Dielectric Sphere in Electric Field
Dielectric sphere in uniform field
A classic problem that illustrates the mechanics of matching boundary conditions across an interface is the dielectric sphere of relative permittivity εr = ε / ε0 > 1.0 immersed in an otherwise constant electric field (Seen in Figure 2).
Given the symmetry of the sphere about the z-axis, variations in the azimuthal angle φ can be neglected. The Laplace Equation in the remaining spherical coordinates is
The potential can be expressed as a product
where is a function of radial distance r alone and is a function of elevation angle alone. Taking the derivatives and rearranging, we get
This expression can hold only if both sides are equal to the same constant, which for convenience, we can call . Hence,
The equation for has the solution
The equation for is not so obvious. However, by using a series solution for , one can verify that the solutions are given by the Legendre polynomials
where . The first few are tabulated below:
Going back to the problem at hand, we recgnise that the constant E-field directed along the z-axis has a potential (that must represent the behaviour at infinity)
The contribution to the field due to the presence of the dielectric sphere must be computed in two parts: one for the region outside the sphere and another for the potential inside the sphere. Outside the sphere, we select the solution such that the perturbation in field vanishes as distance tends to infinity:
Note that this solution has the correct behaviour at infinity.
Within the sphere, if the solution is to "behave" as , we have
The key to the solution is to "match" the potential and the normal electric flux density across the air-dielectric boundary, such that
and, since there are no surface charges on the sphere,
- at .
Equating terms with the same dependence gives
The zeroth coefficient is clearly zero. Coefficients with indices greater than 1 are also zero. We are left with a contribution only from the first coefficients
The total potential outside the sphere is
Notice how the the presence of the dielectric sphere introduces a "dipole field" contribution (the term that varies as 1 / r2) to the overall field solution outside the sphere. Indeed, the dielectric sphere will look like an electric dipole as it is becomes polarised as a result of the external field.
The potential inside the sphere is found to be
The figures show the equipotentials and the field lines for spheres of dielectric constant 2.2 and 10.
Observe how the higher dielectric constant causes a strong concentration of flux lines within the sphere. In fact, it is easy to see the the flux achieves a maximum of
inside the sphere as the relative dielectric constant .