Problem 1 (10 marks )
Consider a wire loop, with resistance R and radius ra, in a
magnetic field as shown in Figure 1. At some time t1 the flux
of the magnetic field through the loop is given by �1. Between
time t1 and a later time t2 the flux changes to �2. Determine
the total charge that flows through the loop over the time
interval between t1 and t2.
Problem 2 (20 marks)
Figure 2 shows a conducting spherical shell with inne
adius Rb and outer radius Rc that encloses a concentric
solid conducting sphere of radius Ra. The solid sphere
has a charge of Q. Determine the capacitance of the
spheres.
Problem 3 (20 marks)
Consider the two wire loops shown in Figure 3. Loop 1
has length L and width h and ca
ies a cu
ent I. Loop
2 has length ` and width a and is located a distance b from Loop 1, in the same plane as
Loop 1. Determine the mutual inductance of the two loops. You may assume that the sides
of Loop 1 of length L are very long and that L � `.
1
Problem 4 (25 marks)
Consider two thin cylindrical conducting shells as shown
in Figure 4. The inner shell has a radius Ra,
length L, and charge Q1 uniformly distributed ove
its surface. The outer shell has a radius Rb, length
L, and charge Q2 uniformly distributed over its sur-
face. The two shells are rotating in opposite direc-
tions with the same angular velocity !. As seen from
above, the outer shell is rotating in a clockwise direc-
tion.
(a) Determine the surface cu
ent densities K1 and K2. (10
marks)
(b) Find the magnetic field B in the three regions: (i) s < Ra, (ii) Ra < s < Rb, and (iii)
s > Rb. You can assume that the cylinders are very long and neglect edge e↵ects. (15 marks)
Problem 5 (25 marks)
Figure 5 shows an infinitely long cylindrical wire
with radius Rw in which the cu
ent density in-
creases over a time interval ⌧ as follows:
J(s, t) =
8
:
0 for t 0
J0 s t/⌧ ẑ for 0 t ⌧
J0 s ẑ for t � ⌧
where J0 is a constant and is positive. Here s is the distance from the center of the cylindrical
wire. At a distance b (with b > Rw) from the center of the cylindrical wire is a square wire
loop with sides of length a and resistance R. Both the cylindrical wire and the square loop
are fixed in space.
(a) During the interval 0 t ⌧ , find the magnetic field B of the long cylindrical wire in the
two regions: (i) s < Rw and (ii) s > Rw. You may neglect inductance e↵ects in answering
this part of the problem. (10 marks)
(b) Determine the magnitude and direction of the cu
ent induced in the square wire loop
during the interval 0 t ⌧ . (10 marks)
(c) Determine the magnitude and direction of the net force on the square wire loop during
the interval 0 t ⌧ . (5 marks)
2
Possibly Useful Constants and Equations
Cartesian Coordinates
dl = dx x̂+ dy ŷ + dz ẑ; d⌧ = dx dy dz
Gradient: rt = @t
@x
x̂+
@t
@y
ŷ +
@t
@z
ẑ Divergence: r · v = @vx
@x
+
@vy
@y
+
@vz
@z
Curl: r⇥ v =
✓
@vz
@y
� @vy
@z
◆
x̂+
✓
@vx
@z
� @vz
@x
◆
ŷ +
✓
@vy
@x
� @vx
@y
◆
ẑ
Laplacian: r2t = @
2t
@x2
+
@2t
@y2
+
@2t
@z2
Spherical Coordinates
dl = d
̂+ rd✓ ✓̂ + r sin ✓d��̂; d⌧ = r2 sin ✓ dr d✓ d�; da = r2 sin ✓ d✓ d� r̂
8
:
x = r sin ✓ cos� x̂ = sin ✓ cos�r̂+ cos ✓ cos�✓̂ � sin��̂
y = r sin ✓ sin� ŷ = sin ✓ sin�r̂+ cos ✓ sin�✓̂ + cos��̂
z = r cos ✓ ẑ = cos ✓r̂� sin ✓✓̂
8
:
=
p
x2 + y2 + z2 r̂ = sin ✓ cos� x̂+ sin ✓ sin� ŷ + cos ✓ ẑ
✓ = tan�1
⇣p
x2 + y2/z
⌘
✓̂ = cos ✓ cos� x̂+ cos ✓ sin� ŷ � sin ✓ ẑ
� = tan�1(y/x) �̂ = � sin� x̂+ cos� ŷ
Gradient: rt = @t
@
̂+
1
@t
@✓
✓̂ +
1
sin ✓
@t
@�
�̂
Divergence: r · v = 1
2
@
@
(r2vr) +
1
sin ✓
@
@✓
(sin ✓v✓) +
1
sin ✓
@v�
@�
Curl: r⇥ v = 1
sin ✓
@
@✓
(sin ✓v�)�
@v✓
@�
�
̂+
1
1
sin ✓
@v
@�
� @
@
(rv�)
�
✓̂ +
1
@
@
(rv✓)�
@v
@✓
�
�̂
Laplacian: r2t = 1
2
@
@
✓
2
@t
@
◆
+
1
2 sin ✓
@
@✓
✓
sin ✓
@t
@✓
◆
+
1
2 sin2 ✓
✓
@2t
@�2
◆
3
Cylindrical Coordinates
dl = ds ŝ+ sd� �̂+ dz ẑ; d⌧ = s ds d� dz
8
:
x = s cos� x̂ = cos� ŝ� sin� �̂
y = s sin� ŷ = sin� ŝ+ cos� �̂
z = z ẑ = ẑ
8
:
s =
p
x2 + y2 ŝ = cos� x̂+ sin� ŷ
� = tan�1(y/x) �̂ = � sin� x̂+ cos� ŷ
z = z ẑ = ẑ
Gradient: rt = @t
@s
ŝ+
1
s
@t
@�
�̂+
@t
@z
ẑ
Divergence: r · v = 1
s
@
@s
(svs) +
1
s
@v�
@�
+
@vz
@z
Curl: r⇥ v =
1
s
@vz
@�
� @v�
@z
�
ŝ+
@vs
@z
� @vz
@s
�
�̂+
1
s
@
@s
(sv�)�
@vs
@�
�
ẑ
Laplacian: r2t = 1
s
@
@s
✓
s
@t
@s
◆
+
1
s2
@2t
@�2
+
@2t
@z2
Vector Identities
A · (B⇥C) = B · (C⇥A) = C · (A⇥B)
A⇥ (B⇥C) = B(A ·C)�C(A ·B)
(fg) = f(rg) + g(rf)
(A ·B) = A⇥ (r⇥B) +B⇥ (r⇥A) + (A ·r)B+ (B ·r)A
· (fA) = f(r ·A) +A · (rf)
· (A⇥B) = B · (r⇥A)�A · (r⇥B)
⇥ (fA) = f(r⇥A)�A⇥ (rf)
⇥ (A⇥B) = (B ·r)A� (A ·r)B+A(r ·B)�B(r ·A)
· (r⇥A) = 0
4
⇥ (rf) = 0
⇥ (r⇥A) = r(r ·A)�r2A
Fundamental Theorems
R
a (rf) · dl = f(b)� f(a)
R
(r ·A) d⌧ =
H
A · da
R
(r⇥A) · da =
H
A · dl
Electrodynamics
· E = 1
✏0
⇢
I
E · da = 1
✏0
Qenc r⇥ E = �
@B
@t
·B = 0
⇥B = µ0J+ µ0✏0
@E
@t
I
B · dl = µ0Ienc B = r⇥A
F = Q[E+ (v ⇥B)] E = �rV � @A
@t
· J = �@⇢
@t
V (r) = �
Z
Ref
E · dl
E(r) =
1
4⇡✏0
Z
⇢(r0)
2 r̂d⌧
0 E(r) =
1
4⇡✏0
Z
�(r0)
2 r̂da
0 E(r) =
1
4⇡✏0
Z
�(r0)
2 r̂dl
0
V (r) =
1
4⇡✏0
Z
�(r0)
da
0 V (r) =
1
4⇡✏0
Z
⇢(r0)
d⌧
0 V (r) =
1
4⇡✏0
Z
�(r0)
dl
0
Eabove � Ebelow =
�
✏0
n̂ W =
1
2
Z
V
⇢V d⌧ =
✏0
2
Z
all
E2d⌧
C =
Q
V
W =
1
2
CV 2 =
1
2
Q2
C
W =
1
2
nX
i=1
qiV (ri)
I =
dQ
dt
I =
Z
J · da J = ⇢v K = �v I = �v Fmag = I
Z
dl⇥B
B(r) =
µ0
4⇡
Z
J(r0)⇥ r̂
2 d⌧
0 B(r) =
µ0
4⇡
Z
K(r0)⇥ r̂
2 da
0 B(r) =
µ0I
4⇡
Z
dl0 ⇥ r̂
2
A(r) =
µ0I
4⇡
Z
dl0
A(r) =
µ0
4⇡
Z
K(r0)
da
0 A(r) =
µ0
4⇡
Z
J(r0)
d⌧
0
J = �E V = IR � =
Z
B·da emf: E =
I
E·dl = �d�
dt
�1 = LI1 �2 = M21I1
5
Magnetic field of a wire segment: B =
µ0I
4⇡s
(sin ✓2 � sin ✓1)�̂
Magnetic field of a very long solenoid: Bin = µ0nIẑ Bout = 0
Fundamental Constants
✏0 = 8.85⇥ 10�12 C2/N m2
µ0 = 4⇡ ⇥ 10�7 N/A2
c = 3.00⇥ 108 m/s
e = 1.60⇥ 10�19 C
m = 9.11⇥ 10�31 kg
6