Required math: calculus
Required physics: electrostatics
Reference: Griffiths, David J. (2007) Introduction to Electrodynamics, 3rd Edition; Prentice Hall – Problem 4.34.
Another example of solving a boundary value problem in a system with linear dielectrics. Suppose we have an ideal dipole at the centre of a sphere of dielectric (dielectric constant ) and radius . What is the potential at any point (inside or outside the sphere)?
We can use a similar approach to that of the problem of a dielectric cylinder in an electric field. We first specify the boundary conditions. At the surface of the sphere, the potential must be continuous, so we have
As we saw in the cylinder problem, the condition on the normal derivative of the potential is, since on the outside of the sphere, :
Since there is no external field, we must have as . Finally, as , the potential must behave like that of an ideal dipole, so, assuming that points in the direction:
Note that we’re using rather than in this formula, since the dipole is inside the dielectric and the potential is reduced by a factor of , where .
With these conditions, we must solve Laplace’s equation in spherical coordinates, so we can quote the general form of the solution:
Inside the sphere, we have
Outside, we have
Applying the continuity condition 1 and equating coefficients of the , we get
We can therefore express the in terms of :
Using the condition on the derivatives 2 and equating coefficients as before, we get
Substituting for into the second equation gives
The only way this can be satisfied is if for , so we get for .
For the case, we can solve the two equations in and to get (using ):
The potential is