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