How many boundary conditions (on the potential $V$) do you use to find $V$ inside the spherical plastic shell?
<img src="./images/plastic_shell_vtheta.png" align="right" style="width: 350px";/>
1. 1
2. 2
3. 3
4. 4
5. It depends on $V_0(\theta)$
Note:
* CORRECT ANSWER: B
* Good for discussion; obviously you need the surface BC, but what about at r=0? Is that technically a BC?
$$V(r,\theta) = \sum_{l=0}^{\infty} \left(A_l r^l + \dfrac{B_l}{r^{l+1}}\right)P_l(\cos \theta)$$
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
* Avoid blowup and match cosine
$$V(r,\theta) = \sum_{l=0}^{\infty} \left(A_l r^l + \dfrac{B_l}{r^{l+1}}\right)P_l(\cos \theta)$$
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
* Avoid blowup and match cosine
Consider a solid sphere of charge that has a charge density that varies with $\cos \theta$. What can we say about the terms in the general solution to Laplace's equation outside there sphere?
$$V(r,\theta) = \sum_l\left(A_l\,r^l + \dfrac{B_l}{r^{(l+1)}}\right)P_l(\cos \theta)$$
1. All the $A_l$'s are zero
2. All the $B_l$'s are zero
3. Only $A_0$ should remain
4. Only $B_0$ should remain
5. Something else
Note: Correct answer E because B0 and B1 remain
<img src="./images/dipole_moment.png" align="left" style="width: 300px";/>
Two charges are positioned as shown to the left. The relative position vector between them is $\mathbf{d}$. What is the value of of the dipole moment? $\sum_i q_i \mathbf{r}_i$
1. $+q\mathbf{d}$
2. $-q\mathbf{d}$
3. Zero
4. None of these
Note:
* CORRECT ANSWER: A
## Multipole Expansion
<img src="./images/universe_multipole.jpg" align="center" style="width: 300px";/>
Multipole Expansion of the Power Spectrum of CMBR
Note: The radiation from cosmic microwave background can be described in terms of contributions using a basis of functions with increasing smaller contributions.
<img src="./images/dipole_setup.png" align="left" style="width: 300px";/>
Two charges are positioned as shown to the left. The relative position vector between them is $\mathbf{d}$. What is the dipole moment of this configuration?
$$\sum_i q_i \mathbf{r}_i$$
1. $+q\mathbf{d}$
2. $-q\mathbf{d}$
3. Zero
4. None of these; it's more complicated than before!
Note:
* CORRECT ANSWER: A
For a dipole at the origin pointing in the z-direction, we have derived:
$$\mathbf{E}_{dip}(\mathbf{r}) = \dfrac{p}{4 \pi \varepsilon_0 r^3}\left(2 \cos \theta\;\hat{\mathbf{r}} + \sin \theta\;\hat{\mathbf{\theta}}\right)$$
<img src="./images/small_dipole.png" align="right" style="width: 200px";/>
For the dipole $\mathbf{p} = q\mathbf{d}$ shown, what does the formula predict for the direction of $\mathbf{E}(\mathbf{r}=0)$?
1. Down
2. Up
3. Some other direction
4. The formula doesn't apply
Note:
* CORRECT ANSWER: D
* The formula works far from the dipole only.
### Ideal vs. Real dipole
<img src="./images/dipole_animation.gif" align="center" style="width: 450px";/>