Laplace Equation for Prism

A rectangular prism has width $a$, height $b$ and length $c$ as show at right.

a) What general type of function should you use to find a solution of the Laplace equation?

b) Apply the Laplacian to this function.

c)  Rearrange the equation such that each term only depends on one variable, and write down the ordinary differential equation for each term.

d) Are these differential equations related to each other? If so, how? Explain.

e) Write a possible solution for $V(x,y,z)$. (Without applying boundary conditions there are several solutions – just write one). Verify your solution by applying the laplacian to it.

f) The box is grounded on 5 faces, with the front face $(z=c)$ held at a constant potential $V_0$. Given these boundary conditions, what is the form of $V(x,y,z)$? Explain.

In tutorial you had several initial constants ($A,B,k,$ etc) that you used the boundary conditions to solve for.What are the values of any such constants in your expression?

g) $V(x,y,z)$ can be written as a sum of solutions to Laplace’s equation. Write the complete solution $V(x,y,z)$ inside the box. What are you summing over?

Use the Cartesian system to write the Laplace equation ($\rho$ is the local density of charge):

$\nabla^2V(x,y,z)=-\rho/\epsilon$ or for simplicity $\nabla^2V(x,y,z)=0$

Choose potential like:

$V(x,y,z)=X(x)Y(y)Z(z)$

and obtain

$YZ\frac{d^2X}{dx^2}+XZ\frac{d^2Y}{dy^2}+XY\frac{d^2Z}{dz^2}=0$

Divide by $V=XYZ$:

$(1/X) frac{d^2X}{dx^2}+(1/Y)\frac{d^2Y}{dy^2}+(1/Z)\frac{d^2Z}{dz^2}=0$

Since there are three separate functions $X,Y,Z$, each of a separate variable $x$, $y$, and $z$ we can choose  each of the terms equal to a constant.

$(1/X)\frac{d^2X}{dx^2}=a(=constant)$  that is $\frac{d^2X}{dx^2}-aX=0$

$\frac{d^2Y}{dy^2}-bY=0$  and the remaining part in $z$ is equal to $c=-a-b=(constant)$

$\frac{d^2Z}{dz^2}+(a+b)Z=0$

the above 3 equations are simple 2-nd order homogenous differential equations . Solutiond for each are $X(x)=A\exp(\pm\sqrt{a}x)$,  $Y(y)=B\exp(\pm\sqrt{b}y)$ and $Z=C\exp(\pm\sqrt{a+b}z)$

A possible solution is

$V(x,y,z)=Z(z)$   with   $\nabla^2V=(1/Z)\frac{d^2Z}{dz^2}=-\rho/\epsilon$

The conditions for the values of the constants above is

$(a+b)=\rho/\epsilon$

The general solution of the Laplace equation in 3D is a general linear superposition of the particular solutions

$V(x,y,z)=\sum C_{ab}\exp(\pm\sqrt{a}x)\exp(\pm\sqrt{b}y)\exp(\pm i*\sqrt{a+b}z)$

Where $C_ab$ are different constants and the sum is performed over all possible values of $a$ and $b$.