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- Problem Statement
- A point charge ##q## is located at a distance ##l## from the center of a conducting grounded sphere with radius ##a < l##. Find the potential on the points outside the sphere and determine the charge distribution on the surface of the sphere (Hint: use a Legendre polynomial expansion with the origin at the center of the sphere, choosing the polar axis to be along the direction of the point charge.)

- Relevant Equations
- ##\phi(\mathbf{r}) = \frac{q}{|\mathbf{r} - \mathbf{r'}|}##

After looking around a bit, I found that, considering the polar axis to be along the direction of the point charge as suggested by the exercise, the following Legendre polynomial expansion is true:

$$\begin{equation}\frac{1}{|\mathbf{r} - \mathbf{r'}|} = \sum_{n=0}^\infty \frac{r^{n}_{<}}{r^{n+1}_{>}}\end{equation}P_n(\cos{\theta}).$$

where ##P_l## are the Legendre polynomial and ##r_>## is the larger distance between ##\mathbf{r}## (observation point) and ##\mathbf{r'}## (charge position), while ##r_<## is the smaller distance between the two. In that case, the potential due to a point charge, for the case where we are considering an observation point at a distance ##r < l##, would be:

$$\begin{equation} \phi(r, \theta) = q \sum_{n=0}^\infty \frac{r^n}{l^{n+1}}P_n(\cos{\theta}) \end{equation}.$$

Since the conducting sphere is grounded, we have that its surface will be an equipotential surface where ##\phi = 0##. This means that, at r = a, where ##a < l##, we would have:

$$\begin{equation}\phi(a, \theta) = q \sum_{n=0}^\infty \frac{a^n}{l^{n+1}}P_n(\cos{\theta}) = 0\end{equation}$$

for every ##\theta##. This is where I'm stuck. I don't see how Equation 3 could be satisfied. I'm aware that this problem can be and has been solved using the method of image charges, but I was wondering if anyone had any insight as to how to find the solution using the hint given by the exercise. Thanks in advance!

$$\begin{equation}\frac{1}{|\mathbf{r} - \mathbf{r'}|} = \sum_{n=0}^\infty \frac{r^{n}_{<}}{r^{n+1}_{>}}\end{equation}P_n(\cos{\theta}).$$

where ##P_l## are the Legendre polynomial and ##r_>## is the larger distance between ##\mathbf{r}## (observation point) and ##\mathbf{r'}## (charge position), while ##r_<## is the smaller distance between the two. In that case, the potential due to a point charge, for the case where we are considering an observation point at a distance ##r < l##, would be:

$$\begin{equation} \phi(r, \theta) = q \sum_{n=0}^\infty \frac{r^n}{l^{n+1}}P_n(\cos{\theta}) \end{equation}.$$

Since the conducting sphere is grounded, we have that its surface will be an equipotential surface where ##\phi = 0##. This means that, at r = a, where ##a < l##, we would have:

$$\begin{equation}\phi(a, \theta) = q \sum_{n=0}^\infty \frac{a^n}{l^{n+1}}P_n(\cos{\theta}) = 0\end{equation}$$

for every ##\theta##. This is where I'm stuck. I don't see how Equation 3 could be satisfied. I'm aware that this problem can be and has been solved using the method of image charges, but I was wondering if anyone had any insight as to how to find the solution using the hint given by the exercise. Thanks in advance!