Charged Conducting Sphere in Constant Electric Field

A conducting sphere of radius on whose surface resides a total charge Q is placed in a uniform electric field E₀ (see Figure 1.1). Find the potential

 Figure 1.1

at all points in space exterior to the sphere. What is the surface charge density?

SOLUTION

Look for a solution of the form

where ϕ₀ = -E₀ · r is the potential due to the external field and ϕ₁ is the change in the potential due to the presence of the sphere. The constant vector E₀ defines a preferred direction, and therefore the potential ϕ­₁ may depend only on this vector. Then, the only solution of Laplace’s equation which goes to zero at infinity is a dipole potential

(1)

where A is some constant (alternatively, we may write the solution in terms of Legendre polynomials and obtain the same answer from the boundary conditions). So

(2)

On the surface of the sphere, ϕ is constant:

(3)

where θ is the angle between E₀ and r (see Figure 1.2). From (3), we find that A = a³, and finally

(4)

The surface charge density

Figure 1.2