Questions

Questions 1 & 2 do not require the use of a computer.

1.

Find a homogeneous solution for the electric potential

$$\nabla^2 \psi = 0,$$

and show that one can use this solution to satisfy the zero potential difference constraint between the conductor and the shielding.

2.

Suppose the spatial density in the conductor changes as follows

$$\sigma(r,t) = 1+ \frac{1}{r} \sin\left[\frac{2\pi (r-8)}{42}\right] \sin\left(\frac{2 \pi t}{100}\right)$$

where $\sigma$ is the density of the insulating material and the equilibrium density is 1. Notice that the total volume does not change. Compute the location of the boundary between insulator A and insulator B as a function of time.

3.

Find the current $\frac{dQ}{dt}$ per unit length generated by the changes in density. Note: As the material deforms $\epsilon$ changes too. If $\epsilon'$ is the deformed dielectric constant, then

$$\epsilon' = 1+\sigma(\epsilon-1).$$