ELECTROMAGNETIC FIELD THEORY
FINAL EXAM - FALL 2002
NAME: _________________________________________
Problem 1:
____________________________________ (30 pts)
Problem 2:
____________________________________ (25 pts)
Problem 3:
____________________________________ (30 pts)
Problem 4:
____________________________________ (30 pts)
Problem 5:
____________________________________ (20 pts)
Problem 6:
____________________________________ (20 pts)
Problem 7:
____________________________________ (15 pts)
Problem 8:
____________________________________ (10 pts)
Problem 9:
____________________________________ (10 pts)
Problem 10: ____________________________________ (10 pts)
Problem 11: ____________________________________ (10 pts)
Problem 12: ____________________________________ (10 pts)
Problem 13: ____________________________________ (10 pts)
Problem 14: ____________________________________ (10 pts)
Problem 15: ____________________________________ (10 pts)
Problem 16: ____________________________________ (10 pts)
Problem 17: ____________________________________ (10 pts)
Problem 18: ____________________________________ (10 pts)
Problem 19: ____________________________________ (10 pts)
Total:
____________________________________ (290 pts)
Grade:
____________________________________
1.
In the figure below, a conducting cone is at a potential V0, and a small gap
separates its vertex from a conducting plane. The axis of the cone is perpendicular
to the conducting plane, which is maintained at zero potential. The angle of the
cone is 1. Because of symmetry of this problem and the fact that the boundary
conditions on the potential V involve  only, V is independent of r and  when
spherical coordinates are used.
z
Vo
Vo
1
A.
Find the electric potential in the region 1    90.
  θ 
dθ
 ln  tan   
Hint: 
sin θ
  2 
1A.
Continued
B.
What is the electric field at the outer surface of the cone?
2.
Two solenoids are centered along a common axis of symmetry as shown below.
The first solenoid has a radius of A with N1 turns per meter while the second
solenoid has a radius of B with N2 turns per meter. Assuming that a current of I
flows in both solenoids in the directions shown, find the magnetic field in each of
the regions requested below.
y
B
z-axis
I
A
x
A.
Region where r < A
2B.
Region where A < r < B
C.
Region where r > B
3.
The Van de Graaff generator is not only useful for performing electrostatics
demonstrations in Freshman physics but also forms the high voltage source for
electrostatic accelerators used in physics research. A Van de Graaff generator
with a spherical dome of radius a = 0.5 m is shown below.
Metal
Conducting
Dome
a
Insulating
Column
A.
Given that the breakdown electric field for air is 3.0 x 106 V/m, what is the
maximum charge that stored on the dome of the Van de Graaff generator.
3B.
What is the maximum voltage obtainable with the generator?
3C.
Graph the voltage and electric field’s magnitude as a function of r.
4.
A.
Starting with Maxwell’s equations, show that the electric field in free space obeys

 1 2E
2
the wave equation,  E  2
with a speed of c.
v  t2
4B.
The electric field of a plane EM wave traveling in free space along the +z-axis is

E  E 0x î  E 0y ĵ sin ω t  k z  φ  where E0x and E0y are constants. Show that the

associated magnetic field must be B  - E 0y î  E 0x ĵ sin ω t  k z  φ / c .




8.
Which of the following Maxwell’s equations imply that there are no magnetic
monopoles?
a)
 ρ
E
ε0
c)


B
 E 
t
e)
magnetic monopoles have recently been found
9.
Two parallel conductors separated by a distance x carry currents I1 and I2 in the
same direction as shown below. What is the force per unit length exerted on the
right conductor by the left conductor?
b)

 B 0
d)



E
  Bμ0 J  μ0 ε0
t
y
I2
I1
x
x
a)
- μ o I1 I 2
î
2πx
b)
μ o I1 I 2
î
2πx
c)
μ o I1 I 2
î
πx
d)
- μ o I1 I 2
î
πx
e)
μ o π I1 I 2
î
x
10.
A toroid with N turns carrying current I has an inner radius a and outer radius b.
What is the magnitude of the magnetic field at a distance midway between a and
b?
b
a
r
a)
μ0 N I
2 π a  b 
b)
μ0 N I
π a  b 
c)
μ0 N I
πb
d)
μ0 I
π a  b 
e)
4μ0 N I
π a  b 
11.
A capacitor is constructed from two square metal plates of area L2 separated by a
distance d. One half of the space between the plates is filled with a dielectric
constant K1 and the other half filled with a dielectric constant K2. Calculate the
capacitance of the device assuming that the free space capacitance is C0.
K1
K2
a)
C 0 K1 K 2
2 K 1  K 2 
b)
K1  K 2 C 0
c)
C 0 K1 K 2
K1  K 2 
d)
2 C 0 K1 K 2
K1  K 2 
e)
K1  K 2 C 0
2
12.
A material that reduces the total magnetic field when placed it is placed inside a
solenoid is called ____________________________________________.
13.
A material that has a positive magnetic susceptibility is called
_____________________________________.
14.
A permanent magnet is made from _____________________________ material.
15.
The source of a radial diverging electric fields is ________________________.
16.
The non-zero magnetic moment of the neutron is evidence for internal
_________________________ inside the neutron.
6
17.
Evaluate the integral:
 (3x
4
3
 2x  1) δ 3x  3 dx
18.

Show that E  x 2  k̂ can not represent an electrostatic field.
19.
Prove that the plane wave f x, t   A e i (k x  ω t ) is traveling in the -x direction.
5.
Two infinite non-conducting sheets of charge are parallel to each other as shown.
Each sheet has a positive uniform charge density .
y


x
A.
Find the electric field to the right of the two sheets.
B.
Find the electric potential at a distance x to the right of the right plate assuming
that the right plate is our zero electric potential reference point.
6.
Two wires are bent in a semicircle of radius a as shown. If the upper half has a
resistance 2R  and the lower half has a resistance R , then find the magnetic
field at the center of the circle in terms of the total current I. (Hint: Biot-Savart)
2R 
I
a
I
R
7.
Prove that the electric potential of a circular annulus of inner radius a and outer
radius b (in the yz plane) of charge density  = charge/area < 0 at point P a
σ
b 2  x 2  a 2  x 2 where x =  is our
distance x along the x-axis is V 
2ε0
zero electric potential reference point. (Hint: Consider the potential due to an
infinitesimal point charge).


b
a
x
P
+x axis