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  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