What is the value of:
$$\int_{-\infty}^{\infty} x^2 \delta(x-2)dx$$
1. 0
2. 2
3. 4
4. $\infty$
5. Something else
Note:
* CORRECT ANSWER: C
A point charge ($q$) is located at position $\mathbf{R}$, as shown. What is $\rho(\mathbf{r})$, the charge density in all space?
<img src ="./images/pt_charge_at_R.png" align="right" style="width: 300px";/>
1. $\rho(\mathbf{r}) = q\delta^3(\mathbf{R})$
2. $\rho(\mathbf{r}) = q\delta^3(\mathbf{r})$
3. $\rho(\mathbf{r}) = q\delta^3(\mathbf{R}-\mathbf{r})$
4. $\rho(\mathbf{r}) = q\delta^3(\mathbf{r}-\mathbf{R})$
5. Something else??
Note:
* CORRECT ANSWER: E
* This one is a curious one because a delta function is always positive, both C and D are correct.
* Expect most everyone to pick C
<img src ="./images/dipole_gauss.png" align="right" style="width: 300px";/>
An electric dipole ($+q$ and $–q$, small distance $d$ apart) sits centered in a Gaussian sphere.
What can you say about the flux of $\mathbf{E}$ through the sphere, and $|\mathbf{E}|$ on the sphere?
1. Flux = 0, E = 0 everywhere on sphere surface
2. Flux = 0, E need not be zero *everywhere* on sphere
3. Flux is not zero, E = 0 everywhere on sphere
4. Flux is not zero, E need not be zero...
Note:
* CORRECT ANSWER: B
* Think about Q enclosed; what can we say about E though?
Which of the following two fields has zero curl?
| I | II |
|:-:|:-:|
| <img src ="./images/cq_left_field.png" align="center" style="width: 400px";/> | <img src ="./images/cq_right_field.png" align="center" style="width: 400px";/> |
1. Both do.
2. Only I is zero
3. Only II is zero
4. Neither is zero
5. ???
Note:
* CORRECT ANSWER: C
* Think about paddle wheel
* Fall 2016: 9 0 [89] 3 0
<img src ="./images/zappa.jpeg" align="right" style="width: 100px";/>
Can superposition be applied to electric potential, $V$?
$$V_{tot} \stackrel{?}{=} \sum_i V_i = V_1 +V_2 + V_3 + \dots$$
1. Yes
2. No
3. Sometimes
Note:
As long as the zero potential is the same for all measurements.
<img src="./images/graph_shell.png" align="center" style="width: 400px";/>
Could this be a plot of $\left|\mathbf{E}(r)\right|$? Or $V(r)$? (for SOME physical situation?)
1. Could be $E(r)$, or $V(r)$
2. Could be $E(r)$, but can't be $V(r)$
3. Can't be $E(r)$, could be $V(r)$
4. Can't be either
5. ???
A point charge $+q$ sits outside a **solid neutral conducting copper sphere** of radius $A$. The charge q is a distance $r > A$ from the center, on the right side. What is the E-field at the center of the sphere? (Assume equilibrium situation).
<img src="./images/copper_1.png" align="left" style="width: 300px";/>
1. $|E| = kq/r^2$, to left
2. $kq/r^2 > |E| > 0$, to left
3. $|E| > 0$, to right
4. $E = 0$
5. None of these
Note:
* CORRECT ANSWER: D
* Net electric field inside of a metal in static equilibrium is zero
* Talk about the net field versus the field due to the charges on the metal.
A neutral copper sphere has a spherical hollow in the center. A charge $+q$ is placed in the center of the hollow. What is the total charge on the outside surface of the copper sphere? (Assume Electrostatic equilibrium.)
<img src="./images/coppersphere_hole_and_charge.png" align="left" style="width: 350px";/>
1. Zero
2. $-q$
3. $+q$
4. $0 < q_{outer} < +q$
5. $-q < q_{outer} < 0$
**True or False:** The electric field, $\mathbf{E(\mathbf{r})}$, in some region of space is zero, thus the electric potential, $V(\mathbf{r})$, in that same region of space is zero.
1. True
2. False
Note:
* CORRECT ANSWER: B
* The electric potential is a constant in the region; it might be zero, but doesn't have to be.
**True or False:** The electric potential, $V(\mathbf{r})$, in some region of space is zero, thus the electric field, $\mathbf{E(\mathbf{r})}$, in that same region of space is zero.
1. True
2. False
Note:
* CORRECT ANSWER: A
* If the potential is zero in that space is zero, then it's gradient is zero in that space, so E must be zero also.
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.
$$\mathbf{p} = \sum_i q_i \mathbf{r}_i$$
<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
You have a physical dipole, $+q$ and $-q$ a finite distance $d$ apart. When can you use the expression:
$$V(\mathbf{r}) = \dfrac{1}{4 \pi \varepsilon_0}\dfrac{\mathbf{p}\cdot \hat{\mathbf{r}}}{r^2}$$
1. This is an exact expression everywhere.
2. It's valid for large $r$
3. It's valid for small $r$
4. No idea...
Note:
* CORRECT ANSWER: B
You have a physical dipole, $+q$ and $-q$ a finite distance $d$ apart. When can you use the expression:
$$V(\mathbf{r}) = \dfrac{1}{4 \pi \varepsilon_0}\sum_i \dfrac{q_i}{\mathfrak{R}_i}$$
1. This is an exact expression everywhere.
2. It's valid for large $r$
3. It's valid for small $r$
4. No idea...
Note:
* CORRECT ANSWER: A
Which charge distributions below produce a potential that looks like $\frac{C}{r^2}$ when you are far away?
<img src="./images/multipole_charge_configs_1.png" align="center" style="width: 600px";/>
E) None of these, or more than one of these!
(For any which you did not select, how DO they behave at large r?)
Note:
* CORRECT ANSWER: E (Both C and D)