<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";/>
$$\mathbf{p} = \sum_i q_i \mathbf{r}_i$$
What is the magnitude of the dipole moment of this charge distribution?
<img src="./images/2q_dipole.png" align="right" style="width: 200px";/>
1. qd
2. 2qd
3. 3qd
4. 4qd
5. It's not determined
Note:
* CORRECT ANSWER: B
$$\mathbf{p} = \sum_i q_i \mathbf{r}_i$$
<img src="./images/dipole_2q_and_q.png" align="right" style="width: 200px";/>
What is the dipole moment of this system?
(BTW, it is NOT overall neutral!)
1. $q\mathbf{d}$
2. $2q\mathbf{d}$
3. $\frac{3}{2}q\mathbf{d}$
4. $3q\mathbf{d}$
5. Someting else (or not defined)
Note:
* CORRECT ANSWER: B