What do you expect for direction of $\mathbf{B}(P)$? How about direction of $d\mathbf{B}(P)$ generated JUST by the segment of current $d\mathbf{l}$ in red? <img src="./images/curvy_wire_current.png" align="center" style="width: 400px";/> 1. $\mathbf{B}(P)$ in plane of page, ditto for $d\mathbf{B}(P$, by red$)$ 2. $\mathbf{B}(P)$ into page, $d\mathbf{B}(P$, by red$)$ into page 3. $\mathbf{B}(P)$ into page, $d\mathbf{B}(P$, by red$)$ out of page 4. $\mathbf{B}(P)$ complicated, ditto for $d\mathbf{B}(P$, by red$)$ 5. Something else!! Note: * CORRECT ANSWER: C
<img src="./images/parallel_currents.png" align="right" style="width: 250px";/> I have two very long, parallel wires each carrying a current $I_1$ and $I_2$, respectively. In which direction is the force on the wire with the current $I_2$? 1. Up 2. Down 3. Right 4. Left 5. Into or out of the page Note: * CORRECT ANSWER: D
What is the magnitude of $\dfrac{d\mathbf{l}\times\hat{\mathfrak{R}}}{\mathfrak{R}^2}$? <img src="./images/ringcurrent_R.png" align="right" style="width: 400px";/> 1. $\frac{dl \sin\phi}{z^2}$ 2. $\frac{dl}{z^2}$ 3. $\frac{dl \sin\phi}{z^2+a^2}$ 4. $\frac{dl}{z^2+a^2}$ 5. something else! Note: * CORRECT ANSWER: D
What is $d\mathbf{B}_z$ (the contribution to the vertical component of $\mathbf{B}$ from this $d\mathbf{l}$ segment?) <img src="./images/ringcurrent_R.png" align="right" style="width: 400px";/> 1. $\frac{dl}{z^2+a^2}\frac{a}{\sqrt{z^2+a^2}}$ 1. $\frac{dl}{z^2+a^2}$ 1. $\frac{dl}{z^2+a^2}\frac{z}{\sqrt{z^2+a^2}}$ 1. $\frac{dl \cos \phi}{\sqrt{z^2+a^2}}$ 5. Something else! Note: * CORRECT ANSWER: A
What is $\oint \mathbf{B}\cdot d\mathbf{l}$ around this purple (dashed) Amperian loop? <img src="./images/two_loops_ampere.png" align="left" style="width: 400px";/> 1. $\mu_0 (|I_2 | +|I_1 |)$ 2. $\mu_0 (|I_2|-|I_1|)$ 3. $\mu_0 (| I_2 | + | I_1 | \sin \theta)$ 4. $\mu_0 (| I_2 | - | I_1 | \sin \theta)$ 5. $\mu_0 (| I_2 | + | I_1 | \cos\theta)$ Note: * CORRECT ANSWER: A
Stoke's Theorem says that for a surface $S$ bounded by a perimeter $L$, any vector field $\mathbf{B}$ obeys: <img src="./images/balloon_surface.png" align="right" style="width: 300px";/> $$\int_S (\nabla \times \mathbf{B}) \cdot dA = \oint_L \mathbf{B} \cdot d\mathbf{l}$$ Does Stoke's Theorem apply for any surface $S$ bounded by a perimeter $L$, even this balloon-shaped surface $S$? 1. Yes 2. No 3. Sometimes Note: * CORRECT ANSWER: A
Rank order $\int \mathbf{J} \cdot d\mathbf{A}$ (over blue surfaces) where $\mathbf{J}$ is uniform, going left to right: <img src="./images/current_surfaces.png" align="center" style="width: 600px";/> 1. iii > iv > ii > i 2. iii > i > ii > iv 3. i > ii > iii > iv 4. Something else!! 5. Not enough info given!! Note: * CORRECT ANSWER: D * They are all the same!
<img src="./images/loop_infinite_wire.jpg" align="right" style="width: 250px";/> Much like Gauss's Law, Ampere's Law is always true (for magnetostatics), but only useful when there's sufficient symmetry to "pull B out" of the integral. So we need to build an argument for what $\mathbf{B}$ looks like and what it can depend on. For the case of an infinitely long wire, can $\mathbf{B}$ point radially (i.e., in the $\hat{s}$ direction)? 1. Yes 2. No 3. ??? Note: * CORRECT ANSWER: B * It violates Gauss's Law for B
<img src="./images/loop_infinite_wire.jpg" align="right" style="width: 250px";/> Continuing to build an argument for what $\mathbf{B}$ looks like and what it can depend on. For the case of an infinitely long wire, can $\mathbf{B}$ depend on $z$ or $\phi$? 1. Yes 2. No 3. ??? Note: * CORRECT ANSWER: B * By symmetry it cannot
<img src="./images/loop_infinite_wire.jpg" align="right" style="width: 250px";/> Finalizing the argument for what $\mathbf{B}$ looks like and what it can depend on. For the case of an infinitely long wire, can $\mathbf{B}$ have a $\hat{z}$ component? 1. Yes 2. No 3. ??? Note: * CORRECT ANSWER: B * Biot-Savart suggests it cannot
For the infinite wire, we argued that $\mathbf{B}(\mathbf{r}) = B(s)\hat{\phi}$. For the case of an infinitely long **thick** wire of radius $a$, is this functional form still correct? Inside and outside the wire? 1. Yes 2. Only inside the wire ($s<a$) 3. Only outside the wire ($s>a$) 4. No Note: * CORRECT ANSWER: A