## MP204: Electricity and Magnetism

### Spring 2020

#### (2nd semester of 2019-2020)

Final Exam, Saturday May 16th, 2020

Here is the Final Exam for 16th May, available after 2:30PM.

Important equations of electromagnetism

Here is a list of the main equations and results we encounter in MP204.

Practice Problems

• Problem bank (problem set 12'')
This is marked as Problem set 12', but it is really a large collection of problems covering all the module material.
Should be useful for practice and for learning the material more thoroughly.
• Many of the textbooks have collections of problems and worked-out exercises.
In particular, the textbook by Griffiths and that by Purcell & Morin both have large numbers of excellent exercises and problems.

Problem sets

Problem set 11.   Due Wednesday May 6th, 8PM.

Problem set 10.   Due Wednesday April 29th, 8PM.

Problem set 09.   Due after the Easter break, Wednesday April 22nd, 8PM.

Problem set 08.   Due Wednesday April 8th, 8PM.

Problem set 07.   Due tuesday March 31st, 6PM.

Problem set 06.   Was due tuesday March 24th. Extended to Thursday March 26th, 6PM
For the Study Break; twice as long as the usual weekly problems.

Problem set 05.   Due tuesday March 10th, 5:30PM.

Problem set 05.   Due tuesday March 10th.

Problem set 04.   Due tuesday March 3rd, 5:30PM.

Problem set 03.   Due tuesday February 25th, 5:30PM.

Problem set 02.   Due tuesday February 18th.

Problem set 01.   Due tuesday February 11th.

Problems 3 and 4 in this problem set involve finding the total charge of objects from the linear charge density or from the surface charge density. This will require breaking up the object into infinitesimal pieces and then adding up the contributions of each piece, which amounts to an integration. We will have to do variants of this procedure many times during this module; so please practice until you are fully comfortable.
The relationship between charge density and total charge is the same as that between the usual (mass) density and total mass. For guidance, you could try
this video,   this video,   this discussion,   these notes.

Energy flow, Poynting vector

The material on electromagnetic energy and the Poynting vector is discussed in pages 81-83 in my hand-written notes (p.73-83).

You can hear someone describe this material to you, in this video titled "Poynting vector".

A more enthusiastic description of properties of EM waves:
* video "Energy Density of Electromagnetic Waves (Light)";
* video "Light Has Energy and (GASP!) Momentum";
* video "Intro to Polarization Filters!";
* video "Accelerating Charges Emit Electromagnetic Waves".

These might complement my notes. In addition, Chapter VI of Prof. Nash's notes provides supplementary reading.

Virtual lecture'' for Friday April 24th

In this lecture we look at the nature of electromagnetic waves.

In my hand-written notes (p.73-83), we will work through the first three pages.

The first page gives an example solution of the wave equations we derived previously.

This solution is slightly pathological: the arguments of the sine function are not unitless. (The argument has the dimensions of length.) We will correct this in the next page. For now, we will learn a lot by trying to visualize this solution.

This is a traveling wave solution. It has a pattern that moves rightward at speed $$c = 1/\sqrt{\mu_0\epsilon_0}$$. The figure is a snapshot': it should help you visualize the directions of the magnetic and electric fields. In this picture, the y-axis points into the plane and the z-axis points upward.

Very useful exercise: consider having the y-axis upward, so that the z-axis points outward from the paper. Plot a snapthot of the electric and magnetic fields.

Show that, because the eletric and magnetic fields must satisfy Maxwell's equations in free space, the amplitudes of the oscillations must be related as $$B_0=E_0/c$$.

The amazing thing is that, we found a solution of Maxwell's equations that involves propagation of the electromagnetic field by themselves, without any charge or current or matter! The electric and magnetic fields drive each other forward. The change of an electric field creates a magnetic field, and the change of a magnetic field in turn creates an electric field. They sustain each other and drive each other forward. This is what an electromagnetic field is. It can travel through vacuum, in the absence of any charge or matter. This is how, for example, light and radiation from the sun reach the earth.

On page 74, we note that the electric and magnetic fields are perpendicular to each other. And also, the direction of propagation is in the direction of $$\vec{E}\times\vec{B}$$. Please check that this is true for the solution we studied on page 73. This is a general feature of electromagnetic waves. Worth remembering.

As I noted above, our previous solution has to be corrected so that the argument of the sine function is dimensionless. This is done by introducing the wavelength and frequency. The expressions can also be written in terms of the wavenumber, $$k=2\pi/\lambda$$, and the angular frequency, $$\omega=2\pi{c}/\lambda=2\pi{f}$$.

The expressions for the traveling wave look cumbersome after we've introduced wavelength/frequency, but you should have met expressions like this in your Vibrations+Waves class.

On page 75, we note that the electromagnetic wave does not have to be sinusodial: you can also have pulses that satisfy the wave equation and Maxwell's equations. An example is constructed. Note that the argument of the exponential should be dimensionless: This is achieved using the constant $$l_0$$, which must have the dimensionsl of length.

What we are learning today is probably one of the top 5 important things you've been learning in university: the nature of light itself!

* Chapter VI of Prof. Nash's notes: Sections 1 and 2. In Section 2, the transversality of the fields is nicely proved.
* Subsection 4.3.1, and the beginning of subsection 4.3.2, of these lecture notes.
* video describing eletromagnetic waves.
* enthusiastic video including an animation of oscillating transverse E and B fields.
* The wikipedia page on electromagnetic waves is a bit heavy, but has a nice animation of transverselly oscillating fields.

Virtual lecture'' for Tuesday April 21st

Today we will
(1) look at physical manifestations of the displacement current density,
(2) start our study of electromagnetic waves.

Displacement current density

We introduced the displacement current density as Maxwell's correction to Ampere's law. Please review how this correction allows Ampere's law to be consistent with the continuity equation.

We did not look at any physical consequences of this law. Time to do that, now.

In my my hand-written notes (p.61-72), this material starts halfway on p.66.

Please go through Example 1, which runs to the end of p.67.

You might want to supplement by reading p.36-37 of Prof. Nash's notes, where the same situation is analyzed.

Example 2 is optional; you can skip it so that we can move to electromagnetic waves.

Electromagnetic waves

Starts on p.70 in my my hand-written notes (p.61-72), halfway down the page.

First we write down Maxwell's equations in free space or in vacuum. This means that there is no charge or current: you can set the charge density $$\rho$$ and the current density $$\vec{J}$$ to zero.

Note that the combination $$\mu_0\epsilon_0$$ is set equal to $$\frac{1}{c^2}$$, where the constant $$c$$ will later turn out to be the speed of light.

In the first half of p.71, there is a brief review of the wave equation and the forms of its solutions.
Please show that: any function of the form $$\phi(x-vt)$$ is a solution of the wave equation. Here $$\phi$$ is an arbitrary function.
Convince yourself that $$\phi(x-vt)$$ represents a shape traveling to rightward with speed $$v$$.

Next, follow the derivation of wave equations of the electric and magnetic fields, starting from Maxwell's equations. (p.71-72)

These are wave equations, but they are for vectors $$\vec{E}$$ and $$\vec{B}$$. Also, they depend on three spatial dimensions, not one. So they are more complicated than the wave equations you might have seen in your Vibrations & Waves'' class. We will think about the solutions to these equations (electromagnetic waves) in the following lectures.

Here is a video of someone working out the derivation of wave equations from Maxwell's equations.
Of course, this derivation is also worked out in almost any textbook on Electromagnetism, for example, try Section 4.3 (page 82) of these lecture notes.

Virtual lecture'' for Tuesday April 7th

Today we look at the vector potential and the scalar potential, and the gauge freedom enjoyed by these potentials.

In my my hand-written notes (p.61-72), this material starts on p.63 and ends on p.66, about one-third into page 66.

The magnetic field can be written as the curl of a vector potential, because the divergence of the magnetic field is zero.

In electrostatics, the electric field could be written as the gradient of a scalar function, which is the negative of the electric (scalar) potential. This was because the curl of the electric field is zero in electrostatics. But, because $$\vec{\nabla}\times\vec{E}$$ is not zero for non-static situations, as you know from Faraday's law. So, we need a correction to $$\vec{E}=-\vec{\nabla}V$$! The correction term is explained in p.63. At the top of p.64 are the important equations expressing the fields in terms of the potentials.

Adding certain kinds of terms to the potentials might keep the fields unchanged. These are known as gauge transformations. Please make sure you show yourselves that the fields are unchanged when the potentials are changed in this way.

The freedom can be restricted by choices of gauge. Two popular choices are the Coulomb gauge and the Lorentz gauge, p. 65. Also shown is what the 1st and 4th Maxwell's equations look like, in the cases of these two gauges.

If the fields are derived from potentials, than the 2nd and 3rd Maxwell's equations are automatically satisfied. Please make sure you convince yourself of this statement.

Of course, the scalar and vector potentials are described in many textbooks and notes. As additional reading, you could try
- Lecture notes describing Maxwell's equations and then describing the potentials. (uses $$\phi$$ instead of V for the scalar potential, which is quite common). There is also a page describing gauge transformations.
- Section 3.4 (p.22-23) of these notes.

Virtual lecture'' for Friday April 3rd

Here are pages 61 to 72 of my hand-written notes. These contain material for this lecture and the next few lectures.

Today's lecture will be a review: (1) of electromagnetic induction (2) of Maxwell's equations.

Review electromagnetic induction: Faraday's law and Lenz's law

Please review how changing magnetic fields create curly electric fields. Look up or tell yourself:
- How Faraday's law gives you the EMF around a circuit;
- How Faradays's law can be expressed as an integral equation that can be used even when there is no wire circuit physically present;
- How you can derive Maxwell's 3rd equation, which is Faraday's law in differential form, from Faraday's law in integral form;
- How Lenz's law gives you the direction of the induced current or the induced electric field.

You could use the material I posted or pointed to in for the last two lectures. Or, if you want something fresh, you can try one/some of the following.
- Chapter 8 from these notes (shows lots of nice examples with diagrams);
- The last 4 pages (23-26) from these lecture slides;
- Section 4.1 and Subsection 4.1.1 from these notes;
- Chapter 16 of Feynman II.

Review Maxwell's equations

This is the first 2+1/2 pages of the my hand-written notes (p.61-72), i.e., up to the first paragraph of p.63.

You should now be extremely familiar with these equations. Please go through them again. Make sure you know exactly where each equation comes from. Please also make sure you can go from integral form to differential form, for each equation. The displacement current density (Maxwell's correction to Ampere's equation) is now given its own symbol, $$\vec{J}_D$$.

Next time we will look at scalar and vector potentials. These are intimately related to the structure of Maxwell's equations.

Virtual lecture'' for Tuesday March 31st

For this lecture, we want to
(2) Appreciate that there are two types of electric fields;
(3) Summarize and review Maxwell's equations;
(4) Introduce and study Lenz's law.

Review of electromagnetic induction, Faraday's law

Electromagnetic induction is the phenomenon that a changing magnetic field can create a curly' electric field. We will be building on electromagnetic induction and Faraday's law; so please review. Make sure you are familiar with both the integral form and differential form of Faraday's law.

On page 55, we note that Faraday's law in integral form has a very similar form to the Ampere's law in integral form. Ampere's law quantifies how a curly magnetic field is created due to a current density $$\mu_0\vec{J}$$. In contrast, Faraday's law tells you how a curly electric field is created due to a changing magnetic field, $$\left(-\frac{\partial\vec{B}}{\partial{t}}\right)$$.

Two types of electric field

We have encountered two types of electric fields!

In electrostatics, we had electric fields created by charges, pointing straight out or straight in from the charges. The electrostatic field had zero curl!

But electric fields can also be created by induction, i.e., by a changing magnetic field. This can even work in a complete vacuum, without any charges around. The electric field created in this way curls around. The curl is definitely not zero.

Maxwell's equations

Time to review all of Maxwell's equations. Please name or describe, to yourself, the law or phenomenon leading to each equation.

An exception is the last term in the 4th equation. This did not come from an observed phenomenon. Rather, Maxwell derived it theoretically in order to make Ampere's law consistent with the continuity equation. Please make sure you can repeat that derivation. This term (Maxwell's correction) is known for historical reasons as the displacement current.

Lenz's law

After all the reviewing, now we come to the new' part of this lecture. Lenz's law helps determine the direction of the induced electric field.

Please go through the examples. The law is a bit tricky and clumsy to state, so this will take some effort.

Note, you don't need a physical wire loop to have electromagnetic induction. The example with the solenoid discusses this. If the magnetic field inside the solenoid is changing (due to changing current through the solenoid) then there will be electric fields generated outside (and inside) the solenoid. You could calculate the magnitude using Faraday's law in integral form. You can infer the direction by imagining a wire loop surrounding the solenoid.

You can watch someone go through the geomerty of several examples of Lenz's law, in this video, in this video, in this video.

Virtual lecture'' for Friday March 27th

In this lecture, we introduce the phenomenon of electromagnetic induction and learn Faraday's law.

Here are my typed notes for March 27th.

This is reasonably self-contained. Please let me know if you spot typographical or spelling errors, or something seems unclear.

A couple of example situations for induction are described and sketched in the notes.

However, there are many other thought experiments (or real experiments) one can think of that leads to electromagnetic induction. It might be instructive to go through a few. For additional examples, you could try:

Also, here is a webpage covering similar material as my typed notes.

Virtual lecture'' for Tuesday March 24th

For this lecture, we want to (1) go through applications of Ampere's law, (2) Introduce the magnetic vector potential.

Applying Ampere's law: long current-carrying wires

This is in p.43-44 of my hand-written notes for March 13th.

These calculations are sort-of reminiscent of what we did using Gauss' law.
In that case we found electric fields outside and inside charged objects, using the surface integrals over closed surfaces that we called Gaussian surfaces.
Now, we are finding magnetic fields outside and inside a current-carrying wire, using the line integrals along closed curves. Such closed curves, on which Ampere's law is used, are called Amperean loops.

If you want to watch/listen to someone discuss this: you can try

Applying Ampere's law: solenoid

This is in p.45 of my hand-written notes for March 13th.
This page is not as clearly written/drawn as I would have wished. So, please supplement this by reading the argument in a textbook or typed notes!
You can try
Feynman lectures II: Section 13-5,
Chapter IV of Prof. Nash's Notes,
These notes.

The result is that, if current flows through a long solenoid, it creates a magnetic field that is zero outside the solenoid, and uniform everywhere inside the solenoid. Please make sure you are able to derive this result, and the value of the magnetic field.

The vector potential

You can find roughly equivalent material typed up
in these notes,
Feynman lectures II: Section 14-I,
this video.

Virtual lecture'' for Friday March 13th

On the lecture of March 13th, we were planning to (1) introduce Ampere's law; (2) review the main equations/laws we have learned till now; (3) start with applications of Ampere's law.

Here is a scan of my hand-written notes.

Please work through pages 39 through 44. Most of my handwriting should be legible. If not, please ask.

Ampere's law:

Ampere's law is another way of expressing how currents create magnetic fields.
(In electrostatics, Gauss's law complements Coulomb's law and expresses the same physics more elegantly. In the same way, Ampere's law complements the Biot-Savart law and expresses the same physics more elegantly.)

My handwritten notes do not always have complete sentences. For a textual discussion of the material on Ampere's law, you could try working through:
Feynman lectures II: Sections 13-4 and 13-5.
Chapter IV of Prof. Nash's Notes.

This material is also discussed in many other books and lecture notes. For example, here are writeups on
applications of Ampere's law and on applications of the Biot-Savart law

Review of Equations/Laws + extension of Ampere's law:

In my handwritten notes, on page 40 and 41, I provide a list of all the equations and laws we have encountered in electrostatics and magnetostatics. Please make sure you go through all of them and explain to yourself the meaning of each. A good practice is to draw the relevant situation in each case. You already met almost all of these equations, except for the equation expressing the magnetic field as a curl of the so-called vector potential''. More on that in later lectures.

On page 42, Ampere's law is argued to be not consistent with the continuity equation when the system is not in steady state. In this case, a correction is needed. This is called Maxwell's correction. In Feynman II, this is discussed in the beginning of Chapter 18. Please make sure you understand and can explain the inconsistency. Also, you should understand and be able to explain how the correction term resolves the inconsistency.

Finally, if you want to have someone explaining some of this on the board, you could try:
this video,   this video,   this video,   this video.

(Solutions to) previous exams   +   Sample Exams

Here is a sample exam for practice:   Sample exam 1, for 2018-2019

Here is the 2019 May exam and here is the 2019 Repeat (August) exam. (Solutions are not available; sorry.)

Below are solutions to some past exams.
(The length of exams has changed since 2017.)

Below are old sample exams for practice. They are in the style of previous (2017-2018) exams. The 2018-2019 exams was structured slightly differently (divided into 4 questions instead of 3), but the material covered and the level of difficulty should be similar.

Material covered in Class

I will point to relevant chapters in Prof. Nash's Notes and in Vol. II of the Feynman lectures (referred to as Feynman II below).
Of course, equivalent material is available in many other textbooks, or in online material such as those linked to further down on this page.

Chapters 16 and 17 in Feynman II.   Nash notes: chapter V.

Vector Potential.
Chapter 15 in Feynman II.   Unfortunately, Nash notes do not discuss the vector potential.

Magnetic field.
Chapters 13 and 14 in Feynman II.   Nash notes: chapter IV.

Electric Currents.
Chapter 13 in Feynman II.   Nash notes: chapter III.

Applications of Gauss' law.
In Feynman II, Chapter 5 is highly recommended reading.   In Nash notes, this is chapter 2.

Electric flux. Gauss' dielectric flux theorem (a.k.a. Gauss' law).
In Feynman II, the discussion of flux begins in Chapter 4 Section 5, and continues through Chapter 5.   Nash-notes covers this material in Chapter 2.

Continuous charge distributions. In class, we worked out how to calculate the potential and the electric field due to a continuous distribution of charge by first calculating the contribution due to an infinitesimal element and then integrating (adding up''). This is an important technique which will recur throughout this module; please make sure you are able to set up integrals like this yourself.

Coulomb's Law, Electric Fields, Electric Potentials. Chapter 1 of Nash-notes. The introduction to the electric potential in Nash-notes Chapter 1 Section 3 is more detailed than we had time for in class; you might want to read this carefully.
In Feynman lectures Vol. II, you will find similar material in the first 4 sections of Chapter 4.

Overview and Background. In Feynman lectures Vol. II, Chapter 1 gives an overview of what we will learn this semester.
Chapters 2 and 3 introduces grad-div-curl and vector integration. You are supposed to know most of this material already. Working through them will be a great help for MP204.

Prerequisite: Vector Calculus

This module requires you to be very familiar with Vector Calculus. You should be comfortable with grad/div/curl, Stokes' theorem and the divergence theorem, and of course vector addition and components.

If you need a review, you can try working through some of the following. I strongly suggest making time to do this at the beginning of semester.

Material, sources

Lecture notes from a previous lecturer

MP204 lecture notes of Prof. Charles Nash --- this is roughly the material to be covered in the module, with some additions. It is recommended that you work through these notes, and in addition spend significant time working through at least one textbook.

Textbooks

There are many, many textbooks on introductory electromagnetism or electrodynamics. You are strongly encouraged to read through one or more textbooks.

In particular, I suggest working through the Feynman lectures (Volume II), which are free to read on this website. The material we will cover in MP204 is mostly contained within the first 20 chapters of Volume II. (Specifically: Chapters 1, 4--6, 13--18, 20.) This will be very close to what we will cover. However, the material is very standard and you will find the same topics in many other texts.

Other texts:

• Fleisch, A Student's Guide to Maxwell's Equations.
Student-friendly, as the title suggests.

• Griffiths, Introduction to Electrodynamics.
Slightly more advanced than the level of this module, but reading through this text and working out exercises is very worthwhile.

• Purcell & Morin, Electricity and Magnetism.
Slightly more advanced than the level of this module.

• Edminister & Nahvi, Schaum's Outline of Electromagnetics.
Many worked-out examples.

• Panofsky & Phillips, Classical Electricity and Magnetism.

• Grant & Phillips, Electromagnetism.

• Shankar, Fundamentals of Physics II: Electromagnetism, Optics, and Quantum Mechanics.

Material available online:

Lecture notes from various places.
Of course, I didn't check in detail for correctness and/or how closely these notes are aligned to the matter we cover in MP204, so please use at your own discretion.