## Welcome to PHY 482
### Electrodynamics
Prof. Danny Caballero
### Contacting Danny
Office: 1310-A BPS
Email: <caballero@pa.msu.edu>
Cell phone: 517-420-5330 (texting is fine)
### Important Sites
* Course Webpage: [dannycab.github.io/phy482msu/](http://dannycab.github.io/phy482msu/)
* Discussion Forum: [
piazza.com/msu/spring2017/phy482](https://
piazza.com/msu/spring2017/phy482) (*Check your email*)
### Course Activities
* Projects:
* 2 of them; Mar 3 & May 1 - 20% each
* In-Class Quizzes:
* 7 of them; Every other Friday; 1 dropped - 20%
* Homework:
* 14 of them; Due on Mondays; 1 dropped - 40%
* Clickers:
* Pure Extra Credit - up to 5% bonus
[Much more detail on website](http://dannycab.github.io/phy482msu/)
Learning is a social and collaborative act!
### Homework Help Session
**Evening session once per week (Location TBD)**
Question to you: When should we do this?
1. Wednesdays 4-5pm
2. Wednesdays 5pm-6pm
3. Thursdays 4pm-5pm
4. Thursdays 5-6pm
*Times restricted by classroom availability*
Reminder: Homework is due on Mondays (expect this first one).
## This Week!!!
* Homework 1 is already up (Due Fri. Jan. 13 at 5pm)
* Read (seriously do this!)
* Griffiths Ch 7.1.1-7.1.2 (Review? Chs 1-6)
* [Download Anaconda distribution of Python](https://www.continuum.io/downloads)
**Stay up-to-date by checking website, calendar, and discussion forum regularly.**
### Computational Homework problems
* We will be using Python on homework problems this semester.
* Installation instructions appear on the piazza site.
* Homework solutions should take the form of a Jupyter notebook, which you can print to PDF and turn in.
* If you get stuck somewhere, post on piazza, so your classmates benefit from your question.
### Projects
#### Individual Project (Mar. 3)
* Literature review of some interesting topic in E&M (3-4 pages)
* Homework questions will support you on this
* See syllabus for sample questions
* Paper should be typed, inline references, bibliography, etc.
* Evaluation rubric will be ready in a couple of weeks
### Projects
#### Pair Project (May 1)
* Poster presentation of an original contribution (theory and computation)
* Homework questions will support you on this
* See syllabus for sample questions
* Can be something that has been done before that you just extend
* Evaluation rubric will be ready in a few weeks
* There will be a significant self-evaluation component to this also
## What do you think PHY 482 is about?
### Electromagnetism is the foundational field theory of physics
Think about everything you already know about electromagnetism (it's a lot already!).
Work with a partner to map out the electromagnetism concepts that you know and how they are related to each other.
5 charges, q, are arranged in a regular pentagon, as shown.
What is the E field at the center?
<img src ="./images/5charges.png" align="center" style="width: 250px";/>
1. Zero
2. Non-zero
3. Really need trig and a calculator to decide
Note:
CORRECT ANSWER: A
### Announcements
* We will use GitHub Classroom for [digital submissions of homework](https://dannycab.github.io/phy482msu/assignments/homework1.html)
* Create a [GitHub account](https://github.com/join)
* Download [GitHub Desktop](https://desktop.github.com/)
* Review [Piazza post on usage](https://piazza.com/msu/spring2017/phy482/home)
* Come to help session (or my office) if you need/want help
<img src ="./images/zappa.jpeg" align="right" style="width: 100px";/>
1 of the 5 charges has been removed, as shown. What’s the E field at the center?
<img src ="./images/4charges.png" align="center" style="width: 400px";/>
1. $+(kq/a^2)\hat{y}$
2. $-(kq/a^2)\hat{y}$
3. 0
4. Something entirely different!
5. This is a nasty problem which I need more time to solve
Note:
CORRECT ANSWER: B
Superposition!
To find the E-field at P from a thin line (uniform charge density $\lambda$):
<img src ="./images/linecharge.png" align="right" style="width: 400px";/>
$$ \mathbf{E}(\mathbf{r}) = \dfrac{1}{4\pi\varepsilon_0}\int \dfrac{\lambda dl'}{\mathfrak{R}^2}\hat{\mathfrak{R}}$$
What is $\mathfrak{R}$?
1. $x$
2. $y'$
3. $\sqrt{dl'^2 + x^2}$
4. $\sqrt{x^2+y'^2}$
5. Something else
Note:
CORRECT ANSWER: D
What do you expect to happen to the field as you get really far from the rod?
$$E_x = \dfrac{\lambda}{4\pi\varepsilon_0\}\dfrac{L}{x\sqrt{x^2+L^2}}$$
1. $E_x$ goes to 0.
2. $E_x$ begins to look like a point charge.
3. $E_x$ goes to $\infty$.
4. More than one of these is true.
5. I can't tell what should happen to $E_x$.
Note:
CORRECT ANSWER: D (A and B)
Given the location of the little bit of charge ($dq$), what is $|\vec{\mathfrak{R}}|$?
<img src ="./images/sphereintegrate.png" align="left" style="width: 300px";/>
1. $\sqrt{z^2+r'^2}$
2. $\sqrt{z^2+r'^2-2zr'\cos\theta}$
3. $\sqrt{z^2+r'^2+2zr'\cos\theta}$
4. Something else
Note:
CORRECT ANSWER: B
Which of the following are vectors?
(I) Electric field, (II) Electric flux, and/or (III) Electric charge
1. I only
2. I and II only
3. I and III only
4. II and III only
5. I, II, and II
Note:
* CORRECT ANSWER: A
A positive point charge $+q$ is placed outside a closed cylindrical surface as shown. The closed surface consists of the flat end caps (labeled A and B) and the curved side surface (C). What is the sign of the electric flux through surface C?
<img src="./images/ABC_cylinder.png" align="center" style="width: 600px";/>
1. positive
2. negative
3. zero
4. not enough information given to decide
Note:
* CORRECT ANSWER: B
* This is meant to be hard to visualize, next slide illustrates it better.
Let's get a better look at the side view.
<img src="./images/ABC_cylinder_side.png" align="center" style="width: 350px";/>
Which of the following two fields has zero divergence?
| I | II |
|:-:|:-:|
| <img src ="./images/cq_left_field.png" align="center" style="width: 400px";/> | <img src ="./images/cq_right_field.png" align="center" style="width: 400px";/> |
1. Both do.
2. Only I is zero
3. Only II is zero
4. Neither is zero
5. ???
Note:
* CORRECT ANSWER: B
* Think about dE/dx and dE/dy
* Fall 2016: 7 [34] 13 43 3; (Asked them to consider dvx/dx and dvy/dy) 3 [90] 3 4 0
What is the value of:
$$\int_{-\infty}^{\infty} x^2 \delta(x-2)dx$$
1. 0
2. 2
3. 4
4. $\infty$
5. Something else
Note:
* CORRECT ANSWER: C
A point charge ($q$) is located at position $\mathbf{R}$, as shown. What is $\rho(\mathbf{r})$, the charge density in all space?
<img src ="./images/pt_charge_at_R.png" align="right" style="width: 300px";/>
1. $\rho(\mathbf{r}) = q\delta^3(\mathbf{R})$
2. $\rho(\mathbf{r}) = q\delta^3(\mathbf{r})$
3. $\rho(\mathbf{r}) = q\delta^3(\mathbf{R}-\mathbf{r})$
4. $\rho(\mathbf{r}) = q\delta^3(\mathbf{r}-\mathbf{R})$
5. Something else??
Note:
* CORRECT ANSWER: E
* This one is a curious one because a delta function is always positive, both C and D are correct.
* Expect most everyone to pick C
<img src ="./images/dipole_gauss.png" align="right" style="width: 300px";/>
An electric dipole ($+q$ and $–q$, small distance $d$ apart) sits centered in a Gaussian sphere.
What can you say about the flux of $\mathbf{E}$ through the sphere, and $|\mathbf{E}|$ on the sphere?
1. Flux = 0, E = 0 everywhere on sphere surface
2. Flux = 0, E need not be zero *everywhere* on sphere
3. Flux is not zero, E = 0 everywhere on sphere
4. Flux is not zero, E need not be zero...
Note:
* CORRECT ANSWER: B
* Think about Q enclosed; what can we say about E though?
Which of the following two fields has zero curl?
| I | II |
|:-:|:-:|
| <img src ="./images/cq_left_field.png" align="center" style="width: 400px";/> | <img src ="./images/cq_right_field.png" align="center" style="width: 400px";/> |
1. Both do.
2. Only I is zero
3. Only II is zero
4. Neither is zero
5. ???
Note:
* CORRECT ANSWER: C
* Think about paddle wheel
* Fall 2016: 9 0 [89] 3 0
<img src ="./images/zappa.jpeg" align="right" style="width: 100px";/>
Can superposition be applied to electric potential, $V$?
$$V_{tot} \stackrel{?}{=} \sum_i V_i = V_1 +V_2 + V_3 + \dots$$
1. Yes
2. No
3. Sometimes
Note:
As long as the zero potential is the same for all measurements.
<img src="./images/graph_shell.png" align="center" style="width: 400px";/>
Could this be a plot of $\left|\mathbf{E}(r)\right|$? Or $V(r)$? (for SOME physical situation?)
1. Could be $E(r)$, or $V(r)$
2. Could be $E(r)$, but can't be $V(r)$
3. Can't be $E(r)$, could be $V(r)$
4. Can't be either
5. ???
A point charge $+q$ sits outside a **solid neutral conducting copper sphere** of radius $A$. The charge q is a distance $r > A$ from the center, on the right side. What is the E-field at the center of the sphere? (Assume equilibrium situation).
<img src="./images/copper_1.png" align="left" style="width: 300px";/>
1. $|E| = kq/r^2$, to left
2. $kq/r^2 > |E| > 0$, to left
3. $|E| > 0$, to right
4. $E = 0$
5. None of these
Note:
* CORRECT ANSWER: D
* Net electric field inside of a metal in static equilibrium is zero
* Talk about the net field versus the field due to the charges on the metal.
A neutral copper sphere has a spherical hollow in the center. A charge $+q$ is placed in the center of the hollow. What is the total charge on the outside surface of the copper sphere? (Assume Electrostatic equilibrium.)
<img src="./images/coppersphere_hole_and_charge.png" align="left" style="width: 350px";/>
1. Zero
2. $-q$
3. $+q$
4. $0 < q_{outer} < +q$
5. $-q < q_{outer} < 0$
**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.
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