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(PHYS223)[2008](f)final~ph_cxxaa^_10543.pdf
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PHYS 223
Intermediate Electricity and Magnetism I
Final Examination
19/Dec/2008
Time allowed: 3 hours
1. (~20 pts)
A uniformly charged sphere with radius R and volume charge density . has its center located at the origin. Outside the sphere there is an isolated spherical conducting shell. The shell carries no net charge and is also centered at the origin, with inner radius a and outer radius b, where b > a > R.

(a) Find the E field in different regions from r = 0 to r = b.

(b) What is the volume charge density inside the conducting shell?

(c) Find the surface charge density on the inner surface of the shell by Gausss law.

(d) Find the surface charge density on the outer surface of the shell.

(e) Find the E field outside the shell.

(f) Take infinity as the reference point. Evaluate the potential in all regions.

2. (~10 pts)
Consider a very long cylinder with square cross-section as shown below.

y

a V = 0



V = V0 V = 0

x

O V = 0 a
The three plates y = 0, y = a, and x = a are grounded, while the left plate x = 0 is kept at a constant potential V0. The region inside the cylinder is charge-free.

(a) Verify that the potential
....12/2/01,3,541sinnnxannxanVnVeeeeyna..............

is the unique solution to the above system by showing that it satisfies all requirements of the first uniqueness theorem.

It is given that
. The Laplacian operator in 2D Cartesian coordinates is


. 22222xy.......

. The periodic odd function


............000,,02fyVyafyVyafyafyy..............

can be expressed as the sum of a sine series:
. ..01,3,5,4sinnVnyfyna......


(b) Using the potential in (a) to construct the solution of the system when all conditions remain unchanged except that the plate at y = 0 is kept at a potential 2V0, as shown in the figure below:

y

a V = 0



V = V0 V = 0

x

O V = 2V0 a




3. (~10 pts)
The Maxwells equations read
0000/Gauss' lawFaraday's law0No nameAmpere's law with Maxwell's correctiontt................................EBEBEBJ


(a) Consider a square wire with side length a. The wire is at rest. There is a region of uniform B field with width a moving towards the wire at speed v, as shown in the figure below. The B field has magnitude B and is perpendicular to the plane on which the wire lies, pointing outwards from the page. The wire enters the region at t = 0.

v

a t = 0 ................................................................


B



a

Use the integral form of the Faradays law to obtain the emf along the wire as a function of t. Take emf in the counterclockwise direction as positive.

(b) By Gausss law and the continuity equation , show that 0t.......J

. 0000t...............EJ







4. (~20 pts)
(a) Using Amperes law, fin