸⃗⃗ = 4𝜋𝜀 𝑟2 𝑟̂ 𝑓𝑜𝑟 𝑟 > 𝑏
0
0
𝑄
𝑄
𝑄
(b) 𝐸⃗⃗ = 4𝜋𝜀 𝑟2 𝑟̂ 𝑓𝑜𝑟 𝑟 < 𝑎; 𝐸⃗⃗ = 0 𝑓𝑜𝑟 𝑎 < 𝑟 < 𝑏 ; 𝐸⃗⃗ = 4𝜋𝜀 𝑟2 𝑟̂ 𝑓𝑜𝑟 𝑏 < 𝑟 < 𝑐; 𝐸⃗⃗ = 4𝜋𝜀 𝑟2 𝑟̂ 𝑓𝑜𝑟 𝑟 > 𝑑
0
0
0
𝑄
2𝑄
(c) 𝐸⃗⃗ = 4𝜋𝜀 𝑟2 𝑟̂ 𝑓𝑜𝑟 𝑟 < 𝑏; 𝐸⃗⃗ = 0 𝑏 < 𝑟 < 𝑐; 𝐸⃗⃗ = 4𝜋𝜀 𝑟2 𝑟̂ 𝑓𝑜𝑟 𝑟 > 𝑑
0
0
𝑄
2𝑄
(d) 𝐸⃗⃗ = 0 𝑓𝑜𝑟 𝑟 < 𝑎; 𝐸⃗⃗ = 0 𝑓𝑜𝑟 𝑎 < 𝑟 < 𝑏 ; 𝐸⃗⃗ 4𝜋𝜀 𝑟2 𝑟̂ 𝑓𝑜𝑟 𝑏 < 𝑟 < 𝑐; 𝐸⃗⃗ = 4𝜋𝜀 𝑟2 𝑟̂ 𝑓𝑜𝑟 𝑟 > 𝑑
0
0
8. In order to have equal surface charge density on the outer surface of both the shells, the following
conditions should be satisfied
(a) 𝑑 = 4𝑏 𝑎𝑛𝑑 𝑐 = 2𝑎
(b) 𝑑 = 2𝑏 𝑎𝑛𝑑 𝑐 = √2𝑎
(c) 𝑑 = √2𝑏 𝑎𝑛𝑑 𝑐 > 𝑎
(d) 𝑑 > 𝑏 𝑎𝑛𝑑 𝑐 = √2𝑎
9. Avery long wire carrying charge ‘q’ per unit length is held parallel to an infinite conducting plane at a
distance ℎ from i. What is the force of attractions per unit length ? [H.C.U.2015]
(a)
𝑞2
4𝜋∈0 ℎ
–𝑞
(b) 4𝜋∈
0ℎ
𝑞2
(c) 2𝜋∈
0ℎ
–𝑞2
(d) 2𝜋∈
0ℎ
10. A hollow, conducting spherical shell of inner radius 𝑅1 and outer radius 𝑅2 encloses a charge 𝑞 inside,
which is located at a distance 1 𝑑 (< 𝑅1 ) from the centre of the spheres. The potential at the centre of the
shell is [JAM 2015]
11. An arbitrarily shaped conductor encloses a charge 𝑞 and is surrounded by a conducting hollow sphere
as shown in the figure. Four different regions of space, 1, 2, 3 and 4, are indicated in the figure. Which one
of the following statements is correct? [JAM 2016]
(a) The electric field lines in region 2 are not affected by the position of the charge q
(b) The surface charge density on the inner wall of the hollow sphere is uniform
(c) The surface charge density on the outer surface of the sphere is always uniform irrespective of the
position of charge q in region 1
(d) The electric field in region 2 has a radial symmetry
12. A steady current in a straight conducting wire produces a surface charge on it. Let 𝐸𝑂𝑢𝑡 and 𝐸𝑖𝑛 be the
magnitudes of the electric fields just outside and just inside the wire, respectively. Which of the following
statements is true for these fields? [JAM 2014]
(a) 𝐸𝑜𝑢𝑡 is always greater than 𝐸𝑖𝑛
(b) 𝐸𝑜𝑢𝑡 is always smaller than 𝐸𝑖𝑛
(c) 𝐸𝑜𝑢𝑡 could be greater or smaller than Ein
(d) 𝐸𝑜𝑢𝑡 is equal to 𝐸𝑖𝑛
13. A solid spherical conductor has a conical hole made at one end, ending in a point B, and a small conical
projection of the same shape and size at the opposite side, ending in a point A. A cross-section through the
centre of the conductor is shown in the figure on the right. [TIFR 2014]
If now, a positive charge Q is transferred to the sphere, then
(a) The charge density at both A and B will be undefined.
(b) The charge density at A will be the same as the charge density at B.
(c) The charge density at A will be more than the charge density at B.
(d) The charge density at B will be more than the charge density at A.
14. A spherical conductor, carrying a total charge Q, spins uniformly and very rapidly about an axis
coinciding with one of its diamters. In the diagrams given below, the equilibrium charge density on its
surface is represented by the thickness of the shaded region. Which of these diagram is correct?
[TIFR 2014]
(a)
(b)
(c)
(d)
15. Two identical metal plates are given positive charges 𝑄1 𝑎𝑛𝑑 𝑄2 (< 𝑄1) respectively. If they are now
brought close together to form a parallel plate capacitor with capacitance C, the potential difference
between them is
(𝒬 +𝒬 )
(a) 12𝐶 2
(b)
(c)
(d)
(𝒬1 +𝒬2 )
𝐶
(𝒬1 −𝒬2 )
𝐶
(𝒬1 −𝒬2 )
2𝐶
16. In a uniform electric field E, a dielectric (dielectric constant ≠ 1) in the form of a sphere is introduced.
How will the intensity of the field at points A, B and C change?
(a) Field at A will increase
(b) Field at B will decrease
(c) Field at C will increase
(d) Field at all points will remain same
17. A spherical conductor of radius a is placed in a uniform electric field 𝐸⃗⃗ = 𝐸0 𝑘̂ . The potential at a point
𝐸 𝑎3
𝑃(𝑟, 𝜃) 𝑓𝑜𝑟 𝑟 > 𝑎, is given by, 𝜑(𝑟, 𝜃) = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 − 𝐸0 𝑒 cos 𝜃 + 0𝑟2 cos 𝜃
Where ′𝑟′ is the distance of 𝑃 from the centre 𝑂 of the sphere and 𝜃 is the angle OP makes with the z-axis
The charge density on the sphere at 𝜃 = 30° is:
(a) 3√3 𝜀0 𝐸0 /2
(b) 3𝜀0 𝐸0 /2
√3𝜀 𝐸
(c) 20 0
(d) 𝜀0 𝐸0 /2
18. Two metallic cubes of side a and 2a respectively are separated by a large distance D, (D>>a). Initially
the smaller cube carries a charge Q while the larger one is uncharged. If a thin metallic wire is connected
between the two cubes, the ratio of the charges on the smaller cube to the larger one will be : [IISc. 2010]
(a) 1/4
(b) 1/2
(c) 2/3
(d) 4/9
19. Tow plane parallel conducting plates have surface charge density 𝜎 each. What is the force per unit
area between the two plates [IISc. 2010]
𝜎2
(a) 2𝜀 , attractive
0
(b)
(c)
(d)
𝜎2
2𝜀0
𝜎2
, repulsive
, repulsive
𝜀0
2𝜎 2
𝜀0
, repulsive
20. Two metallic sphere of radius 1 cm and 3 cm respectively are separated by a large distance
𝐷, (𝐷 ≫ 1𝑐𝑚). Initially, the smaller sphere carries a charge Q where the larger one is uncharged. If a thin
metallic wire is connected between the two sphere, the ratio of the smaller sphere to the larger on in
equilibrium will be:
(a) 1/3
(b) 1/9
(c) 3
(d) 9
21. A charge +q is brought near an isolated metal cube having no charge initially
(a) the cube becomes positively charged
(b) the cube becomes negatively charged
(c) the external surface becomes negatively charged and the interior becomes positively charged
(d) the interior remains charge free and the surface gets non-uniform charge distribution
22. Two conducting plates P and S with large surface area A are placed as shown in the figure. A charge q
is given to plate P. The electric field between the plates at any point is
𝑞
(a) 3𝐴𝜀
𝑞
0
(b) 2𝐴𝜀
𝑞
(c) 𝐴𝜀
(d)
0
0
2𝑞
𝐴𝜀0
23. A spherical conductor A contains two spherical cavities. The total charge on the conductor itself is
zero. However, there is a point charge 𝑞𝑏 at the centre of one cavity and 𝑞𝑐 at the centre of the other. A
considerable distance r away from the centre of the spherical conductor, there is another charge 𝑞𝑑 .
Force acting on 𝑞𝑏 , 𝑞𝑐 and 𝑞𝑑 are.
𝐹1 𝐹2 and 𝐹3 respectively. [Assume all charges are positive]
(a) 𝐹1 < 𝐹2 < 𝐹3
(b) 𝐹1 = 𝐹2 < 𝐹3
(c) 𝐹1 = 𝐹2 > 𝐹3
(d) 𝐹1 > 𝐹2 > 𝐹3
24. Consider an electric field 𝐸⃗⃗ existing in the interface between a conductor and free space. Then the
electric field 𝐸⃗⃗ is : [GATE 2001]
(a) External to the conductor and normal to the conductor’s surface
(b) Internal to the conductor and normal to the conductor’s surface
(c) External to the conductor and tangential to the conductor’s surface
(d) Both external and internal to the conductor and normal to the conductor’s surface
25. Three infinite plane have a uniform surface charge distribution 𝜎 on its surface. All charges are fixed.
On each of the three finite planes, parallel to the 𝑦 − 𝑧 plane placed at 𝑥 = −𝑎, 𝑥 = 0 and = 𝑎 , there is a
uniform surface charge of the same density, 𝜎 . The potential difference between A and 𝐶 is
𝜎
(a) 2∈ 𝑎
𝜎
0
(b) ∈ 𝑎
0
𝑎𝜎
(c) 2∈
0
(d) zero
26. 26. An electric charge, +𝑄 is placed on the surface of a solid, conduction sphere of radius a. The
distance measured from the centre of the sphere is denoted as 𝑟. Then : [GATE 2003]
(a) The charge gets distributed uniformly through the volume of the sphere
(b) The electrostatic potential has the same value for 𝑟 < 𝑎
(c) An equal and opposite charge gets induced in the bottom half of the sphere
(d) The electric field is given by 1/1 (4𝜋𝜀0 𝑟 2 )𝑓𝑜𝑟 𝑟 > 𝑎
27. A spherical conductor of radius R has two cavities centered at 𝑟1 and 𝑟2 respectively from the centre of
the conductor. The cavities contain point charges +2𝑞 and –q at their respective centers as shown in the
figure. The magnitude of the electric field at a point P is :
(a)
(b)
1
𝑞
4𝜋𝜀0 𝑟2
1
4𝜋𝜀0
1
(c) 4𝜋𝜀
1
2
2𝑞
[((𝑟−𝑟
𝑞
1
)2
2 −1/2
−𝑞
) + ((𝑟−𝑟
1
)2
) ]
2
0 (𝑟−𝑟1 −𝑟2 )
(d) 4𝜋𝜀
𝑞
2
2
0 (𝑟−𝑟1 ) +(𝑟−𝑟2 )
28. A point charge +q is placed outside a grounded conducting sphere of radius a. Which of the following
statements is not true? [JNU 2014]
(a) there is an attractive force between the sphere and the point charge
(b) the induced surface charge density on the sphere is not the same everywhere
(c) the electric field inside the sphere is zero
(d) the total induced charge on the sphere is -q
(e) the electrostatic potential at a large distance d (compared to the distance between the charge and the
sphere) falls off as 1/d.
29. A point charge +q is placed at (0, 0, d) above a grounded infinite conducting plane defined by z=0.
There are no charge present anywhere else. What is the magnitude of electric field at (a, a, -d) ? [JEST
2012]
𝑞
(a)
2
8𝜋𝜀0 𝑑
(b) −∞
(c) 0
𝑞
(d)
(16𝜋𝜀0 𝑑 2 )
30. A point charge 𝑞 of mass 𝑚 is released from rest at a distance 𝑑 from an infinite grounded conducting
plane (ignore gravity). How long does it takes for the charge to hit the plane?
(a)
(b)
(c)
(d)
√2𝜋3 𝜀0 𝑚𝑑 3
𝑞
√2𝜋3 𝜀0 𝑚𝑑 3
𝑞
√𝜋 3 𝜀0 𝑚𝑑 3
𝑞
√2𝜋3 𝜀0 𝑚𝑑
𝑞
31. A charge 𝑞 is placed in front of an infinite metal plate at a distance d. If instead a charge 2q were to
experience the same force, at what distance should it be placed from the plate ?
(a) 𝑑
(b) 2𝑑
(c) 𝑑/2
(d) 4𝑑
32. A point charge 𝑄 is brought without any acceleration form infinity to a distance ‘d’ from an infinite
plane conducting sheet. The work done on the charge is given by [IISc. 2010]
1
𝑄2
1
0 4𝑑
𝑄2
(1) 4𝜋𝜀
(b) 4𝜋𝜀
0 2𝑑
1
(c) − 4𝜋𝜀
𝑄
0 4𝑑
1 𝑄2
(d) − 4𝜋𝜀
0 2𝑑
33. A point charge ‘q’ of mass ‘m’ is kept at a distance ‘d’ below a grounded infinite conducting sheet
which lies in the xy-plane. What is the value of ‘d’ for which the charge remains stationary?
(a) 𝑞/4 √𝑚𝑔𝜋𝜀0
(b) 𝑞/√𝑚𝑔𝜋𝜀0
(c) There is no finite value of ‘d’
(d) √𝑚𝑔𝜋𝜀0 /𝑞
34. A positive charge 𝒬 is located at a distance L above an infinite grounded conducting plane, as shown in
the figure below. What is the total charge induced on the plane? [B.H.U-2012]
(a) 2𝒬
(b) 𝒬
(c) 0
(d) −𝒬
35. A point charge q is placed symmetrically at a distance d from two perpendicularly placed grounded
conducting infinite plates as shown in the figure. The net force on the charge (in units of
1/4𝜋 ∈0) is
𝑞2
(a) 8𝑑 2 (2√2 − 1) away from the corner
𝑞2
(b) 8𝑑 2 (2√2 − 1) toward the corner
𝑞2
(c) 8𝑑 2 2√2𝑑2 towards the corner
3𝑞2
(d) 8𝑑 2 away from the corner
36. A charged particle is at a distance d from an infinite conducting plane maintained at zero potential.
When released from rest, the particle reaches a speed u at a distance d/2 from the plane. At what distance
from the plane will the particle reach the speed 2u?
(a) d/6
(b) d/3
(c) d/4
(d) d/5
37. A charge (-e) placed in vacuum at the point (d, 0, 0), where > 0 . The region 𝑥 ≤ 0 is filled uniformly
𝑑
with a meal. The electric filed at the point ( 2 , 0, 0) is
10𝑒
(a) − 9𝜋𝜀
10𝑒
(b) 9𝜋𝜀
𝑒
(c) 𝜋𝜀
0𝑑
0𝑑
2
0𝑑
2
2
(1, 0, 0)
(1, 0, 0)
10𝑒
(d) − 𝜋𝜀
0𝑑
2
(1, 0, 0)
38. A Charge +q is kept at a distance of 2𝑅 form the center of a grounded conducting sphere of radius 𝑑 in
front of a semi-infinite metal surface is:
𝑞2
(a) 4𝜋𝜀
(b)
0 (𝑑)
𝑞2
4𝜋𝜀0 (2𝑑)
𝑞2
(c) 4𝜋𝜀
0 (4𝑑)
𝑞2
(d) 4𝜋𝜀
0 (6𝑑)
40. Two charges q and 2q are placed along the x-axis in front of a grounded, infinite conducting plane, as
shown in the figure. They are located respectively at a distance of 0.5 m and 1.5 m from the plane. The
force acting on the charge q is:
(a)
1
7𝑑 2
4𝜋𝜀0 2
(b)
1
2𝑑2
4𝜋𝜀0
1
(c) 4𝜋𝜀 𝑞 2
0
1 𝑞2
(d)
4𝜋𝜀0 2
41. Tow point charge are placed in front of a conducting sheet (infinite) as shown in the figure. Electric
force on charge 𝑞 is
(a)
(b)
(c)
(d)
𝑘𝑞2
2𝑥 2
3𝑘𝑞2
4𝑥 2
7𝑘𝑞2
2𝑥 2
5𝑘𝑞2
2𝑥 2
42. Let point charge +q be at a distance d from an infinite conducting plate which is grounded .
The force experience by the charge is given by
1 𝑞2
̂
(a) 𝐹⃗ = +
2𝑘
4𝜋𝜀0 𝑑
1 𝑞2
(b) 𝐹⃗ = − 4𝜋𝜀
𝑘̂
2
0𝑑
1
𝑞2
(c) 𝐹⃗ = − 4𝜋𝜀
2
0 (2𝑑)
1
𝑞2
(d) 𝐹⃗ = + 4𝜋𝜀
2
0 (2𝑑)
𝑘̂
𝑘̂
43. A point charge q is kept at the mid-point between two large parallel grounded conducting plates.
Assume no gravity. The charge is displaced a little towards the right plate. The charge will now,
(a) Stay where it is
(b) Move towards the right plate
(c) Move towards the left plate
(d) Oscillate between the plates
44. A grounded conducting sphere of radius a is placed with its centre at the origin. A point dipole
moment 𝑝⃗ = 𝑝𝑘̂ is placed at a distance d along the x-axis, where 𝑖̂, 𝑗̂ are the units vector along the x and zaxes respectively. This leads to the formation of an image dipole of strength 𝑝⃗′ at a distance 𝑑′ from the
centre along the x-axis. If 𝑑′ = 𝑎2 /𝑑, then 𝑝⃗ = [TIFR 2016]
𝑎4𝑝
(a) – 4 𝑘̂
𝑑
(b) –
(c) −
(d) –
𝑎3𝑝
𝑑3
𝑎2𝑝
𝑘̂
𝑘̂
𝑘̂
𝑑2
𝑎𝑝
𝑑
45. Two equal and opposite charges are placed at a distance r from each other. A large metallic sheet is
placed at a distance d from them as shown in the figure. Due to the presence of the sheet, the attractive
force between the charges along the direction joining them
(a) decreases
(b) increases
(c) remains unchanged
(d) decreases or increases depending on the conductivity of the sheet.
46. A point charge is placed between two semi-infinite conducting plates which are inclined at an angle of
30° with respect to each other. The number of image charges is_______
47. A positive point charge Q is located at a distance d from two grounded conducting half planes
perpendicular to each other, as shown in the figure. The force on Q caused by the charges induced on the
planes is
(a) 0
(b)
𝒬2
4𝜋∈0 𝑑 2
𝒬2
(c) 4𝜋∈
2
0𝑑
𝒬2
(d) 4𝜋∈
2
0𝑑
(
(
−1
4
−1
1
𝑖̂ − 𝑗̂)
4
1
𝑖̂ + 2√2 𝑗̂)
2√2
−1
(
1
2√2
𝑖̂ − 2√2 𝑗̂)
48. Two points charges +Q and –Q are placed away from a grounded conducting plane as shown in the
figure. The net force on the charge –Q is
(a)
𝒬2
4𝜋∈0 𝑑 2
𝒬2
(b) 4𝜋∈
0𝑑
2
away from the conducting plane
(
137
144
) towards the conducting plane
𝒬2
(c) 4𝜋∈
0𝑑
𝒬2
(d) 4𝜋∈
31
2
( ) away from the conducting plane
2
(
0𝑑
36
137
144
) away from the conducting plane
49. A uniformly Charged thin spherical shell of radius 𝑅 carries surface charge density of 𝜎 per unit area.
It is made of two hemispherical shells, held together by pressing them with force F (see figure, F is
proportional to)
1
(a) 𝜀 𝜎 2 𝑅2
0
(b)
(c)
1
𝜎2𝑅
𝜀0
1 𝜎2
𝜀0 𝑅
1 𝜎2
(d) 𝜀
2
0𝑅
50. A spherical metal shell A of radius RA and a solid metal sphere B of radius
RB RA are kept far
apart and each is given charge ‘+Q’. Now they are connected by a thin metal wire. Then
(a) 𝐸𝐴𝑖𝑛𝑠𝑖𝑑𝑒 = 0
(b) 𝑄𝐴 > 𝑄𝐵
𝜎
𝑅
(c) 𝜎𝐴 = 𝑅𝐵
𝐵
𝐴
𝑜𝑛 𝑠𝑢𝑟𝑓𝑎𝑐𝑒
(d) 𝐸𝐴
𝑜𝑛 𝑠𝑢𝑟𝑓𝑎𝑐𝑒
< 𝐸𝐵
51. In a coaxial cable, the radius of the inner conductor is 2 mm and that of the outer one is 5 mm. The
inner conductor is at a potential of 10V, while the outer conductor is grounded. The value of the potential
at a distance of 3.5 mm from the axis is _________________ [JAM 2017]
(Specify your answer in volts to two digits after the decimal point)
52. Out of the following statements, choose the correct option(s) about a perfect conductor.
(a) The conductor has an equipotential surface [JAM 2019]
(b) Net charge, if any, resides only on the surface of conductor
(c) Electric field cannot exist inside the conductor
(d) Just outside the conductor, the electric field is always perpendicular to its surface
53. A point charge q=0 is brought in front of a grounded conducting sphere. If the induced charge density
on the sphere is plotted such that the thickness of the black shading is proportional to the charge density,
the correct plot will most closely resemble [TIFR 2019]
DIELECTRICS & CAPACITORS :
1. The dielectric constant of a material at optical frequencies is mainly due to [B.H.U-2016]
(a) Ionic polarizability
(b) Electric polarizability
(c) Dipolar polarizability
(d) Ionic and dipolar polarizabilities
2. Where a monatomic gas atom is placed in a uniform electric field E, the resulting induced dipole
moment is proportional to [B.H.U-2010]
(a) 𝐸
(b) 𝐸 2
(c) 𝐸 3
(d) independent of 𝐸
3. The shape of a dielectic lamina is defined by the two curves 𝑦 = 0 and 𝑦 = 1 − 𝑥 2 . If the charge
density of the lamina = 15 𝑦 𝐶/𝑚2 , then the total charge on the lamina is ________C [JAM 2016]
4. A cylindrical rod of length L and radius r, made of an inhomogeneous dielectric, is placed with its axis
along the z direction with one end at the origin as shown below :
If the rod carries a polarization, 𝑃⃗⃗ = (5𝑧 2 + 7)𝑘̂, the volume bound charge inside the dielectric is :
(a) Zero
(b) 10𝜋𝑟 2 𝐿
(c) −5𝜋𝑟 2 𝐿
(d) −5𝜋𝑟 2 𝐿2
5. A dielectric sphere has polarization 𝑃⃗⃗ = 𝑘𝑟̂ , bound charge density inside the sphere is
(a) −3𝑘
−2𝑘
(b) 𝑟
−𝑘
(c)
𝑟
(d) – 𝑘𝑟
6. Two spheres 𝑆1 and 𝑆2 have polarization 0
𝑃0 𝑘̂ and 𝑃0 𝑟̂ respectively. Which of the following statements is / are correct? Where 𝑘̂ is unit vector in z
direction and rˆ is unit vector radially outward from centre.
(a) Surface bound charge density is non-uniform on 𝑆1 uniform on 𝑆2
(b) Electric field is zero at centre of 𝑆1 and non-zero at centre of 𝑆2
(c) Volume bound charge density is zero for 𝑆1 and non-zero for 𝑆2
(d) Electric field in outside region is non-zero for 𝑆1 and zero for 𝑆2
7. A dielectric cube lying in first octant with its one corner at origin has polarization 𝑃⃗⃗ = 𝑘𝑥 2 𝑖̂. If side of
the cube is ‘a’, total surface bound charge on cube is
(a) 𝑘𝑎4
(b) – 𝑘𝑎4
(c) zero
(d) 8𝑘𝑎4
8. A long, solid dielectric cylinder of radius a is permanentely polarized so that the polarization is
everywhere radially outward, with a magnitude proportional to the distance from the axis of the cylinder,
1
i.e., 𝑃⃗⃗ = 𝑃0 𝑟𝑟̂ The bound charge density in the cylinder is given by [TIFR 2016]
2
(a) – 𝑃0
(b) 𝑃0
–𝑃
(c) 20
(d)
𝑃0
2
9. A charge q is placed at the centre of an otherwise neutral dielectric sphere of radius ‘a’ and relative
permittivity
𝑞
𝜀𝑟 . We denote the expression 4𝜋𝜀 𝑟2 𝑏𝑦 𝐸(𝑟). Which of the following statements is false?
0
(a) The electric field inside the sphere, r < a, is given by
𝐸(𝑟).
𝜀1
[JEST 2013]
(b) The field outside the sphere, r > a, is given by E(r)
(c) The total charge inside a sphere of radius r > a is given by q.
(d) The total charge inside a sphere of radius r < a is given by q.
10. A spherical shell of inner and outer radii a and b, respectively, is made of a dielectric material with
𝑘
frozen polarization 𝑃⃗⃗(𝑟) = 𝑟 𝑟̂ , where k is a constant and r is the distance from the its centre. The electric
field in the region 𝑎 < 𝑟 < 𝑏 is , [JEST 2015]
𝑘
(a) 𝐸⃗⃗ = 𝜀 𝑟 𝑟̂
0
(b) 𝐸⃗⃗ = −
𝑘
𝜀0 𝑟
𝑟̂
(c) 𝐸⃗⃗ = 0
𝑘
(d) 𝐸⃗⃗ = 𝜀 𝑟2 𝑟̂
0
11. An infinite slab of insulating material with dielectric constant K and permittivity 𝜀 = 𝑘𝜀0is placed at a
uniform electric field of magnitude 𝐸0. The field is perpendicular to the surface of the material, as shown
in the figure below. The magnitude of the electric field inside the material is [B.H.U-2012]
(a)
𝐸0
𝐾
𝐸
(b) 𝐾𝜀0
0
(c) 𝐸0
(d) 𝑘𝜀0 𝐸0
12. A parallel plate capacitor is filled with two dielectrics as shown in the figure below. A is the area of a
plate and d is the separation between the plates. The capacitance of the capacitor is [B.H.U-2012]
𝜀0 𝐴 𝑘1 +𝐾−2
(
)
𝑑
2
𝜀0 𝐴
(𝑘1 − 𝑘)
=
𝑑
𝜀0 𝐴 𝑘1 −𝑘2
(a) 𝐶 =
(b) 𝐶
(c) 𝐶 =
(d) 𝐶 =
𝑑
𝜀0 𝐴
𝑑
(
2
)
(𝑘1 + 𝑘2 )
13. The displacement vector 𝐷 and electric field strength 𝐸 are related by [B.H.U-2013]
𝐸
(a) 𝐷 = 𝜀
(b) 𝐷 = 𝜀𝐸
(c) 𝐷 = 𝜀𝐸 2
(d) 𝐷 = 𝜀𝐸1/2
Stement for Linked Answer Q.14 and Q.15
14. A sphere of radius R carries a polarization 𝑃⃗⃗ = 𝑘𝑟̂ , where k is a constant and 𝑟̂ is measured from the
centre of the sphere
(a) −𝑘 |𝑟̂ | 𝑎𝑛𝑑 3𝑘
(b) −𝑘|𝑟̂ | 𝑎𝑛𝑑 − 3𝑘
(c) −𝑘 |𝑟̂ | 𝑎𝑛𝑑 − 4𝜋𝑘𝑘
(d) −𝑘|𝑟̂ | 𝑎𝑛𝑑 4𝜋𝑘𝑘
15. The electric surface field 𝐸⃗⃗ at point 𝑟̂ outside the sphere is given by
(a) 𝐸⃗⃗ = 0
𝑘𝑅(𝑅 2 −𝑟2 )
(b) 𝐸⃗⃗ =
𝑟̂
3
𝜀0 𝑟
2 −𝑟2 )
𝑘𝑅(𝑅
(c) 𝐸⃗⃗ =
𝜀0 𝑟 3
𝑘𝑅(𝑅 2 −𝑟2 )
(c) 𝐸⃗⃗
(d) 𝐸⃗⃗
𝜀0 𝑟 5
𝑘𝑅(𝑟−𝑅)
4𝜋𝜀0 𝑟4
𝑟̂
𝑟̂
𝑟̂
16. A dielectric sphere of radius 𝑅 carries a polarization 𝑃⃗⃗ = 𝑘𝑟⃗ throughout its volume. The volume
charge density 𝜌𝑏 of the bound charge is
(a) −3𝑘
(b) 3𝑘
(c) – 𝑘
(d) 𝑘
Statement for Linked Answer Q. 17 and Q. 18 :
A dielectric sphere of radius R carries polarization 𝑃⃗⃗ = 𝑘𝑟 2 𝑟̂ wherer 𝑟̂ is the distance from the centre and
𝑘 is a constant. In the spherical polar coordinate system, 𝑟̂ . 𝜃̂and 𝜑̂ are the unit vectors.
17. The bound volume charge density inside the sphere at a distance 𝑟 from the centre is :
(a) −4𝑘𝑅
(b) −4𝑘𝑟
(c) −4𝑘𝑟 2
(d) −4𝑘𝑟 3
18. The electric field inside the sphere at a distance d from the centre is;
(a)
(b)
(c)
(d)
–𝑘𝑑 2
𝑟̂
𝜀0
–𝑘𝑅 2
𝜀0
–𝑘𝑑 2
𝜀0
–𝑘𝑅
𝜀0
𝑟̂
𝜃̂
𝜃̂
19. A sphere of radius 𝑅 carries a polarization 𝑃⃗⃗ = 𝑘𝑟̂ electric field inside the sphere is
−𝑘𝑟̂
(a)
(b)
(c)
𝜀0
−2𝑘𝑟̂
3𝜀0
–𝑘𝑟⃗
3𝜀0
(d) zero
20. In previous questions electric field outside the sphere is
−𝑘𝑟⃗
(a) 𝜀
0
(b)
(c)
−2𝑘𝑟⃗
3𝜀0
–𝑘𝑟⃗
3𝜀0
(d) zero
𝛽𝑟̂
21. A dielectric shell o inner and outer radii 𝑅1 and 𝑅2 has polarisation 𝑃⃗⃗ = 𝑟2 , where 𝑟 is radial distance
of any point from centre of shell Electric field at any point in the region 𝑅1 < 𝑟 < 𝑅2 is
−𝛽𝑟̂
(a) 𝜀 𝑟2
0
(b)
(c)
–𝑅22 𝛽𝑟̂
𝑅12 𝑟2
–𝑅22 𝛽𝑟̂
𝑅22 𝑟2
(d) zero
22. A dielectric sphere has uniform polarization 𝑃⃗⃗ electric field inside the sphere is
(a)
−𝑃⃗⃗
𝜀0
−𝑃⃗⃗
(b) 2𝜀
0
−𝑃⃗⃗
(c) 3𝜀
0
(d) zero
23. A very long cylinder of radius 𝑅 has uniform polarization 𝑃⃗⃗ perpendicular to its axis. Electric field
inside the cylinder is
(a)
−𝑃⃗⃗
𝜀0
−𝑃⃗⃗
(b) 2𝜀
0
−𝑃⃗⃗
(c) 3𝜀
0
(d) zero
24. A long straight wire, carrying uniform charge of density 𝜆 is surround by long coaxial dielectric
cylindrical cable. If relative permittivity of cable is 𝜀𝑟 , Polarization induced inside the cylinder due to the
line charge is
𝜆𝑠̂
(a) 2𝜋𝜀 𝑆
𝑟
(b)
−𝜆𝑠̂
2𝜋𝜀𝑟𝑆
𝜆𝑠̂
1
(c) 2𝜋𝑆 (1 − 𝜀 ) 𝑆̂
𝑟
𝜆𝑠̂
1
(d) 2𝜋𝑆 (𝜀 − 1) 𝑆̂
𝑟
25. A point charge q is lying at the centre of a dielectric sphere of radius 𝑅 and relative permittivity 𝜀𝑟 .
Electric flux through a concentric sphere 𝑟 > 𝑅 is
𝑞
(a) 𝜀
0
(b) 𝜀0 𝜀𝑟
(c)
(d)
1
𝜀𝑟
𝑞(1− )
𝑞
𝜀0
𝜀0
1
(1 + )
𝜀𝑟
26. In Previous question flux through sphere radius 𝑟 > 𝑅 is
𝑞
(a) 𝜀
0
(b) 𝜀0 𝜀𝑟
(c)
(d)
1
𝜀𝑟
𝑞(1− )
𝑞
𝜀0
𝜀0
1
(1 + )
𝜀𝑟
27. A metal sphere of radius 𝑅 carries a charge 𝑄. It is surround out to radius 2R by linear dielectric
material of relative permittivity 4. Electric potential at the centre of sphere is
5𝑄
(a) 4𝜋∈ 𝑅
0
5𝑄
(b) 32𝜋∈
5𝑄
0𝑅
(c) 16𝜋∈
0𝑅
(d)
3𝑄
16𝜋∈0 𝑅
28. A dielectric sphere of radius ‘R’ has polarization 𝑃⃗⃗ = 𝛽𝑟̂ . The electric potential at centre of sphere is
𝛽𝑅
(a) ∈
0
𝛽𝑅
(b) – ∈
0
𝛽
(c) 3∈
𝛽
0
(d) ∈
0
29. A dielectric sphere of radius R and dielectric constant 𝜀𝑟 has uniform charge of density 𝜌 embded in it.
Electric potential at the centre of sphere is
𝜌𝑅 2
(a) 3∈
0∈𝑟
𝜌𝑅 2
1
0
𝜌𝑅 2
1
(b) 3∈ (1 + ∈ )
(c)
𝑟
(1 +
6∈ 0
𝜌𝑅 2
)
∈𝑟
1
(d) 3∈ (1 + 2∈ )
0
𝑟
30. A dielectric is filled in the space between two concentric conducting shells as shown in figure. If a
charge Q is given to the inner sphere. Electric field region 𝑅1 < 𝑟 < 𝑅2 is
𝒬
(a) 4𝜋∈
0𝑟
(b) 2𝜋∈
(c) 2𝜋∈
2
𝒬
2
0 (∈𝑟 +1)𝑟
∈1 𝒬
0 (∈ 𝑟 +1)𝑟
(d)
2
𝒬
2𝜋∈0 ∈𝑟 𝑟2
31. 31. Consider a spherical dielectric material of radius ‘a’, centered at origin. If the polarization vector
𝑃⃗⃗ = 𝑃0 𝑒̂𝑥 , where 𝑃0 is a constant of appropriate dimensions, then [JAM 2016]
(𝑒̂𝑥 , 𝑒̂𝑦 and 𝑒̂𝑧 are unit vectors in Cartesian - coordinate system)
(a) the bound volume charge density is zero
(b) the bound surface charge density is zero at (0, 0, a)
(c) the electric field is zero inside the dielectric
(d) the sign of the surface charge density changes over the surface.
32. Assume that z = 0 plane is th interface between two linear and homogeneous dielectrics (see figure).
The relative permitivities are 𝜀𝑟 = 5 for 𝑧 > 0 and 𝜀𝑟 = 4 for 𝑧 < 0. The electric field in the region 𝑧 >
0 is 𝐸⃗⃗1 = (3𝑖̂ − 5𝑗̂, 4𝑘̂)𝑘𝑉/𝑚. If there are no free charges on the interface, the electric field in the region z
< 0 is given by [2010]
3
5
(a) 𝐸⃗⃗2 = (4 𝑖̂ − 4 𝑗̂ + 𝑘̂) 𝑘𝑉/𝑚
(b) 𝐸⃗⃗2 = (3𝑖̂ − 5𝑗̂ + 𝑘̂ )𝑘𝑉/𝑚
(c) 𝐸⃗⃗2 = (3𝑖̂ − 5𝑗̂ − 5𝑘̂)𝑘𝑉/𝑚
(d) 𝐸⃗⃗2 = (3𝑖̂ − 5𝑗̂ − 5𝑘̂)𝑘𝑉/𝑚
33. Consider a spherical cavity in an infinite, homogeneous and isotropic dielectric of permittivity 𝜀. When
placed in an external electric field E, the electric inside the cavity is : [JNU 2010]
(a) 3𝜀𝐸(2𝜀 + 𝜀0 )
(b) (2𝜀0 + 𝜀)𝐸/3𝜀0
(c) – 𝜀𝐸
(d) 𝐸/𝜀
34. Suppose 𝑦𝑧-plane forms the boundary between two linear dielectric medial I and II with dielectric
constant 𝜀𝐼 = 3 and 𝜀𝐼𝐼 = 4, respectively. If the electric field in region. 𝐼 at the interface is given by 𝐸⃗⃗1 =
(4𝑥̂ + 3𝑦̂ + 5𝑧̂ ), then the electric field 𝐸⃗⃗𝐼𝐼 is : [JEST 2016]
(a) 4𝑥̂ + 3𝑦̂ + 5𝑧̂
(b) 4𝑥̂ + 0.75𝑦̂ + 1.25𝑧̂
(c) −3𝑥̂ + 3𝑦̂ + 5𝑧̂
(d) 3𝑥̂ + 3𝑦̂ + 5𝑧̂
35. The capacitance of two concentric spherical metal shells with radii 𝑎 and 𝑏 is [H.C.U.2014]
1
1
1
(a) 4𝜋∈ (𝑎 − 𝑏 )
1
0
𝑎𝑏
(b) 4𝜋∈ (𝑏−𝑎)
0
(c) 4𝜋 ∈0
𝑎𝑏
𝑏−𝑎
1
1
(d) 4𝜋 ∈0 (𝑎 + 𝑏 )
36. Three distinct materials with dielectric constants given by 𝐾1 , 𝐾2 and 𝐾3, respectively are arranged
between two parallel plates of a capacitor of area A, as shown in the figure. The effective capacitance of
the system is two parallel plates of a capacitors of area A, as shown in the figure. The effective capacitance
of the systems is [H.C.U.-2013]
(a) 𝐶 =
(b) 𝐶 =
(c) 𝐶 +
(d) 𝐶 =
2∈0 𝐴 𝐾1 𝐾2 𝐾3
𝑑 𝐾1 +𝐾2 +𝐾3
2∈0 𝐴 𝐾1 +𝐾2 +𝐾3
(
𝑑
∈0 𝐴 𝐾1
(
2𝑑
2
∈0 𝐴 𝐾3
(
2𝑑
2
+
3
𝐾2 𝐾3
)
𝐾2 +𝐾3
𝐾2 𝐾1
+
)
𝐾1 +𝐾2
)
37. The half space regions 𝑥 > 0 and 𝑥 > 0 are filled with dielectric media of dielectric constants 𝜀1 and
𝜀2 respectively. There is a uniform electric field in each part. In the right half, the electric field makes an
angle 1 to the interface. The corresponding angle 𝜃2 in the left half satisfies
(a) 𝜀1 sin 𝜃2 = 𝜀2 sin 𝜃1
(b) 𝜀1 tan 𝜃2 = 𝜀2 tan 𝜃1
(c) 𝜀1 tan 𝜃1 = 𝜀2 tan 𝜃2
(d) 𝜀1 sin 𝜃1 = 𝜀2 sin 𝜃2
38. In the figure shown if surface charge density at the interface of two dielectric is 𝜎0 then value of 𝜎 is
𝜎
(a) 0
6
(b) −6𝜎0
𝜎
(c) 30
(d)
𝜎0
2
39. Two infinitely extended homogeneous isotropic dielectric media ( medium -1 and medium – 2 with
𝜀
𝜀
dielectric 𝜀1 = 2 and 𝜀2 = 5 , respectively) meet at the z = 0 plane as shown in the figure. A uniform
0
0
electric field exists everywhere. For 𝑧 ≥ 0, the electric field is given by 𝐸⃗⃗1 = 2𝑖̂ − 3𝑗̂ + 5𝑘̂.
The interface separating the two media is charge free. The electric displacement vector in the medium-2
is given by
⃗⃗⃗⃗⃗2 = 𝜀0 [10𝑖̂ + 15𝑗̂ + 10𝑘̂]
(a) 𝐷
⃗⃗⃗⃗⃗2 = 𝜀0 [10𝑖̂ − 15𝑗̂ + 10𝑘̂]
(b) 𝐷
⃗⃗⃗⃗⃗
(c) 𝐷2 = 𝜀0 [4𝑖̂ − 6𝑗̂ + 10𝑘̂]
⃗⃗⃗⃗⃗2 = 𝜀0 [4𝑖̂ + 6𝑗̂ + 10𝑘̂ ]
(d) 𝐷
40. The space between two plates of a capacitor carrying charges +Q and –Q is filled with two different
dielectric materials, as shown in the figure. Across the interface of the two dielectric materials, which one
of the following statements is correct ?
⃗⃗ are continuous
(a) 𝐸⃗⃗ and 𝐷
(b) is continuous and is discontinuous
⃗⃗ continuous and is discontinuous
(c) 𝐸⃗⃗ is 𝐷
⃗⃗
⃗⃗ are discontinuous
(d) 𝐸 and 𝐷
41. A parallel plate capacitor is filled with two dielectrics of dielectric constant 𝑘1 and 𝑘2 as shown in the
figure below. A is the area of each plate and d is the separation between the plates.
The capacitance of the capacitor is given by [H.C.U.-2011]
𝜀 𝐴 𝑘 +𝐾
(a) 𝐶 = 0𝑑 ( 1 2 2)
(b) 𝐶 =
(c) 𝐶 =
(d) 𝐶 =
𝜀0 𝐴
𝑑
𝜀0 𝐴
𝑑
𝜀0 𝐴
𝑑
(𝑘1 − 𝑘2 )
(
𝑘1 −𝑘2
2
)
(𝑘1 + 𝑘2 )
42. In a parallel plate capacitor the distance between the plates is 10 cm. Two dielectric slabs of thickness
5cm each and dielectric constants 𝐾1 = 2 and 𝐾2 = 4 respectively, are inserted between the plates. A
potential of 100V is applied across the capacitor as shown in the figure. The value of the net bound
surface charge density at the interface of the two dielectrics is [JAM 2014]
(a) −
2000
𝜀0
3
1000
− 3 𝜀0
(b)
(c) −250𝜀0
2000
(d) 3 𝜀0
43. The space between the plates of a parallel plate capacitor is filled with two dielectric slabs of dielectric
constants 𝑘1 and 𝑘2 as shown in the figure. If the capacitor is charged to a potential V, then at the interface
between the two dielectrics.
⃗⃗ is continuous
(a) 𝐸⃗⃗ is discontinuous and 𝐷
⃗⃗ is discontinuous 𝑘1 𝑘2 𝑉
(b) 𝐸⃗⃗ is discontinuous and 𝐷
⃗⃗
⃗
⃗
(c) 𝐸 is continuous and 𝐷 is continuous
⃗⃗ is discontinuous
(d) 𝐸⃗⃗ is continuous and 𝐷
44. A parallel plate capacitor has a dielectric slab of dielectric constant 𝐾 between its plates that covers
1/3 of the area of its plates, as shown in the figure. The total capacitance of the capacitor is 𝐶 while that of
the portion with dielectric in between in 𝐶1 . When the capacitor is charged, the plate area covered by the
dielectric gets charge 𝒬1 and the rest of the area gets charge 𝒬2 . The electric field in the dielectric is 𝐸1
and that in the other portion is 𝐸2 . Choose the correct option(s), ignoring edge effects.
𝐸
(a) 𝐸1 = 1
2
𝐸
1
2
𝒬1
3
(b) 𝐸1 = 𝐾
(c) 𝒬 = 𝐾
2
(d)
𝐶
𝐶1
=
2+𝐾
𝐾
45. Suppose the 𝑦𝑧- plane forms a chargeless boundary between two media of permittivities 𝑒𝑙𝑒𝑓𝑡 and
𝑒𝑟𝑖𝑔ℎ𝑡 where 𝑒𝑙𝑒𝑓𝑡 : 𝑒𝑟𝑖𝑔ℎ𝑡 = 1.2 If the uniform electric field on the left is 𝐸⃗⃗𝑙𝑒𝑓𝑡 = 𝑐(𝑖̂ + 𝑗̂ + 𝑘̂) (where c is a
constant), then the electric field on the right 𝐸⃗⃗𝑟𝑖𝑔ℎ𝑡 is
(a) 𝑐(2𝑖̂ + 𝑗̂ + 𝑘̂)
̂ + 2𝑘̂)
(b) 𝑐 = (𝑖̂ + 2𝑗
1
(c) 𝑐 = ( 𝑖̂ + 𝑗̂ + 𝑘̂ )
2
1
1
(d) 𝑐 = (𝑖̂ + 2 𝑗̂ + 2 𝑘̂)
46. Which of the following boundary condition is correct at the interface of two dielectrics.
(a) 𝐷1∥ + 𝐷2∥ = 𝑃1∥ + 𝑃2∥
(b) 𝐷1∥ − 𝐷2∥ = 𝑃1∥ − 𝑃2∥
(c) 𝐷1⊥ + 𝐷2⊥ = 𝑃1⊥ + 𝑃2⊥
(d) 𝐷1⊥ − 𝐷2⊥ = 𝑃1⊥ − 𝑃2⊥
⃗⃗1 = 2𝑖̂ +
47. If Electric displacement vectors on the two sides of a dielectric interface (𝑧 = 0 𝑝𝑙𝑎𝑛𝑒) are 𝐷
̂
̂
⃗
⃗
3𝑗̂ + 4𝑘 and 𝐷2 = 2𝑖̂ + 3𝑗̂ + 6𝑘 then free charge density at the interface must be
(a) 2𝐶/𝑚2
2𝐶
(b) − 𝑚2
(c) +2𝐶/𝑚2 𝑜𝑟 − 2𝐶/𝑚2
(d) 1𝐶/𝑚2
48. The half space regions 𝑥 > 0 and 𝑥 < 0 are filled with dielectric media of dielectric constant 𝜀1 = 2
and 𝜀2 = 4 respectively. There is a uniform electric filed in each part. In the right half. the electric
fieldmakes an angle 𝜃1 and to the interface. The corresponding angle in the half 𝜃2 then
tan 𝜃1
is ____________
tan 𝜃2
49. Two parallel plates of metal sandwich a dielectric pad of thickness d, forming an ideal capacitor of
capacitance 𝐶. The dielectric pad is elastic, having a spring constant 𝑘. If an ideal battery of voltage 𝑉
across its terminals is connected to the two plates of this cpacitor, the fractional change 𝛿𝑑/𝑑 << 1 in
the gap between the plates is [TIFR 2010]
(a) zero
1
(b) + 2
(c) −
𝐶𝑉 2
𝑘𝑑 2
1
𝐶𝑉 2
2
𝑘𝑑 2 +𝐶𝑉 2
1
𝐶𝑉 2
(d) − 𝑘𝑑22−𝐶𝑉 2
50. A capacitor with capacitance C is charged to a voltage V. If it is fully discharged by shorting through a
resistor R, the total heat generated in the resistor is
1
(a) 2 𝐶𝑉 2
1
(b) 𝐶𝑉 2
4
(c) 𝐶𝑉 2
(d) dependent on 𝑅
51. A capacitor is made of two conducting each of radius R and covered by a thin insulating sheet. They
carry charge Q and –Q respectively and their centres are separated by a distance d. If the capacitance of
this capacitor is given by C = Q/V, where V is the potential difference between the two spheres then
(a) C is maximum when d is infinity
(b) C is maximum when d approaches 2 R
(c) C does not depend on d
(d) C is maximum for d = 4R [IISc. 2011]
52. A dielectric slab of thickness d is inserted in a parallel plate capacitor whose negative plate is at 𝑥 =
0 and positive plate is at 𝑥 = 3𝑑. The slab is equidistant from the plates. The capacitor is given some
charge. As 𝑥 goes from 0 to 3𝑑.
(a) the magnitude of the electric field remains the same
(b) the direction of the electric field remains the same
(c) the electric potential increases continuously
(d) the electric potential increases at first, then decreases and again increases
53. A parallel plate capacitor of area A, plate separation d and capacitance 𝐶 is filled with three different
dielectric materials having dielectric constants 𝑘1 , 𝑘2 and 𝑘3 as shown. If a single dielectric material is to
be used to have the same capacitance C in this capacitor then its dielectric constant k is given by
1
1
1
1
(a) 𝑘 = 𝑘 + 𝑘 + 2𝑘
1
(b) =
1
2
1
𝑘
1
𝑘1 +𝑘2
𝑘1 𝑘2
𝑘
+𝑘2
𝑘1 𝑘3
(c) =
(c) 𝑘 =
+
+
𝑘1 +𝑘2
1
3
2𝑘3
1
2𝑘3
𝑘2 𝑘3
+
𝑘2 +𝑘3
54. Consider the situation shown in the figure. The capacitor A has charge q on it whereas B is uncharged.
The charge appearing on the capacitor B a long time after the switch is closed is
(a) zero
𝑞
(b)
2
(b) 𝑞
(d) 2𝑞
55. The capacitance of two condensers in parallel is four times their capacitance when they are connected
in series. The ratio of their individual capacitances will be
(a) 1:2
(b) 1:1
(c) 2:1
(d) 4:1
56. Two identical capacitors, have the same capacitance 𝐶. One of them is charged to potential 𝑉1 and the
other to 𝑉2 . The negative ends of the capacitors are connected together. When the positive ends are
connected, the decrease in energy of the combined system is
1
(a) 𝐶(𝑉12 − 𝑉22 )
4
1
(b) 𝐶(𝑉12 + 𝑉22 )
4
1
(c) 4 𝐶(𝑉1 − 𝑉2 )2
1
(d) 4 𝐶(𝑉1 + 𝑉2 )2
57. An infinite ladder of capacitors, each 1𝜇𝐹, is made as shown in figure. The capacitance between A and
B (in 𝜇𝐹) is
(a) 1
(b) 1.3
(c) 1.6
(d) 0
58. The potential at point P is
(a)
(b)
(c)
(d)
59. A parallel plate capacitor C with plates of unit area and separation d is filled with a liquid of dielectric
constant. The level of liquid is 𝑑/3 initially. Suppose the liquid level decreases at a constant speed 𝑣, the
time constant as a function of time 𝑡 is
(a)
(b)
(c)
(d)
60. A 3-microfarad capacitor is connected in series with a 6-microfarad capacitor. When a 300 volt
potential difference is applied across this combination, the total energy stored in the two capacitors is
(a) 0.09 J
(b) 0.18 J
(c) 0.27 J
(d) 0.41 J [B.H.U-2012]
61. Which of the following changes to an ideal parallel plate capacitor, connected to an ideal battery, will
result in an increase in the charge of the capacitor?
(a) Decreasing the potential difference across the plates?
(b) Decreasing the area of the plates
(c) Decreasing the separation of the plates
(d) None of the above
62. Two parallel plate capacitors, separated by distances x and 1.1𝑥 respectively, have a dielectric
material of dielectric constant 3.0 inserted between the plates, and are connected to a battery of voltage
𝑉. The difference in charge on the second capacitor compared to the first is
(a)
(b)
(c)
(d)
63. Find the capacitance of a system of three parallel plates each of area A separated by distances 𝑑1 and
𝑑2 . The space between them is filled with dielectrics of relative permittivities and The dielectric constant
of free space is
(a)
(b)
(c)
(d)
64. A parallel plate capacitor is connected across a source of constant potential difference. When a
dielectric plate is introduced between the two plates, then
(a) some charge from the capacitor will flow back into the source
(b) some extra charge from the source will flow into the capacitor
(c) the electric field intensity between the two plates increases
(d) energy of the capacitor does not change
65. A thin dielectric slab is slowly introduced between the plates of a charged parallel plate capacitor, as
shown in the figure. [NET 2008]
Which of the following is true?
(a) The slab is pushed out of the region between the capacitor plates.
(b) The slab is sucked into the region between the capacitor plates.
(c) The slab moves towards the positively charged plate.
(d) The slab moves towards the negatively charged plate.
66. A charged capacitor is made of two concentric cylinder of radii 𝑟1 and 𝑟2 . The region between the
Cylindrical plates is filled with a medium of dielectric constant, 𝑘. The displacement vector at a distance
(a) Is independent of 𝑟
(b) Varies as In 𝑟.
(c) Is proportional to 𝑟
(d) Varies as 1/𝑟.
67. A 2𝜇𝐹
Capacitor is charged as shown in figure . The percentage of its stored energy dissipated after the switch S
is turned to position 2 is
(a) 0%
(b) 20%
(c) 75%
(d) 80%
68. In the given circuit, a charge of is given to the upper plate of the 𝜇𝐹 Capacitor . Then in the steady
state, the charge on the upper plate of the 3𝜇𝐹 capacitors is
69. Charges are uniformly distributed in three solid sphere 1, 2 and 3 of radii respectively, as shown in
figure. If magnitude of the electric fields at point 𝑃 at a distance 𝑅 from the centre of sphere 1, 2 and 3 are
respectively, then
(a)
(b)
(c)
(d)
70. A dielectric sphere of radius 𝑅 has constant polarization so that inside the sphere is
Then, which of the following is (are) correct?
(a) The bound surface charge density is
(b) The electric field at a distance r on the z-axis varies as for
(c) The electric potential at a distance 2R on the z-axis is
(d) The electric field outside is equivalent to that of a dipole at the origin
71. For an ideal dielectric, polarization is given by [BHU 2018]
(a)
(b)
(c)
(d)
72. During the charging of a capacitor 𝐶 in a series 𝑅𝐶 circuit, the typical variations in the magnitude of
the charge 𝑞(𝑡) deposited on one of the capacitor plates, and the current 𝑖(𝑡) in the circuit, respectively
are best represented by
(a) Fig. I and Fig. II
(b) Fig. I and. IV
(c) Fig. III and Fig. IV
(d) Fig. III and Fig. IV
73. An infinitely long very thin straight wire carries uniform line charge density The magnitude of electric
displacement vector at a point located 20 mm away from the axis of the wire is _________𝐶/𝑚2. [JAM 2019]
74. Consider three identical, ideal capacitors. The first capacitor is charged to a voltage V and then
disconnected from the battery. The other two capacitors, initially uncharged and connected in series, are
then connected across the first. What is the final voltage across the first capacitor? [DU 2019]
(a) V/3
(b) V
(c) 2V/5
(d) 2V/3
ELECTROSTATIC ENERGY :
1. Farad is equivalent to
(a) 𝐶/𝐽
(b) 𝐶 2 /𝐽
(c) 𝐶/𝑁
(d) 𝐶 2 /𝑁
2. Work done in carrying a point charge of 1𝜇𝐶 from point A to B is 5mJ. Potential
(a) 10nV
(b) 5mV
(c) 5kV
(d) 10 kV
3. Three point charges q, q and Q are placed at equal separation as shown in figure. If potential energy of
system is zero, value of q/Q is
(a) -1/2
(b) -2
(c)
(d)
4. A Proton of mass 1.67 × 10−27 𝑘𝑔 and charge 1.6 × 10−19 𝐶 is accelerated from rest through an electrical
potential of 400kV. Its final velocity is
(a) 4.2 × 106 𝑚/ sec
(b) 8.8 × 106 𝑚/𝑠𝑒𝑐
(c) 6.6 × 106 𝑚/𝑠𝑒𝑐
(d) 12.1 × 106 𝑚/𝑠𝑒𝑐
5. Three equal charges 𝒬 are successively brought from infinity and each is placed at one of the three
vertices of an equilateral triangle. Assuming the rest of the Universe as a whole to be neutral, the energy
𝐸0 of the electrostatic field will increase, successively, to where
(a) 1 : 2 : 3
(b) 1 : 1 : 1
(c) 0 : 1 : 1
(d) 0 : 1 : 2
6. Two point charges each of charge +q are fixed at (+𝑎, 0) and (– 𝑎, 0). Another positive point charge q
placed at the origin is free to move along x-axis. The charge q at the origin in equilibrium will have
(a) maximum force and minimum potential energy
(b) minimum force and maximum potential energy
(c) maximum force and maximum potential energy
(d) minimum force and minimum potential energy
7. Charges and are placed at one one of the three vertices of an shown in figure. The work done in
moving a charge 1 × 10−6 𝐶 from corner 𝑅 to 𝑆 is given by
(a) 0.053 J
(b) 0.53 J
(c) 1.06 J
(d) 0.106 J
8. A sphere of radius 𝑅 and uniformly distributed charge 𝒬 has potential energy 𝑢. If charge density is
doubled without changing the value of charge then potential energy will become
(a)
(b)
(c)
(d)
9. Two identical particles, each having a charge of 2.0 × 10−4 𝐶 and mass of 10g, are kept at a separation
of 10cm and then released. Speeds of the particles when the separation becomes large is _______ 𝑚/𝑠
10. A long cylindrical wire carries a positive charge of linear density 2.0 × 10−8 𝐶/𝑚. An electron
revolves around it in a circular path under the influence of the attractive electrostatic force. The kinetic
energy of the electron is _____________ 10−17 𝐽.
11. It requires 1 𝑚𝐽 of work to move two identical positive charge +q from infinity so that they are
separated by distance ‘a’. How much work is required to move four identical positive charges +𝑞 from
infinity so that they are arranged at the vertices of a tetrahedron with edge length 𝑎?
(a) 3 𝑚𝐽
(b) 4 𝑚𝐽
(c) 6 𝑚𝐽
(d) 16 𝑚𝐽
12. Interaction potential energy of two dipoles lying in same plane is minimum when dipole moment are
(a) parallel
(b) antiparallel
(c) perpendicular
(d) at an angle
13. A spherically symmetric gravitational system of particles has a mass density
where is a constant. A test mass can undergo circular motion under the influence of the gravitational field
of particles. Its speed v as a function of distance from the centre of the system is represented by
14. The interaction energy between a dipole having dipole moment in an electric field is [H.C.U.-2016]
(a)
(b)
(c)
(d)
15. Two capacitors of capacitance 6 𝜇𝐹 𝑎𝑛𝑑 4 𝜇𝐹 are connected in parallel and this combination is
connected in series with a capacitor of capacitance 5 𝜇𝐹 across a voltage source of 1000 𝑉. The total
charge stored in the system is [H.C.U.-2014]
(a)
(b)
(c)
(d)
16. Three point charges 𝑞, 2𝑞 and 8𝑞 are to be placed on a straight line 9𝑐𝑚 long. In order that the
potential energy of the system be minimum. Their positions should be
(a) 2𝑞 and 𝑞 at the ends (9cm apart) and 8𝑞 at a distance of 3cm from the 2𝑞 charge
(b) 2𝑞 and 8𝑞 at the ends and 𝑞 at a distance of 3𝑐𝑚 from the 8𝑞 charge
(c) 2𝑞 and 8𝑞 at the ends and 𝑞 at a distance of 6𝑐𝑚 from the 8𝑞 charge
(d) 𝑞 and 8𝑞 at the ends and 2𝑞 at a distance of 6𝑐𝑚 from the q charge
17. Consider the situation depicted in the adjacent figure. The work done in taking a point charge from 𝑃
to 𝐴 is 𝑊𝐴, from 𝑃 to 𝐵 is WB and from P to 𝐶 is 𝑊𝐶. 𝐴, 𝐵, 𝐶 lie on a circle centered at ‘𝑞’. Therefore,
(a)
(b)
(c)
(d)
18. The amount of energy required to charge a metallic sphere of radius 2.0 𝑚 to a potential of 3000 𝑉 in
air is [JNU 2009]
(a)
(b)
(c)
(d)
19. Electric charges 𝑄 and – 𝑄 are distributed uniformly on the surfaces of two concentric shells of radii a
and b, respectively (𝑎 < 𝑏). The energy required to assemble this charge distribution is [JNU 2015]
(a)
(b)
(c)
(d)
20. In the electric field due to a point charge q, a test charge is carried from 𝐴 to the points 𝐵, 𝐶, 𝐷 and E
lying on the same circle around 𝑞. The work done is
(a) the least along 𝐴𝐵
(b) the least along 𝐴𝐷
(c) zero along any one of the paths 𝐴𝐵, 𝐴𝐷, 𝐴𝐶 and 𝐴𝐸
(d) the least along 𝐴𝐸
21. A charged particle 𝑞 is shot towards another charged particle 𝑄 which is fixed, with a speed 𝑣 from a
very large distance. It approaches 𝑄 upto a closest distance 𝑟 and then returns. If 𝑞 were given a speed 2𝑣,
the closest distance of approach would be
(a) 𝑟
(b) 2𝑟
(c) 𝑟/2
(d) 𝑟/4
22. A conducting sphere of radius ‘𝑟’ has charge 𝑄 on its surface. If the charge on there sphere is doubled
and its radius is halved, the energy associated with the electric field will [JEST 2014]
(a) Increase four times
(b) Increase eight times
(c) Remain the same
(d) Decrease four times
23. Two identical thin rings, each of radius R meters, are coaxially placed a distance R meters apart. If Q1
C and Q2 C are the charges uniformly spread on the two rings, the work done in moving a charge q from
the centre of one ring to that of the other is
(a)
24. Two identical point charges of mass m and charge q are separated by a distance d and are moving at a
relative speed u. What is their relative speed when they are at a large distance from each other. [IISc.
2011]
(a)
(b)
(c)
(d)
25. The dimension of is
(a)
(b)
(c)
(d)
26. Three charges Q, +q and +q are placed at the vertices of a right angle triangle (isosceles triangle) as
shown. The net electrostatic energy of the configuration is zero if Q is equal to
(a)
(b)
(c)
(d)
27. Which of the following statements is / are correct
(a) The work done by the electric field of a nucleus in moving an electron around it in a complete orbit is
zero irrespective of whether the orbit is circular or elliptical
(b) The equipotential surfaces corresponding to the electric field of an isolated point charge concentric
spheres with the point change as the common centre
(c) If coulomb’s law involved 1/r3 dependence instead of 1/r2, Gauss law would still hold good.
(d) A single conductor cannot have any capacitance
28. Two equal point charges are fixed at and on the x-axis. Another point charge Q is placed at the origin.
The change in the electrical potential energy of Q, when it is displaced by a small distance x along the xaxis, is approximately proportional to
(a) x
(b) x2
(c) x3
(d) 1/x
29. Positive and negative point charges of equal magnitude are kept at , respectively.
The work done by the electric field when another positive point charge is moved from is :
(a) positive
(b) negative
(c) zero
(d) depends on the path connecting the initial and final positions
30. Two fixed insulating rings A and B carry charges with uniform linear charge density and ,
respectively, as shown in the adjacent figure. The planes of the rings are parallel to each other and their
axes are coinciding. A particle of charge “q” and mass “m” is released with zero velocity from centre P of
the positively charged ring. The kinetic energy of the particle when it reaches centre Q of the negatively
charged ring will be
(a)
(b)
(c)
(d) none of above
31. Three charges are placed at the three corners of an equilateral triangle of side a. The work done in
gathering the system of charges is given by [H.C.U-2013]
(a)
(b)
(c)
(d)
32. Consider a system of three charges placed at points A, B and C, respectively, as shown in the figure.
Take O to be centre of the circle of radius R and angle CAB = 60°.
(a) The electric field at point O is directed along the negative x-axis
(b) The potential energy of the system is zero
(c) The magnitude of the force between the charges at C and B is
(d) The potential at point O is
33. The energy stored in a system of four identical charges of 4 × 10−9 𝐶 at the corners of a square of side
1 m is [H.C.U.-2010]
(a) 97.5 J
(b)
(c) 780 J
(d)
34. Which of the following statement(s) is/are correct?
(a) if the electric field due to a point charge varies as 𝑟 −25 instead of 𝑟 −2 , then the Gauss law will still be
valid
(b) the Gauss law can be used to calculate the field distribution around an electric dipole
(c) if the electric field between two point charges is zero somewhere, then the sign of the two charges is
the same.
(d) the work done by the external force in moving a unit positive charge from point A at potential VA to
ponit 𝐵 𝑉𝐵 is (𝑉𝐵 – 𝑉𝐴 )
35. A particle gains a momentum 𝑝0 when a accelerated through a potential difference 𝑉. What is
momentum of the particle if it is accelerated through a potential difference of 2𝑉
(a) 2p0
(b) 4p0
(c) (d)
36. Two electric dipoles 𝑃1 and 𝑃2 are placed at (0, 0, 0) and (1, 0, 0) respectively, with both of them
pointing in the +z direction. Without changing the orientations of the dipoles, 𝑃2 is moved to (0, 2, 0). The
ratio of the electrostatic potential energy of the dipoles after moving to that before moving is: [JAM 2006]
(a)
(b)
(c)
(d)
37. Dipole is placed parallel to the electric field. If W is the work done in rotating the dipole 60º, then
work done in rotating it by 180º is
(a) 2 W
(b) 3 W
(c) 4 W
(d) W/2
38. In which of the following states is the potential energy of an electric dipole maximum 3. When a test
charge is brought in from infinity along the perpendicular bisector of an electric dipole, the work done is
[DU 2015]
(a) positive
(b) negative
(c) zero
(d) infinity
(a) (b) (c) (d)
39. An electric dipole of dipole moment is lying in an external uniform electric field. If be, potential
energy, force and torque on dipole then which of the following is certaintly true.
(a)
(b)
(c)
(d)
40. If the electrostatic potential at a point (x, y) is given by by volts, the electrostatic energy density at that
point (𝑖𝑛 𝐽/𝑚3 ) is: [JAM 2008]
(a)
(b)
(c)
(d)
41. A small charged spherical shell of radius 0.01 m is at a potential of 30V. The electrostatic energy of the
shell is [JAM 2014]
(a) 10– 10 𝐽
(b) 5 × 10−10 𝐽
(c) 5 × 10−9 𝐽
(d) 10−9 𝐽
42. The electric fields outside and inside a solid sphere with a uniform volume charge density are given by
and respectively, while the electric field outside a spherical shell with a uniform surface charge density is
given by , q being the total charge. The correct ratio of the electrostatic energies for the second case to
first case is [JEST 2013]
(a) 1 :3
(b) 9 : 16
(c) 3 : 8
(d) 5 : 6
43. A parallel plate capacitor is connected to an ideal battery which provides a fixed potential difference.
Originally, the energy stored in the capacitor is 𝑈0 . If the distance between the plates is doubled, then new
energy stored in the capacitor will be
(a) 4𝑈0
(b) 2𝑈0
(c) 𝑈0
(d) 𝑈0 /2
44. A parallel plate capacitor of circular cross section with radius , where d is the spacing between the
plates, is charged to a potential V and then disconnected from the charging circuit. If, now, the plates are
slowly pulled apart (keeping them parallel) so that their separation is increased from d to , the work done
will be [TIFR 2013]
(a)
(b)
(c)
(d)
45. Three capacitors are connected with the source of electromotive force E as shown in the figure. Then,
the energy, drawn from the source is
(a)
(b)
(c)
(d)
46. A parallel plate capacitor 𝐶0 is connected to battery of emf E. It is then disconnected from the battery
and a dielectric slab of dielectric constant 𝑘 (Completely filling the air gap of the capacitor in inserted in it.
If indicates the change in energy, then)
(a)
(b)
(c)
(d)
47. A parallel plate capacitor of plate area A and plate separation d is charged to potential difference V
and then the battery is disconnected. A slab of dielectric constant k is then inserted between the plates. If
Q, E and n denote respectively, the magnitude of charge on each plate, the electric field between the plates
(after the slab is inserted) and the work done on the system in question, in the process of inserting the
slab, then
(a)
(b)
(c)
(d)
48. A fully charged capacitor has a capacitance 𝐶. It is discharged through a small coil of resistance wire
embedded in a thermal insulated block of specific heat capacity 𝑠 and mass 𝑚. If the temperature of the
block is raised by , the potential V across the capacitance is
(a)
(b)
(c)
(d)
49. The work done in carrying a charge 𝑞 around a circle of radius 𝑟 with a charge 𝒬 at the centre is
[B.H.U-2010]
(a)
(b)
(c)
(d)
50. A positively charged particle of mass 𝑚 𝑘𝑔 and charge 𝒬 coulomb travels from through a potential
difference of 𝑉. Its kinetic energy in Joules is
(a)
(b)
(c)
(d)
51. Four equal charges of each are kept at the vertices of a square of side 𝑅. A particles of mass 𝑚 and
charges is placed in the plane of the square at a short distance from the centre. If the motion of the articles
is confined to the plane, it will undergo small oscillations with an angular frequency
(a)
(b)
(c)
(d)
52. For spherical symmetrical charge distribution variation of electric potential with distance from centre
is given in diagram.
Given that : . Then which options(s) are correct:
(a) Total charge within is 𝑞
(b) Total electrostatic energy for is zero
(c) At electric field is discontinuous
(d) There will be no charge anywhere except
53. The electrostatic energy (in units of a uniformly charged spherical shell of total charge 5C and radius 4
m is _____________(Round off to 3 decimal places) [JAM 2019]
SUBJECTIVE QUESTIONS
1. A thin hollow cylinder of radius and length both equal to L is closed at the bottom A disc of radius L/2 is
removed from the bottom as shown in figure. This object carries a uniform surface-charge density.
Calculate late the electrostatic potential at the point P on the axis of the cylinder as shown in the figure.
[JAM 2008]
2. (a) Two concentric, conducting spherical shells of radii 𝑅1 (𝑅1 < 𝑅2 ) are maintained at potentials 𝑉1 and
𝑉2 , respectively. Find the potential and electric field in the region 𝑅1 < 𝑟 < 𝑅2.
(b) A polarize dielectric cube of side/is kept on the x-y plane s shown. If the polarization in the cube is
where ;k; is a positive constant, then find all the bound surface charge densities and volume charge
density [JAM 2012]
3. A conducting spherical shell of radius 𝑅1 carries a total charge Q. A spherical layer of a liner,
homogeneous and isotropic dielectric of dielectric constant K and outer radius 𝑅2 (> 𝑅1 ) covers the shell
as shown in the figure. [JAM 2010]
4. A conducting sphere of radius 𝑅𝐴 has a charge Q. It is surrounded by a dielectric spherical shell of inner
radius 𝑅𝐴 and outer RB (as shown in the figure below) having electrical permittivity.
(a) Find the surface bound charge density at
(b) Find the total electrostatic energy stored in the dielectric (region B)
5. Two identical parallel plate capacitors are connected across terminals A and B as shown Each of the
capacitors is made of square of side 𝑙 with a distance d between them. A dielectric slab (relative
permittivity k) of thickness ‘d’ is kept between the plates. The slab covers only half of the length of the
plates in each of the capacitors as shown Find the total capacitance of the assembly. The capacitors are
charged by a battery and then the battery is disconnected. If the slab is now displaced slightly by a
distance, show that it will perform harmonic oscillations. [JAM 2007]
6. A conducting solid sphere of radius a, carrying a charge q is kept in a dielectric of dielectric constant k,
such that half the sphere is surrounded by the dielectric as shown in figure. Find the surface charge
densities in the upper and lower hemispherical surfaces. [JAM 2013]
ANSWER KEY
COLUMB FORCE & FIELD :
1
2
3
(b)
(d)
(a)
11
12
13
(c)
(b, d)
(c)
21
22
23
(c)
(d)
(a)
31
32
33
(a)
a)
(ad)
41
42
43
(a)
(c)
(c)
51
52
53
(d)
(d)
(a)
61
62
63
(b)
(d)
(a)
71
72
73
(a, c, d) (a)
(c)
81
82
83
(a)
(b)
(c)
91
92
93
(a)
(c)
(a, c)
4
(d)
14
(a)
24
(b)
34
(3)
44
(d)
54
(c)
64
(b)
74
(a)
84
(b)
94
(a)
5
(c)
15
(a)
25
(b)
35
(c)
45
(b)
55
(a)
65
(a)
75
(b)
85
(a)
95
(d)
6
(a)
16
(a, d)
26
(a)
36
(a)
46
(d)
56
(a)
66
(c)
76
(b)
86
(a)
96
(c)
7
(b)
17
(c)
27
(d)
37
(a)
47
(a)
57
(a)
67
(a)
77
(4)
87
(c)
97
(a, c)
8
(c)
18
(c)
28
(d)
38
(b)
48
(c)
58
(c)
68
(b)
78
(a)
88
(d)
98
(b)
9
(b)
19
(a)
29
(c)
39
(d)
49
(d)
59
(c)
69
(a)
79
(b)
89
(b)
99
(d)
10
(b)
20
(a)
30
(d)
40
(b)
50
(a)
60
(a)
70
(a)
80
(c)
90
(1.5)
100
(b)
ANSWER KEY
ELECTRICITY FLUX & GAUSS LAW :
1
2
3
4
(a)
(b)
(c)
(a)
11
12
13
14
(a, b)
(c)
(a)
(d)
21
22
23
24
(b)
(b)
(b)
(c)
31
32
33
34
(d)
(a, c, d)
(d)
(b)
41
42
43
44
(d)
(b)
(c)
(d)
51
52
53
54
(c)
(a)
(a)
(d)
61
62
63
64
(b)
(c)
b
1
71
72
73
74
5
(b)
15
(c)
25
(b)
35
(a)
45
(c)
55
(c)
65
b
75
6
(c)
16
(b)
26
(d)
36
(c)
46
(d)
56
(c)
66
(a)
76
7
(d)
17
(c)
27
(c)
37
(d)
47
(c)
57
(a, c)
67
(a)
8
(c)
18
(d)
28
(a)
38
(a)
48
(b)
58
(b)
68
b
9
(b)
19
(a)
29
(d)
39
(a)
49
(b)
59
(a, c, d)
69
(a)
10
(d)
20
(b)
30
(c)
40
(c)
50
(c)
60
(a)
70
6
4
(c, d)
(c)
(8)
(b)
(a)
ANSWER KEY
ELECTRIC DIPOLE/MULTI POLE & ASYMPTOTIC VARIATION
1
2
3
4
5
6
(b)
(b)
(c)
(a)
all
(b)
11
12
13
14
15
16
(b)
(b)
(b)
(c)
(c)
(c)
21
22
23
24
25
26
(c)
(a)
(d)
(c)
(a, c, d)
(a)
31
32
33
34
35
36
(b)
(b)
(a, b, d)
(c)
(c, d)
(d)
7
(b)
17
(c)
27
(c)
37
(c)
8
(a)
18
(b)
28
(c)
38
1
9
(a)
19
(a, c), d)
29
(d)
10
(c)
20
(b)
30
(c)
ANSWER KEY
CONDUCTORS & IMAGE PROBLEMS:
1
2
3
4
(d)
(24)
(c)
(b)
11
12
13
14
(d)
(b)
(b)
(a)
21
22
23
24
(c)
(b)
(1)
(a, b, c
31
32
33
34
(a)
(a)
(c)
(a)
41
42
43
44
(d)
(b)
(b)
(a)
51
52
53
54
(c)
(b)
(c)
(a)
61
62
63
64
(c)
(c)
(d)
(c)
71
72
73
74
(d)
(c)
(a)
(b)
5
(b)
15
(d)
25
(a, b, d)
35
(d)
45
(b)
55
(b)
65
(a)
75
(d)
6
(b)
16
2
26
(d)
36
(c)
46
(c)
56
(b)
66
(b)
7
(b)
17
(b)
27
(a)
37
(b)
47
(a)
57
(d)
67
(d)
8
(c)
18
(d)
28
(d)
38
(b)
48
(c)
58
(a)
68
(b)
9
(c)
19
(b)
29
(d)
39
(a)
49
(b)
59
(d)
69
(a)
10
(b)
20
(d)
30
(c)
40
(d)
50
(b)
60
(d)
70
(c)
ANSWER KEY
DIELECTRICS & CAPACITORS :
1
2
3
(d)
(c)
(c)
11
12
13
(c)
(a)
(c)
21
22
23
(d)
(b)
(b)
31
32
33
(b)
(c)
(a)
41
42
43
(a)
(c)
(b)
51
52
53
(3.50)
(a, b, c, d)
(b)
4
(b)
14
(b)
24
(a)
34
(d)
44
(b)
5
(a)
15
(d)
25
(d)
35
(b)
45
(a)
6
(d)
16
(a, b, c)
26
(b)
36
(d)
46
(11)
7
(d)
17
(a)
27
(a)
37
(b)
47
(d)
8
(c)
18
(b)
28
(d)
38
(a)
48
(c)
9
(b)
19
(b)
29
(c)
39
(c)
49
(a)
10
(d)
20
(a)
30
(a)
40
(a)
50
(a, b, c, d)
ANSWER KEY
ELECTROSTATIC ENERGY :
1
2
3
4
5
6
7
8
9
10
(b)
11
(a)
21
(a)
31
(a, b, c, d)
41
(a)
51
(a)
61
(c)
71
(b)
(a)
12
(a)
22
(c)
32
(d)
42
(a)
52
(b, c)
62
(d)
72
(a)
8
13
(b)
23
(b)
33
(b)
43
(a)
53
(b)
63
(a)
73
(2)
(d)
14
(b)
24
(c)
34
(d)
44
(a, d)
54
(a)
64
(b)
(b)
15
(a)
25
(b)
35
(b)
45
(c)
55
(b)
65
(b)
(a, c, d)
16
(a)
26
(a)
36
(c)
46
(b)
56
(c)
66
(d)
(a)
17
(b)
27
(b)
37
(c)
47
(c)
57
(c)
67
(d)
(a)
18
(a)
28
(b)
38
(a)
48
2
58
d
68
(c)
(d)
19
(a)
29
(d)
39
(b)
49
(b)
59
(a)
69
(c)
(b)
20
(d)
30
(b)
40
(d)
50
(a)
60
(a)
70
(a, c, d)
ANSWER KEY
CONDUCTORS & IMAGE PROBLEMS:
1
2
3
4
(b)
(c)
(d)
(b)
11
12
13
14
(c)
(a)
(a)
(b)
21
22
23
24
(d)
(b)
(b)
(c)
31
32
33
34
(d)
(c)
(d)
(d)
41
42
43
44
(c)
(d)
(d)
(b)
51
52
53
(c)
(all)
(3.125)
5
(d)
15
(c)
25
(c)
35
(c)
45
(c)
6
(d)
16
(c)
26
(b)
36
(d)
46
(d)
7
(a)
17
(c)
27
(a, b)
37
(c)
47
(d)
8
(c)
18
(a)
28
(b)
38
(a)
48
(b)
9
(6000)
19
(b)
29
(c)
39
(a)
49
(d)
10
(2.88)
20
(c)
30
(b)
40
(b)
50
(d)