Solving Laplace's Equation

Students should be familiar with:

• Gauss's law for electric fields.
• How a test charge behaves in the presence of an electric field.

Students should be familiar with:

• Solving PDEs using separation of variables (e.g., for the wave equation, heat equation, Schrodinger equation, etc.)
• Expressing a function in terms of a Fourier series, finding the expansion coefficients, and writing out a full Fourier series solution.
• How to plot functions of two (or three) variables in a computer software program like Mathematica.

1. (Capacitor) Examine the electric field of a parallel plane capacitor through superposition.

   We know that the electric field everywhere in space due to an infinite plane of charge with charge density located in the $xy$-plane at $z=0$ is \begin{equation*} \EE(z) = \begin{cases}\displaystyle +{\sigma\over2\epsilon_0}\>\zhat & z>0 \cr \noalign{\smallskip}\displaystyle -{\sigma\over2\epsilon_0}\>\zhat & z<0 \end{cases} \end{equation*}

   (Mentally check that this is true for both positive and negative values of $\sigma$.)

   1. Sketch the $z$-component of the electric field as a function of $z$.

   2. Draw a similar picture, and write a function that expresses the electric field everywhere in space, for an infinite conducting slab in the $xy$-plane, of thickness $d$ in the $z$-direction, that has a charge density $+|\sigma|$ on each surface.

   3. Repeat for a charge density $-|\sigma|$ on each surface.

   4. Now imagine two {\bf conductors}, one each of the two types described above, separated by a distance $L$. Use the principle of superposition to find the electric field everywhere. Discuss whether your answer is reasonable. Does it agree with the known fact that the electric field inside a conductor is zero? Has superposition been correctly applied? Is Gauss' Law correct? Try to resolve any inconsistencies.

2. (ConductorsGEM235) A long problem about the charge density, potential, and electric field due to a conducting sphere surrounded by a conducting shell, from Griffiths E&M book.

   A metal sphere of radius $R$, carrying charge $q$ is surrounded by a thick concentric metal shell (inner radius $a$, outer radius $b$, as shown below). The shell carries no net charge.

   \medskip \centerline{\includegraphics[scale=1]{\TOP Figures/vfconductor}} \medskip

   1. Find the surface charge density $\sigma$ at $R$, at $a$, and at $b$.

   2. Find $E_r$, the radial component of the electric field and plot it as a function of $r$. Are the discontinuities in the electric field related to the charge density in the way you expect from previous problems?

   3. Find the potential at the center of the sphere, using infinity as the reference point.

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

1. (LaplacePractice)

   Laplace's equation in two dimensions is: $\frac{\partial^2 V}{\partial x^2} + \frac{\partial^2 V}{\partial y^2} = 0$. Assume the region if interest is a rectangle of width $a$ and height $b$.

   1. Use separation of variables to find the general solution to Laplace's equation in two dimensions.

   2. Suppose three of the boundaries ($x=0$, $x=a$, and $y=0$) are known to have $V=0$. Find the general solution in this case.

   3. Suppose only one boundary ($y=0$) is known to have $V=0$, and that two boundaries ($x=0$ and $x=a$) are known to have $\frac{\partial V}{\partial x} = 0$. Find the general solution in this case.

2. (Laplace)

   Consider the bounded two-dimensional region from class. Three sides are metal and held at $V = 0$ while one is an insulator on which the potential is known to be:

   $V(x, b) = V_0\left(\sin\left(\frac{\pi x}{a}\right) + \sin\left(\frac{2\pi x}{a}\right) - \sin\left(\frac{3\pi x}{a}\right) \right)$

   1. Starting from the general solution from the practice problem, find a symbolic expression for the potential $V(x, y)$.

   2. Make several plots of your solution and discuss any interesting features you find. (I particularly recommend both surface plots and plots of $x$- and $y$-cross sections at several different values.)

   3. Suppose that the fourth side of the region is also a conductor at constant potential $V_0$. Find a symbolic expression for $V(x, y)$, graph your solution, and discuss its features.