Problems on Chapters 24 to 26

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1
Part I: Solve the following 10 problems
1.
An electron escapes from the surface of a conducting sphere with radius 5 cm and
charge -5 µC. What is the highest acceleration which the electron experiences?
(2 points)
F=
a=
2.
kQe
R2
F
= 3.2 ×1018 ms − 2
m
What is the electric flux through the top face of the cube shown? The side of cube is

a = 15 cm and the electric field, in N/C, is E = −500iˆ + 700 ˆj − 200 kˆ . (2 points)
(
=
)
 
φ = ∫ E . dA = ∫ − 500 iˆ + 700 ˆj − 200 kˆ ⋅ kˆ dA
z
top face
y
= − 200 a 2 = − 4.5 N ⋅ m 2 / C
a
x
1
2
3.
In the figure below, the ring with radius R = 6 cm is in the x-y plane and is centered
at the origin. The ring carries a uniformly distributed charge of +2 nC. An electron is
released from rest along the z-axis at z = 8 cm. What is the kinetic energy of the
electron, in eV, as it passes through the origin? (3 points)
z
y
R
∆K = − ∆U →
 kQ
kQ
K =e
−
R
R2 + z2

(
as Kin=0
)
x

 =1.92 ×10 −17 J
1
2 

K =120 eV
4.
The electric potential V in a region of space is given by V = x2 – 3 y2 + z2 where V is
in volts and x, y and z are in meters. What is the magnitude of the electric field at a
point with coordinates x = y = z = 1 m? (3 points)
Ei = −
∂V
so
∂xi
(

E = − 2 x iˆ − 6 y ˆj + 2 z kˆ
)

E ( 1,1,1) = − 2 iˆ + 6 ˆj − 2 kˆ
E = 2 2 + 6 2 + 2 2 N / C = 6.6 N / C
5.
The capacitors C1 = 3.00 µF, C2 = 6.00 µF and C3 = 4.00 µF are fully charged; the
capacitor C2 has a plate charge of 18.0 µC. What is the voltage across the capacitor C3?
(3 points)
2
3
C1
C2
Q2 =18 µC →V2 = V1 = 3V and Q1 = 9 µC
C3
Q3 = Q12 = 27 µC
V3 = Q3 C 3 = 6.75V
6.
In the circuit below, ε1 = 24.0 V, R = 12.0 Ω and r = 4.00 Ω. What is ε2 if I1 = 3.50 A?
(3 points)
ε
From upper loop
ε 1 −12 I 2 = 0 → I 2 = 2 A
I1
Junction rule
I3
I 3 = I 1 − I 2 → I 3 = 1.5 A
From big loop
1
I2
R
r
ε
2
ε 1 − ε 2 − rI 3 = 0 → ε 2 =18V
7.
A source of emf, a 10-kΩ resistor and 8-µF capacitor are connected in series. How
long does it take the energy stored in the capacitor to reach 80% of its final value?
(4 points)
3
4
−t


q = Cε 1 − e τ


 → U =U f


−t


0.8U f = U f 1 − e τ

−t

1 − e τ










2
2
and t = 2.25τ = 2.25 RC = 0.18 s
8.
A proton enters a region of uniform magnetic field (B = 0.5 T) with an initial velocity
(v = 105 m/s) which makes an angle θ = 20° with the field. What is the pitch of the
helical path of the proton? (3 points)
v
θ
B
v11 = v cos 20° = 9.4 ×10 4 m / s
T=
2πm
= 1.31×10 −7 s
eB
and p = T v11 = 0.012 m
9.
A wire bent as shown carries current I = 20 A perpendicular to the magnetic field
B = 0.3 T of a solenoid. If R = 12 cm and θ = 120°, what
B is the magnitude of the net
R
magnetic force exerted on the wire? (3 points)
.
R
θ
4
I
5
F = I L' B with L' = 2 R sin 60° = 0.208 m
F =1.25 N
10.
A pair of point charges q1 = 5 µC and q2 = -5 µC are moving with identical speeds
v1 = v2 = 105 m/s in the directions shown. When the charges are at the locations shown
what are the magnitude and direction of the net magnetic field produced at the origin?
(4 points)
y
q1
v1
v2
0.2 m
0.4 m
B1 =
q2
x
µ o q1 v1
= 1.25 µT
4π ( 0.2 ) 2
B2 =
µ o q2 v2
= 0.313 µT
4π ( 0.4 ) 2
Direction for both: − k̂ so

Bnet = − 1.56 µT kˆ
Part II: Conceptual Questions
Tick the most appropriate answer (each question carries 1 point)
1.
Point charges +4q and -2q are held in the X-Y plane as shown. A free charge Q with
coordinates xo and yo is in the same plane and in equilibrium. Then
5
6
a) xo < 0 and yo = 0
b) 0 < xo < a and yo = 0
c) xo > a and yo = 0
d) yo > a and xo = 0
2.
y
-2q
+4q
x
a
A positive charge is distributed uniformly within a non-conducting spherical object.
If the magnitude of the electric field and the electric potential (with respect to infinity)
at the center of the object are denoted by E and V, respectively, then
a) E ≠ 0 and V = 0
b) E ≠ 0 and V > 0
c) E = 0 and V = 0
d) E = 0 and V > 0
3.
Four point charges are held as shown. A, B and C are the mid points on three sides of
the square and D is the center point. A charge Q can be moved with constant speed
from one of these points to the other one while the net work performed is zero. These
two points are
A
-q
a) A and D
b) A and C
c) C and D
d) D and B
4.
D
+q
-q
C
B
+q
Cylindrical wires 1 and 2 shown below are made of the same material and have the
same length L. If I1 = 2I2, then:
a) The electric fields in wires 1 and 2 are equal.
wire 2
b) The current densities in wires 1 and 2 are equal. 2r
c) The charge carrier concentrations in wires 1 and 2 are equal.
d) The drift velocities in wires 1 and 2 are equal.
r
wire 1
I2
I1
5.
In the single loop circuit below, source of emf S is connected to load D by wires bc and
ad. The conventional current I is shown. The electric potential V around the circuit is such
that:
I
a) Vb = Vc > Va = Vd.
c
b
b) Vd > Va > Vb > Vc.
S
D
c) Va = Vb = Vc = Vd.
d) Vc > Vb > Va > Vd.
d
a
6.
A current I flows from b to a through the real source of emf shown below.
b
2
a) Energy is transferred to the charge at the rate εI + rI .
b) Energy is transferred from the charge at the rate εI + rI2.
c) Energy is transferred to the charge at the rate εI − rI2.
d) Energy is transferred from the charge at the rate εI − rI2.
ε
r
6
a
7
7.
The figure shows the paths of two charged particles A and B in a mass spectrometer.
The semicircular paths have radii RA and RB (RA = 2RB). The mass and charge of the
particles are identical. The transit times of the particles in the mass spectrometer are
denoted by tA and tB.
a) tA > tB
b) tA < tB
c) tA = 2tB
d) tA = tB
8.
A
B
The figure shows a long wire and a rectangular loop, both carrying current I in the
directions shown. The direction of the net magnetic force acting on the loop is
a) Upward
b) Downward
c) To the right
d) To the left
9.
I
I
Currents I1, I2 and I3 carried by three wires surrounded by the closed loop C are
1  
B.dl around loop C is equal to
shown below. The value of the line integral
µ0 ∫
I3
I1
a) I1 - I2 + I3
b) -I1 + I2 - I3
I2
c) I1 + I2 + I3
d) -I1 - I2 - I3
C
dl
10.
A ring with radius R carries current I and produces a magnetic field 0.3 mT at its
center. The same current I is passing in the wire shown below where the semicircular section
has the same radius R. The magnetic field at point P, the center of the semicircular
section, is
y
a) -0.30 mT k̂
x
b) +0.60 mT k̂
c) +0.15 mT k̂
I
.
d) -1.60 mT k̂
2R
P
2R
7
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