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";/>