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
<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 line of charge with uniform charge density, $\lambda = \rho \pi a^2$. What is the magnitude of the electric field outside of the line charge (at a distance $s>a$)? 1. $E = \lambda/(4 \pi \varepsilon_0 s^2)$ 2. $E = \lambda/(2 \pi \varepsilon_0 s^2)$ 3. $E = \lambda/(4 \pi \varepsilon_0 s)$ 4. $E = \lambda/(2 \pi \varepsilon_0 s)$ 5. Something else?! *Use Gauss' Law* 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. $\sigma\,\mathbf{\omega}$ 2. $\sigma\,\mathbf{\dot{r}}$ 3. $\sigma\,\mathbf{r} \times \mathbf{\mathbf{\omega}}$ 4. $\sigma\,\mathbf{\mathbf{\omega}} \times \mathbf{r}$ 5. Something else? Note: * CORRECT ANSWER: D