Dielectric Cylinder in Uniform Electric Field

An infinitely long circular cylinder of radius a, dielectric constant ε, is placed with its axis along the z-axis, and in an electric field which would be uniform in the absence of the cylinder, E = E₀ x (see Figure 1.1). Find

Figure 1.1

the electric field at points outside and inside the cylinder and the bound surface charge density.

SOLUTION

First solution: Introduce polar coordinates in the plane perpendicular to the axis of the cylinder (see Figure 1.2). In the same manner as,

Figure 1.2

we will look for a potential outside the cylinder of the form

(1)

where ϕ₀ = -E . r and ϕ₁ is a solution of the two-dimensional Laplace equation, which may depend on one constant vector E

(2)

where A is some constant. Inside the cylinder, the only solution of Laplace’s equation that is bounded in the center of the cylinder and depends on E is

Using the condition on the potential ϕ at r = a, ϕⁱⁿ = ϕ^out we find

(3)

from which we find

We now have

(4)

(5)

Using the boundary condition Dⁱⁿ_n = D^out_n or Eⁱⁿ_n = εE^out_n, we find

So we obtain

(6)

(7)

The polarization is

So the dipole moment per unit length of the cylinder is

which corresponds to the potential

The surface charge density σ is

Second solution: Use the fact that for any dielectric ellipsoid with a dielectric constant ε immersed in a uniform electric field in vacuum, a uniform electric field inside is created. Therefore there must be a linear dependence between E_0x, Eⁱⁿ_x, and Dⁱⁿ_x, where the applied field E₀ is along the x-axis

(8)

where α and β are coefficients independent of the dielectric constant of the ellipsoid and only depend on its shape. For the trivial case in which ε = 1

Therefore

(9)

For a conducting ellipsoid (which can be described by a dielectric constant ε = ∞)

where n_x is the depolarization factor. From (9), we have

Finally (8) takes the form

(10)

For a cylinder parallel to the applied field along the z-axis, n_z = 0, but n_x + n_y +n_z = 1, so n_x = n_y = 1/2. Equation (10) becomes

(11)

and

(12)

as in (6) above.