Spherical Void in Dielectric

Suppose there is a spherical void of radius R in an otherwise homogeneous material of dielectric constant ε (see Figure 1.1). At the center of the void is a point dipole p. Solve for the electric field everywhere.

Figure 1.1

SOLUTION

We expect the dipole to induce some charge in the dielectric which would create a constant electric field inside the void, proportional to the dipole moment p. Therefore the field inside is due to the dipole field plus (presumably) a constant field. The field outside is the “screened” dipole field, which goes to zero at infinity. We look for a solution in the form

(1)

(2)

where n is normal to the surface of the void (see Figure 1.2). Use the

Figure 1.2

boundary conditions to find the coefficients α and β

and D = εE, so

Write (1) and (2) at some point P on the surface of the void

(3)

(4)

where θ is the angle between p and the normal to the surface of the void,

(5)

(6)

We solve for α and β in (5) and (6):

(7)

(8)

and find

(9)

(10)