Is the following mathematical operation ok?
$$\nabla \times \left(\dfrac{1}{4\pi\epsilon_0}\int\int\int_V \dfrac{\rho(\mathbf{r}')d\tau'}{\mathfrak{R}^2}\hat{\mathfrak{R}}\right) = $$
$$\dfrac{1}{4\pi\epsilon_0}\int\int\int_V \left(\nabla \times\dfrac{\rho(\mathbf{r}')d\tau'}{\mathfrak{R}^2}\hat{\mathfrak{R}}\right)
$$
1. Yup. It's just fine and I can say why
2. I think it's fine, but I'm not sure I know why
3. No, we can't exchange the curl and integral!
4. I'm not sure.
## Announcements
* Homework 4 due Wednesday
* Exam 1 next Wednesday
- Topics: Charge, Electric field, $\delta$ functions, Electric potential
- Sections: Ch 1.1-1.5 and 2.1-2.3
* More detailed information coming this Wednesday!
Is it mathematically ok to do this?
$$\mathbf{E} = \dfrac{1}{4\pi\varepsilon_0}\int_V\rho(\mathbf{r}')d\tau'\left(-\nabla\dfrac{1}{\mathfrak{R}}\right)$$
$$\longrightarrow \mathbf{E} =-\nabla\left( \dfrac{1}{4\pi\varepsilon_0}\int_V\rho(\mathbf{r}')d\tau'\dfrac{1}{\mathfrak{R}}\right)$$
1. Yes
2. No
3. ???
Note:
* Correct Answer: A
If $\nabla \times \mathbf{E} = 0$, then $\oint_C \mathbf{E} \cdot d\mathbf{l} =$
1. 0
2. something finite
3. $\infty$
4. Can't tell without knowing $C$
Note:
* Correct Answer: A
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:
* Correct answer: A (usually)
As long as the zero potential is the same for all measurements.
The potential is zero at some point in space.
You can conclude that:
1. The E-field is zero at that point
2. The E-field is non-zero at that point
3. You can conclude nothing at all about the E-field at that point
Note:
* Correct Answer: C
The potential is constant everywhere along in some region of space.
You can conclude that:
1. The E-field has a constant magnitude in that space.
2. The E-field is zero in that space.
3. You can conclude nothing at all about the magnitude of $\mathbf{E}$ along that line.
Note:
* Correct Answer: B
<img src="./images/charged_shell.png" align="right" style="width: 200px";/>
A spherical *shell* has a uniform positive charge density on its surface. (There are no other charges around.)
What is the electric field *inside* the sphere?
1. $\mathbf{E}=0$ everywhere inside
2. $\mathbf{E}$ is non-zero everywhere in the sphere
3. $\mathbf{E}=0$ only that the very center, but non-zero elsewhere inside the sphere.
4. Not enough information given
Note:
* Correct Answer: A
<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. ???
Note:
* Correct Answer: B