## 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
# Questions?
## 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