January 20, 1998

PHY 712 - Problem Set # 3

1.

Consider the Poisson equation for a one-dimensional system with the boundary condition $\Phi(x \rightarrow -\infty) = 0$. Evaluate $\Phi(x)$ for one or both of the charge densities $\rho(x)$, using Maple (or some other software) to plot $\rho(x)$ and $\Phi(x)$.

(a)

$$\rho(x)= \rho_0 \frac {x/a}{\sqrt{\pi}a}e^{-x^2/a^2}$$

(b)

$$\rho(x)=\frac {2 \rho_0}{\pi a} \frac{1}{(x^2/a^2+1)^2}$$

Here, $\rho_0$ and a are constants.

2.

An electrically isolated conducting sheet of area A contains a total charge Q and is located in the z = 0 plane. A point charge q is placed at the point (0,0,d). Negecting edge effects, find the electrostatic potential $\Phi(x,y,z)$ and field ${\bf{E}}(x,y,z)$ for $z \ge 0$. Find the surface charge distribution of the conducting sheet and also the force acting between the sheet and the point charge.

3.

Problem #2.2 in Jackson.