I still have questions about what we are trying to do with separation of variables in spherical coordinates.
1. Yes, definitely, let's talk about what we are trying to do (briefly).
2. I have some questions, but I think I got the gist of it. We can move on.
3. I got it, let's move on.
$$V(r,\theta) = \sum_{l=0}^{\infty} \left(A_l r^l + \dfrac{B_l}{r^{l+1}}\right)P_l(\cos \theta)$$
V everywhere on a spherical shell is a given constant, i.e. $V(R,\theta) = V_0$. There are no charges inside the sphere. 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 $A_0$
4. Just $B_0$
5. Something else!
Note:
* CORRECT ANSWER: C
The general solution for the electric potential in spherical coordinates with azimuthal symmetry (no $\phi$ dependence) is:
$$V(r,\theta) = \sum_{l=0}^{\infty} \left(A_l r^l + \dfrac{B_l}{r^{l+1}}\right)P_l(\cos \theta)$$
Consider a metal sphere (constant potential in and on the sphere, remember). Which terms in the sum vanish outside the sphere? (Recall: $V \rightarrow 0$ as $r \rightarrow \infty$)
1. All the $A_l$'s
2. All the $A_l$'s except $A_0$
3. All the $B_l$'s
4. All the $B_l$'s except $B_0$
5. Something else
Note:
* CORRECT ANSWER: E
* Only B0 will survive.
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. $P_m(\sin \theta) \sin \theta$
Note:
* CORRECT ANSWER: D
$$V(r,\theta) = \sum_{l=0}^{\infty} \left(A_l r^l + \dfrac{B_l}{r^{l+1}}\right)P_l(\cos \theta)$$
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!
Note:
* CORRECT ANSWER: C
$$V(r,\theta) = \sum_{l=0}^{\infty} \left(A_l r^l + \dfrac{B_l}{r^{l+1}}\right)P_l(\cos \theta)$$
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!
Note:
* CORRECT ANSWER: D