**True or False:** The electric field, $\mathbf{E(\mathbf{r})}$, in some region of space is zero, thus the electric potential, $V(\mathbf{r})$, in that same region of space is zero.
1. True
2. False
Note:
* CORRECT ANSWER: B
* The electric potential is a constant in the region; it might be zero, but doesn't have to be.
**True or False:** The electric potential, $V(\mathbf{r})$, in some region of space is zero, thus the electric field, $\mathbf{E(\mathbf{r})}$, in that same region of space is zero.
1. True
2. False
Note:
* CORRECT ANSWER: A
* If the potential is zero in that space is zero, then it's gradient is zero in that space, so E must be zero also.
## Announcements
* Homework 1 due today at 5pm
* After 3:40pm turn in to Kim Crosslan
* 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 spherical symmetry
* Sketch a graph of the resulting electric field
The general solution for the electric potential in spherical coordinates with azimuthal symmetry (no $\phi$ dependence) is:
$$V(r,\theta) = \sum_{l=0}^{\infty} \left(A_l r^l + \dfrac{B_l}{r^{l+1}}\right)P_l(\cos \theta)$$
Consider a metal sphere (constant potential in and on the sphere, remember). Which terms in the sum vanish outside the sphere? (Recall: $V \rightarrow 0$ as $r \rightarrow \infty$)
1. All the $A_l$'s
2. All the $A_l$'s except $A_0$
3. All the $B_l$'s
4. All the $B_l$'s except $B_0$
5. Something else
Note:
* CORRECT ANSWER: E
* Only B0 will survive.
$$\mathbf{p} = \sum_i q_i \mathbf{r}_i$$
<img src="./images/dipole_2q_and_q.png" align="right" style="width: 200px";/>
What is the dipole moment of this system?
(BTW, it is NOT overall neutral!)
1. $q\mathbf{d}$
2. $2q\mathbf{d}$
3. $\frac{3}{2}q\mathbf{d}$
4. $3q\mathbf{d}$
5. Someting else (or not defined)
Note:
* CORRECT ANSWER: B
You have a physical dipole, $+q$ and $-q$ a finite distance $d$ apart. When can you use the expression:
$$V(\mathbf{r}) = \dfrac{1}{4 \pi \varepsilon_0}\dfrac{\mathbf{p}\cdot \hat{\mathbf{r}}}{r^2}$$
1. This is an exact expression everywhere.
2. It's valid for large $r$
3. It's valid for small $r$
4. No idea...
Note:
* CORRECT ANSWER: B
You have a physical dipole, $+q$ and $-q$ a finite distance $d$ apart. When can you use the expression:
$$V(\mathbf{r}) = \dfrac{1}{4 \pi \varepsilon_0}\sum_i \dfrac{q_i}{\mathfrak{R}_i}$$
1. This is an exact expression everywhere.
2. It's valid for large $r$
3. It's valid for small $r$
4. No idea...
Note:
* CORRECT ANSWER: A
Which charge distributions below produce a potential that looks like $\frac{C}{r^2}$ when you are far away?
<img src="./images/multipole_charge_configs_1.png" align="center" style="width: 600px";/>
E) None of these, or more than one of these!
(For any which you did not select, how DO they behave at large r?)
Note:
* CORRECT ANSWER: E (Both C and D)
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
<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