What flexibility do we have in defining the vector potential given the Coulomb gauge ($\nabla \cdot \mathbf{A} = 0$)? That is, what can $\mathbf{A}'$ be that gives us the same $\mathbf{B}$? 1. $\mathbf{A}' = \mathbf{A} + C$ 2. $\mathbf{A}' = \mathbf{A} + \mathbf{C}$ 3. $\mathbf{A}' = \mathbf{A} + \nabla C$ 4. $\mathbf{A}' = \mathbf{A} + \nabla \cdot \mathbf{C}$ 5. Something else? Note: * Correct answer: C

What is the physical interpretation of $\oint \mathbf{A} \cdot d\mathbf{l}$? 1. The current density $\mathbf{J}$ 2. The magnetic field $\mathbf{B}$ 3. The magnetic flux $\Phi_B$ 4. It's none of the above, but is something simple and concrete 5. It has no particular physical interpretation at all Note: * CORRECT ANSWER: C

For a infinite solenoid of radius $R$, with current $I$, and $n$ turns per unit length, which is the current density $\mathbf{J}$? 1. $\mathbf{J} = nI\hat{\phi}$ 2. $\mathbf{J} = nI\delta(r-R)\hat{\phi}$ 3. $\mathbf{J} = \frac{I}{n}\delta(r-R)\hat{\phi}$ 4. $\mathbf{J} = \mu_0 nI\delta(r-R)\hat{\phi}$ 5. Something else?! Note: * CORRECT ANSWER: B

<img src="./images/long_wire_A.png" align="right" style="width: 200px";/> The vector potential A due to a long straight wire with current I along the z-axis is in the direction parallel to: 1. $\hat{z}$ 2. $\hat{\phi}$ (azimuthal) 3. $\hat{s}$ (radial) *Assume the Coulomb Gauge* Note: * CORRECT ANSWER: A

Consider a fat wire with radius $a$ with uniform current $I_0$ that runs along the $+z$-axis. We can compute the vector potential due to this wire directly. What is $\mathbf{J}$? 1. $I_0/(2 \pi)$ 2. $I_0/(\pi a^2)$ 1. $I_0/(2 \pi a) \hat{z}$ 4. $I_0/(\pi a^2) \hat{z}$ 5. Something else!? Note: * CORRECT ANSWER: D

Consider a fat wire with radius $a$ with uniform current $I_0$ that runs along the $+z$-axis. Given $\mathbf{A}(\mathbf{r}) = \dfrac{\mu_0}{4\pi}\int \dfrac{\mathbf{J}(\mathbf{r}')}{\mathfrak{R}}d\tau'$, which components of $\mathbf{A}$ need to be computed? 1. All of them 2. Just $A_x$ 3. Just $A_y$ 4. Just $A_z$ 5. Some combination Note: * CORRECT ANSWER: D

Consider a shell of charge with surface charge $\sigma$ that is rotating at angular frequency of $\mathbf{\omega}$. Which of the expressions below describe the surface current, $\mathbf{K}$, that is observed in the fixed frame. 1. $\sigma\,\vec{\omega}$ 2. $\sigma\,\mathbf{\dot{r}}$ 3. $\sigma\,\mathbf{r} \times \vec{\omega}$ 4. $\sigma\,\vec{\omega} \times \mathbf{r}$ 5. Something else? Note: * CORRECT ANSWER: D