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A grounded conducting sphere of radius $a$ is placed in an (effectively) infinite uniform electric field $\mathbf{E} = E\hat{\mathbf{z}}$. The potential for a uniform electric field in the $z$-direction is given by $Er\cos\theta$. The boundary condition at the surface of the grounded sphere is that

$$

u(a,\theta) = 0.

$$

Use a perturbation scheme for the total potential

$$

u(r,\theta) = Er\cos\theta + u'

$$

to solve for the perturbation potential $u'$.

Using our giving condition, we have $u(a,\theta) = Ea\cos\theta + u'(a,\theta) = 0$. That is,

$$

u'(a,\theta) = -Ea\cos\theta = \sum_{n = 0}^{\infty}\frac{A_n}{a^{n + 1}}P_n(\cos\theta).

$$

From our previous work, we know that we only need the $n = 1$ term. Therefore, $-Ea\cos\theta = \frac{A_1}{a^2} P_1(\cos\theta)\Rightarrow -Ea^3 = A_1$. We have that the perturbation potential is

$$

u'(r,\theta) = -Ea^3\frac{\cos\theta}{r^2}

$$

and that the total potential is

$$

u(r,\theta) = Er\cos\theta\left(1 - \frac{a^3}{r^3}\right).

$$