Physics 3323, Electromagnetism I, Spring 2004 EXAM 1 Wednesday by jal11416


									                       Physics 3323, Electromagnetism I, Spring 2004

                                                EXAM 1

                                      Wednesday February 11, 2004
Useful facts:
dl = dx x + dy y + dz z (cartesian)
        ˆ      ˆ      ˆ
dl = dr r + rdθ θˆ + r sin θ dφ φ (spherical)

Please do all 3 of the problems. Remember to work each problem as fully as possible. SHOW
ALL WORK as partial credit will be given.

                                                                           −x      y   z
   1. (a) (10 pts) A potential is given by the expression V = Vo exp(         ) sin cos . What is
                                                                           L       L   L
       the charge density, ρ associated with this potential?

       (b) (15 pts) A vector field is given by A = y 2 ( x 2 + y 2 ) x + xy 3 y . Calculate the line
                                                                     ˆ        ˆ
       integral    ∫ Aidl around the path and direction in the x-y plane shown in the figure below.
       Verify explicitly that Stoke’s Theorem holds for this vector field and region.


       (c) (9 pts) Which of the following vector fields could be electrostatic fields and why?

        A1 = y x + xy
               ˆ ˆ

        A2 = y 2 x + x 2 y
                 ˆ       ˆ

                               1      ˆ
        A3 (r ,θ , φ ) =             φ
                           r sin(θ )
2. (a) (15 pts) Four charges with values of +q, +2q, +q and +2q are placed at the corners of
   a square with side of length L as shown in the figure. What is the total work done in
   assembling all the charges?

   (b) (10 pts) If a fifth charge of value +q is brought in from infinity to the center of the
   square (point p) what is the ADDITIONAL work done?

   (c) (8 pts) What should the value of the fifth charge (in part b) be so that the TOTAL
   work in assembling all the charges is 0?

            +q                                L                        +2q

                  L                            P

            +2q                               L                          +q
3. A conducting sphere of radius R , carrying charge q, is surrounded by a thick concentric
   conducting shell (inner radius A, outer radius B) as shown in the figure below. The shell
   carries no net charge.

       a) (9 pts) Find the surface charge density σ at R, at A, and at B.

       b) (8 pts) Find the electric field for the four regions: i) r < R, ii) R<r<A; iii)
          A<r<B; and iv) B<r.

       c) (8 pts) Find the potential at the center, using infinity as the reference point.

       d) (8 pts) Now the outer surface is touched to a grounding wire, which lowers its
          potential to zero (same as infinity). How do your answers to (a) and (c) change?


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