Charge and Dielectric

A charge e is situated at the point x = h > 0, y = z = 0 outside a homogeneous dielectric which fills the region x < 0 (see Figure 1.1).

Figure 1.1

a) Write the electric fields E(0⁺, y, z) and E(0^-, y, z)  just outside and just inside the dielectric in terms of the charge e and surface charge density σ_b of bound charges on the surface of the dielectric.

b) Express σ_b in terms of E(0^-, y, z). Denote by the dielectric constant of the dielectric.

c) By using the equations obtained in (a) and (b), show that

d) Calculate the electric field E' due σ_b to at the position (h, 0, 0) of the charge e. Show that it can be interpreted as the field of an image charge e' situated at the point (-h, 0, 0).

e) Show that the charge experiences the force

SOLUTION

a) The electric fields just above and below the dielectric due to the charge e and the surface charge σ_b(y, z) are, respectively,

(1)

(2)

b) From D = E + 4πP = εE, we have for the polarization P and its normal component to the boundary P_n

(3)

c) Using (2) in (3), we have

(4)

So

(5)

d) The field at the position of the charge due to the surface charge σ_b is

(6)

where dA is the area. We can rewrite this integral in cylindrical coordinates:

(7)

This can be interpreted as an image charge

(8)

at a distance 2h from the charge e (see Figure 1.2).

Figure 1.2

e) The force on the charge e is

(9)