Conformal Mapping Using Maple

This Maple worksheet will illustrate some methods for plotting the solutions of electrostatics problems in two dimensions. Let's start with the charged conducting strip, which is located on the x-axis between -1 and 1. The solution for the complex potential is

> F:=C*I*arcsin(z);

where C is a constant to be determined and the complex variable z is

> z:=x+I*y;

Now Laplace's equation in two dimensions is

> laplace:=diff(Phi(x,y),x$2) + diff(Phi(x,y),y$2)=0;

Let's substitute F into Laplace's equation and verify that it is indeed a solution:

> simplify(subs(Phi(x,y)=F,laplace));

This is certainly reassuring. Both the real and the imaginary parts of F are solutions of Laplace's equation. We can also see that on the conductor (y=0, -1<x<1), Re(F) = 0, so that the Re(F) can be associated with the potential and it satisfies the correct boundary conditions. Now let's plot the potential. We first need to load the plots package.

> with(plots);

We have many options. The best in this case is the two dimensional contour plot. First plot the real part, which gives the equipotential surfaces. The solution will be symmetric about y=0, so we only need to plot the top half.

> contourplot(subs(C=1,Re(F)),x=-2..2,y=0.01..2,contours=20,coloring=[red,blue],numpoints=1000);

Next, plot the imaginary part, which will give the field lines.

> contourplot(subs(C=1,Im(F)),x=-2..2,y=0.01..2,contours=20,coloring=[red,blue],numpoints=1000);

Now let's plot both on the same plot. We can see how the equipotentials and the field lines are orthogonal.

> contourplot({subs(C=1,Re(F)),subs(C=1,Im(F))},x=-2..2,y=0.01..2,contours=20,coloring=[red,blue],numpoints=1000);

We can now find the surface charge density on the conductor. To do this we need the y-component of the electric field evaluated on the strip. Clearly the value will depend upon how we take the limit-from the top or the bottom. Let's take the limit from the top. First, calculate the y-component of the electric field (we actually want to take the real part):

> E_y := -diff(F,y);

The surface charge density, from Gauss's Law, is sigma=E/2*epsilon_0, where the factor of 2 comes from the two sides of the strip. Evaluate E_y on

the strip, y=0.

> sigma:=2*epsilon*simplify(subs(y=0,E_y));

Plot the result.

> plot(subs(C=1,epsilon=1,sigma),x=-1..1);

We see that the surface charge density has a square root divergence, which is typical for sharp edges. To calculate the constant C, we need to integrate the surface charge density across the strip and set the integral equal to the charge per unit length lambda.

> eqn:=lambda= int(sigma,x=-1..1);

> solve(eqn,C);

We've now calculated C; let's assign this value to C, and display the surface charge density once again.

> assign(C=%);

> sigma;

> unassign('C');

>