Potential of Charged Rod

A thin non-conducting rod of length L carries a uniformly distributed charge Q and is oriented as shown in Figure 1.1.

Figure 1.1

a) Find the potential ϕ due to the charged rod for any point on the x-axis with z > L/2.

b) Find ϕ(r, θ, φ) for all |r| > L/2 where r, θ, φ are the usual spherical coordinates.

Hint: The general solution to Laplace’s equation in spherical coordinates is

(i)

SOLUTION

a) The potential along the z-axis may be computed by integrating along the rod:

(1)

where u ≡ L/2z.

b) Since the problem has azimuthal symmetry, we may use the Legendre polynomials P_l (cosθ) and rewrite the expansion in (i) (see Figure 1.2)

Since we consider r > L/2, all the A_l are zero, and

(2)

Figure 1.2

We wish to equate (2) along the with z-axis (1) where P_l(1) = 1 for all l. We must rewrite ln|(1+u)/(1-u)| as a sum. Now,

So

(3)

Rewriting (1) using (3), we find

Replacing n by l + 1,

(4)

Using (2) and (4), we have