PHY531 Problem Set 5. Due November 8, 2001

1. Solve for the potential $\Phi(x,y)$ and the ${\vec {E}}$ and ${\vec {D}}$ fields for the region $0 \leq y \leq d$, $-\infty < x < \infty$ in two-dimensions (i.e. everything is $z$ independent) of the following two systems. The potential satisfies the boundary conditions
   

   $$\Phi(x,0) = 0$$

   $$\Phi(x,d) = V$$ (1)

   
   
   in both cases. The permittivity as a function of position is

   a.

   
   

   $$\epsilon(x,y) = \left \{ \begin{array}{ll} K & x \leq 0 \\ 1 & x > 0\\ \end{array}\right . \,.$$ (2)

   
   

   b.

   
   

   $$\epsilon(x,y) = \left \{ \begin{array}{ll} K & 0 < y < \frac... ...eq 0 \\ 1 & \frac{d}{2} \leq y < d\\ \end{array}\right . \,.$$ (3)

   
   

   Notice that the volume containing permittivity $K$ is the same. If we use these results to approximate the capacitance of a parallel plate capacitor where we fill half of the interior volume with dielectric and ignore edge effects, by what factor do we multiply the vacuum capacitance in each case?

2. If an uncharged, conducting, spherical shell of mass $m$ floats with one fourth of its volume submerged in a liquid dielectric of permittivity $\epsilon$, what must the potential difference between the sphere and infinity be to make the sphere float half submerged? Assume that the liquid remains flat in both cases.

3. Jackson problem 4.5

4. Jackson problem 4.6

5. Jackson problem 4.10

6. Jackson problem 4.13

   In the first edition of his book Jackson left out the word ``average.'' Why is this word necessary to describe the physical system?