January 27, 1998

PHY 712 - Problem Set # 5

1.

Convince yourself that Eqs. 3.38 and 3.62 are correct by expanding the expression to second order. That is, verify the following:

$$\frac{1}{{\bf{\vert r - r^{\prime}\vert}}} = \sum_{l=0}^{\in... ...r^l_<}{r^{l+1}_\gt} P_l({\bf{\hat{r} \cdot \hat{r}^{\prime}}})$$ (1)

and

$$P_l({\bf{\hat{r} \cdot \hat{r}^{\prime}}}) = \frac{4 \pi}{2 ... ...-l}^l Y^*_{lm}({\bf{\hat{r}}})Y_{lm}({\bf{\hat{r}^{\prime}}}).$$ (2)

2.

Consider a charge distribution of the form:

$$\rho(r, \theta, \phi) = \rho_0 r^2 exp^{-\alpha r} \cos^2(\theta),$$ (3)

where $\rho_0$ and $\alpha$ are constants. Find the corresponding electrostatic potential which vanishes at $\infty$.

3.

Problem #3.8 in Jackson.