Conformal Mapping-a Slit in a Conducting Plane

Here we will consider the problem of an infinite, grounded, conducting plane with a slit between -1<x<1. There is an electric field E in the upper half plane which is uniform far from the plane, and which points downward. We want to find how the field ''leaks'' through the slit into the lower half plane.

The complex potential is

> F(z):= (I*E/2) * (z + (z^2 -1)^(1/2));

To obtain the real potential takes a bit of algebra, which I actually did by hand. The result is (taking E=1):

> Phi(x,y):=(1/2)*(y + (1/2)^(1/2)*( ( (x^2-y^2 -1)^2 + (2*x*y)^2)^(1/2) - (x^2 - y^2 -1) )^(1/2));

We can verify that this is a solution of Laplace's equation by direct substitution:

> simplify(diff(Phi(x,y),x$2) + diff(Phi(x,y),y$2));

We can also obtain the imaginary part, or the stream function, which will give the field lines:

> Psi(x,y):=(1/2)*(x*signum(x) + (1/2)^(1/2)*( ( (x^2-y^2 -1)^2 + (2*x*y)^2)^(1/2) +(x^2 - y^2 -1) )^(1/2)*signum(y));

Now plot the results, by first loading the plots package.

> with(plots);

> contourplot({Phi(x,y),Psi(x,y)},x=-2.5..2.5,y=-5..1,contours=15,numpoints=5000,scaling=CONSTRAINED);

We can see how the field leaks out into the other side. From the potential we can also calculate the field components:

> E_y:=-diff(Phi(x,y),y);

> E_x:=-diff(Phi(x,y),x);

What is the asymptotic behavior of the field as y->-infinity?

> asympt(subs(y=-y1,E_y),y1,4);

> asympt(subs(y=-y1,E_x),y1,4);

We see that this is a two-dimensional dipole field, with a dipole moment which points in the -y direction with strength

> p:= (4*Pi*epsilon)*(E*a^2/4);

>