Consider a cube of constant charge density centered at the origin.
**True or False**: I can use Gauss' Law to find the electric field directly above the center of the cube.
1. True and I can argue how we'd do it.
2. True. I'm sure we can, but I don't see how to just yet.
3. False. I'm pretty sure we can't, but I can't say exactly why.
4. False and I can argue why we can't do it.
Consider a spherical Gaussian surface. What is the $d\mathbf{A}$ in $\int\int\mathbf{E}\cdot d\mathbf{A}$?
1. $r d\theta d\phi \hat{r}$
2. $r^2 d\theta d\phi \hat{r}$
3. $r \sin \theta d\theta d\phi \hat{r}$
4. $r^2 \sin \theta d\theta d\phi \hat{r}$
5. Something else
Note:
* Correct Answer: D
Consider an infinite sheet of charge with uniform surface charge density $+\sigma$ lying in the $x-y$ plane. From symmetry arguments, we can argue that $\mathbf{E}(x,y,z)$ can be simplified to:
1. $\mathbf{E}(x,y)$; direction undetermined
2. $E_z(x,y)$
3. $\mathbf{E}(z)$; direction undetermined
4. $E_z(z)$
5. Something else
We derived that the electric field due to an infinite sheet with charge density $\sigma$ was as follows:
$$\mathbf{E}(z) = \begin{cases} \dfrac{\sigma}{2\varepsilon_0}\hat{k} & \mbox{if} & \mbox{ z > 0} \cr \dfrac{-\sigma}{2\varepsilon_0}\hat{k} & \mbox{if} & \mbox{ z < 0}\end{cases}$$
What does that tell you about the difference in the field when we cross the sheet, $\mathbf{E}(+z)-\mathbf{E}(-z)$?
1. it's zero
2. it's $\frac{\sigma}{\varepsilon_0}$
3. it's -$\frac{\sigma}{\varepsilon_0}$
4. it's +$\frac{\sigma}{\varepsilon_0} \hat{k}$
5. it's -$\frac{\sigma}{\varepsilon_0} \hat{k}$
Note:
* CORRECT ANSWER: D
* Makes for a good discussion about cross one direction versus the other
## Electric Potential
<img src="./images/lightning.jpg" align="center" style="width: 600px";/>
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
What is the curl of the vector field, $\mathbf{v}= c\hat{\phi}$, in the region shown below?
<img src="./images/c_phi_field.png" align="right" style="width: 300px";/>
1. non-zero everywhere
2. zero at some points, non-zero at others
3. zero curl everywhere
Note:
* CORRECT ANSWER: A
What is the curl of this vector field, in the red region shown below?
<img src="./images/curl_red_box.png" align="center" style="width: 400px";/>
1. non-zero everywhere in the box
2. non-zero at a limited set of points
3. zero curl everywhere shown
4. we need a formula to decide
Note:
* CORRECT ANSWER: D
* I think it's D because it depends on how the field drops off, which we haven't indicated. If it's drops off like 1/r, then it has no curl.
What is the curl of this vector field, $\mathbf{v} = \dfrac{c}{s}\hat{\phi}$, in the red region shown below?
<img src="./images/curl_red_box.png" align="center" style="width: 400px";/>
1. non-zero everywhere in the box
2. non-zero at a limited set of points
3. zero curl everywhere shown