Given $V_0(\theta) = \sum_l C_l P_l(\cos \theta)$, we want to get to the integral:

$$\int_{-1}^{+1}P_l(u)\;P_m(u)\;du = \dfrac{2}{2+1}\; (\mathrm{for}\;l=m)$$

we can do this by multiplying both sides by:

1. $P_m(\cos \theta)$
2. $P_m(\sin \theta)$
3. $P_m(\cos \theta) \sin \theta$
4. $P_m(\sin \theta) \cos \theta$
5. $P_m(\sin \theta) \sin \theta$

$$V(r,\theta) = \sum_{l=0}^{\infty} \left(A_l r^l + \dfrac{B_l}{r^{l+1}}\right)P_l(\cos \theta)$$

Suppose V on a spherical shell is:

$$V(R,\theta) = V_0 \left(1+\cos^2\theta\right)$$

Which terms do you expect to appear when finding V(inside)?

1. Many $A_l$ terms (but no $B_l$'s)
2. Many $B_l$ terms (but no $A_l$'s)
3. Just $A_0$ and $A_2$
4. Just $B_0$ and $B_2$
5. Something else!

$$V(r,\theta) = \sum_{l=0}^{\infty} \left(A_l r^l + \dfrac{B_l}{r^{l+1}}\right)P_l(\cos \theta)$$

Suppose V on a spherical shell is:

$$V(R,\theta) = V_0 \left(1+\cos^2\theta\right)$$

Which terms do you expect to appear when finding V(outside)?

1. Many $A_l$ terms (but no $B_l$'s)
2. Many $B_l$ terms (but no $A_l$'s)
3. Just $A_0$ and $A_2$
4. Just $B_0$ and $B_2$
5. Something else!

How many boundary conditions (on the potential $V$) do you use to find $V$ inside the spherical plastic shell?

1. 1
2. 2
3. 3
4. 4
5. It depends on $V_0(\theta)$