A proton ($q=+e$) is released from rest in a uniform $\mathbf{E}$ and uniform $\mathbf{B}$. $\mathbf{E}$ points up, $\mathbf{B}$ points into the page. Which of the paths will the proton initially follow? <img src="./images/proton-in-EandB.png" align="center" style="width: 800px";/> Note: * CORRECT ANSWER: C
## Announcements * Homework 1 due today at 5pm * After class put on my office * Last two questions turn in on Github * Quiz #1 - Next Friday * Last 20 minutes of class * No cheat sheets; all formulas will be provided * Solve a Gauss' Law Problem with cylindrical symmetry * Sketch a graph of the resulting electric field
<img src="./images/v_at_an_angle_to_B.png" align="right" style="width: 300px";/> A proton (speed $v$) enters a region of uniform $\mathbf{B}$. $v$ makes an angle $\theta$ with $\mathbf{B}$. What is the subsequent path of the proton? 1. Helical 2. Straight line 3. Circular motion, $\perp$ to page. (plane of circle is $\perp$ to $\mathbf{B}$) 4. Circular motion, $\perp$ to page. (plane of circle at angle $\theta$ w.r.t. $\mathbf{B}$) 5. Impossible. $\mathbf{v}$ should always be $\perp$ to $\mathbf{B}$ Note: * CORRECT ANSWER: A
Current $I$ flows down a wire (length $L$) with a square cross section (side $a$). If it is uniformly distributed over the entire wire area, what is the magnitude of the volume current density $J$? 1. $J = I/a^2$ 2. $J = I/a$ 3. $J = I/4a$ 4. $J = a^2I$ 5. None of the above Note: * CORRECT ANSWER: A
To find the magnetic field $\mathbf{B}$ at P due to a current-carrying wire we use the Biot-Savart law, $$\mathbf{B}(\mathbf{r}) = \dfrac{\mu_0}{4\pi}I\int \dfrac{d\mathbf{l}\times\hat{\mathfrak{R}}}{\mathfrak{R}^2}$$ In the figure, with $d\mathbf{l}$ shown, which purple vector best represents $\mathfrak{R}$? <img src="./images/linecurrent_r.png" align="center" style="width: 400px";/> Note: * CORRECT ANSWER: A
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
Consider the B-field a distance z from a current sheet (flowing in the +x-direction) in the z = 0 plane. The B-field has: <img src="./images/currentsheet_axes.png" align="left" style="width: 400px";/> 1. y-component only 2. z-component only 3. y and z-components 4. x, y, and z-components 5. Other 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
<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
Gauss' Law for magnetism, $\nabla \cdot \mathbf{B} = 0$ suggests we can generate a potential for $\mathbf{B}$. What form should the definition of this potential take ($\Phi$ and $\mathbf{A}$ are placeholder scalar and vector functions, respectively)? 1. $\mathbf{B} = \nabla \Phi$ 2. $\mathbf{B} = \nabla \times \Phi$ 3. $\mathbf{B} = \nabla \cdot \mathbf{A}$ 4. $\mathbf{B} = \nabla \times \mathbf{A}$ 5. Something else?! Note: * CORRECT ANSWER: D
We can compute $\mathbf{A}$ using the following integral: $\mathbf{A}(\mathbf{r}) = \dfrac{\mu_0}{4\pi}\int \dfrac{\mathbf{J}(\mathbf{r}')}{\mathfrak{R}}d\tau'$ Can you calculate that integral using spherical coordinates? 1. Yes, no problem 2. Yes, $r'$ can be in spherical, but $\mathbf{J}$ still needs to be in Cartesian components 3. No. Note: * CORRECT ANSWER: B * It's subtle. Griffiths discusses this in a footnote, you can't solve for, say, the phi component of A by integrating the "phi component" of J (because the unit vectors in spherical coordinates themselves depend on position, and get differentiated by del squared too)
<img src="./images/magnetic_dipole_oriented.png" align="left" style="width: 300px";/> Two magnetic dipoles $m_1$ and $m_2$ (equal in magnitude) are oriented in three different ways. Which ways produce a dipole field at large distances? 1. None of these 2. All three 3. 1 only 4. 1 and 2 only 5. 1 and 3 only Note: * CORRECT ANSWER: E