PHYS 196 Home Work 3A
1. A thin rod carrying electric charge lies on the x-axis, with one end at the origin, and another at the point
x  2 . The linear charge density on the rod is given by  x   C x3  3x 2 . Find (1) the total charge
q carried by the rod in terms of C and (2) the x-component of the electric field E x at the origin in terms
of q .
2. A thin rod carrying electric charge lies on the x-axis, with one end at the origin, and another at the point
x  1 . The linear charge density on the rod is given by  x   C x3  3x  2 . Find (1) the total charge
q carried by the rod in terms of C and (2) the x-component of the electric field E x at the point x  1 in
terms of q . (Hint: factorize  x  )
3. A thin rod carrying electric charge lies on the x-axis, with one end at the origin, and another at the point
x  a . The linear charge density on the rod is given by  x  Ax . Find (1) the total charge q carried by
the rod in terms of A and a and (2) the x-component of the electric field E x at the point x  2a in terms
of q and a .
4. A 90 arc of a circle of radius a is placed in the first quadrant of the x-y plane with its center at the
origin. It carries a uniform charge distribution of linear density  . Find the x and y components of the
electric field at the origin. Hence find the magnitude and direction of the electric field.
5. A 270 arc of a circle of radius a is placed on the x-y plane, occupying the second, third and fourth
quadrants, with its center at the origin. It carries a uniform charge distribution of linear density  . Find
the magnitude and direction of the electric field at the origin. (Hint: use the result from the previous
problem.)
6. A thin wire of length 20cm carries a total charge of 24nC uniformly distributed over its length. Find the
electric field at a point 6.0mm from the wire. (The distance of the point is so small the wire can be
considered infinite in length.)
7. The diagram shows two parallel infinite line charges (charge density  C / m ) a distance a apart. Find
the magnitude of the electric field at the point P, which is on a plane perpendicular to the lines and is at
distance a from one of the lines.





a
a
90º
P
8. A 25cmx25cm square sheet carries a total charge of 40nC uniformly distributed over its area. Find the
electric field at a point 3.0mm from the sheet. Find also the force on a 6.0nC charge placed at the point.
(The point is so close to the sheet that the latter can be considered infinite.)
1
9. The diagram shows two parallel infinite sheet charges with surface charge densities   and
 3 respectively. Find the electric fields in the regions A, B and C.
-σ
-3σ
A
C
\
B
10. A ring of radius a carrying a total charge q lies on the x-y plane with center at the origin. Find the
direction and magnitude of the electric field at a point (0,0,z) on the z-axis. Find the value of z where
this electric field is maximum.
11. A line charge of linear density  extends to infinity from a perpendicular plane as shown. A point P on
the plane is at a distance r from the point of intersection of the line and the plane. Use integration to find
the components of the electric field at P parallel and perpendicular to the line.

r
P
12. (hard) A hemispherical shell of radius a carries a uniform surface charge density  . Use integration to
find the electric field at the center. (Hint: divide the hemisphere into thin rings using planes
perpendicular to the axis, and use the formula for electric field due to the rings and integrate)
Answers:
2kq
3
3C 10kq
,
2.
4
3
2
Aa
2kq
,
(1  ln 2)
3.
2
a2
1. 12C , 
8. 3.6  104 N / C 0.22mN
9.   0 ,
10.
a
2  0 ,   0 ,
kqz
2
z

2 32
,
2k r 135 with
x  axis
11. k r  k r
5. 2k r 45 with
6. 3.6  105 N / C
k
7. 10
a
x  axis
12. k
4.


2
a
2
PHYS 196 Home Work 3B (Gauss Law)
1. A uniform electric field of magnitude 720N/C passes through a circular disk of radius 13cm . Find the
electric flux if its face is (a) perpendicular to the field line, (b) at 30 to the field line, and (c) parallel to
the field line.

2. A uniform electric field E is parallel to the axis of a hollow hemisphere of radius r . (a) What is the

electric flux through the hemispherical surface? (b) What is the result if E is perpendicular to the axis?
3. A point charge q is at the center of a cube of side a . What is the electric flux through one of the faces?
4. A spherical shell of radius 25cm has a surface charge density 48nC / m 2 . Find the electric field at a point
(a) just outside the surface, (b) in the interior, and (c) at a distance 50cm from the center.
5. A solid sphere of radius 25cm carries a total charge 72nC uniformly distributed over its volume. Find
the electric field (a) at a point just outside the sphere, and (b) at an interior point 10cm from the center.
6. Two concentric thin spherical shells of radius R1 and R2 with R1  R2 carry uniformly distributed charge
Q1 and Q2 respectively. Use Gauss law to find the electric field at a point a distance r from the center, in
the three cases r  R2 , R2  r  R1 and R1  r .
7. A solid sphere of radius a carries a uniform charge distribution over its volume. The total charge is Q .
Use Gauss law to find the electric field at a point a distance r from the center in the two cases r  a and
ra.
8. Two coaxial infinitely long thin cylindrical shells of radius R1 and R2 with R1  R2 carry uniformly
distributed charge per unit length of 1 and 2 respectively. Use Gauss law to find the electric field at a
point a distance r from the axis, in the three cases r  R2 , R2  r  R1 and R1  r .
9. From a solid sphere of radius a with a uniform charge density  , a spherical volume of radius a 2 is
removed as shown. Find the magnitude and direction of the electric field at the center of the original
sphere.
10. The electron cloud in a hydrogen atom corresponds to a spherically symmetric charge distribution with
volume charge density    0e  2 r a where r is the distance from the center. Find the electric field as a
function of r .
3
Answers:
Q
1 Qr
,
2
40 r
40 a 3
1 1  2
1 1
,
, 0
8.
20 r
20 r
a
9.
6 0
1. 38.2 Nm2 / C, 19.1Nm2 / C , 0
7.
2. r 2 E , 0
3.
q
6 0
1
 0 a 3   2r 2 2r   2 r a 

 1e
10.
1  

4 0 r 2   a 2
a


4. 5400 N / C , 0, 1350 N / C
5. 1.04  104 N / C, 4140 N / C
1 Q1  Q2
1 Q1
,
, 0
6.
2
40 r
40 r 2
4