# Welcome to the course!

Welcome to the FYS1120 course, where we will guide you through the main topics of electromagnetism. To prepare you for the coming weeks, we’re starting out building your tool box with most of the mathematical background you’ll need in this course.

Every week we’re posting a post like this with exercises and some information about this week’s topic. We’re also sometimes including a zip file containing the Python scripts you’ll need to perform this week’s numerical and visualization exercises.

Knowledge about vectors and vector fields are important in the study of electromagnetism, as well as knowing how to analyze them. We’ll encounter concepts such as flux, divergence and curl which might not be familiar to all of you, but are essential to our exploration of electromagnetic phenomena. Three notes have been written as an introduction to these concepts and to prepare you for this week’s problem set:

This year we have chosen to use Python and Mayavi as core tools in this course, so you will also need to learn how to install and use these tools. Therefore, we’ve written a note on how to install Python and Mayavi and a note on how to use them:

While the problem set itself is found here:

We will also post exercises for week 1 later this week, so it is very important that you start on the mathematical warm-up set, as soon as possible.

# Note on RLC circuits

Atle has written very helpful note on RLC circuits that could be pretty useful as a repetition before the exam. The book is not soo thorough on how to work with RLC circuits and does some parts of the text on specialized cases that are not necessarily working out in general. Reading about how to solve such circuits in the final pages of this note can turn out extremely valuable on the exam.

Enjoy!

# Online lectures for repetition from MIT

If you’ve missed out on some lectures this semester or need some repetition for central subjects, we recommend having a look at Walter Lewin’s lectures from MIT:

Walter Lewin’s lectures on Electricity and Magnetism (MIT)

They do not coincide to our syllabus in all areas, but some lectures are very relevant. For instance there are lectures on magnetism, Lorentz’ force, LRC circuits and much more. I guess you could tell which ones you need or want to watch from the titles.

Spending some time watching a few lectures when you have spare time or are tired of reading can help you grasp concepts that you find hard to understand, and could be very inspiring!

Should you find some lectures very useful, don’t forget to tip your fellow students about it by leaving a comment below this post.

Enjoy!

# Superconduction

As promised by Finn in the lectures, here is his note on superconduction (in Norwegian):

# Motivational note for the labs

The lab is just around the corner and many of you will maybe experience lab work at the University for the first time. As with any lab, preparation will help you understand more and give you more out of the day at the lab. Therefore, Jørgen Høgberget has written this motivational note for the labs.

In addition, we encourage you to have a look at the lab texts that are published here.

### General thoughts

The idea behind this note is to clarify the concepts introduced in the lab, as well as to tie the perhaps cryptic values and concepts to the more familiar ones in electromagnetism. Some important questions to ask are: Why are we interested in these values? Why are we performing the measurements, and what do they tell us?

The lab is just around the corner.

Electromagnetism is, like gravity, a fundamental force of nature. You may find gravity obvious; you feel it every day. It is what keeps you grounded. Electromagnetism, however, is slightly harder to wrap your head around; charge is less intuitive than mass! You are not surprised when an apple falls from a tree, but when a beam of water is bent in the presence of a glass rod, there is a more magical mood to it.

Is gravity what keeps your skin together, your bones from breaking, etc.? No. It is the charge of molecules, properties of atoms etc. A lot of the explanations to these things have their roots in electromagnetism. Electromagnetism is more than shining light, it explains a lot of the properties of i.e. the materials most of our modern technology is based on. This is the main reason why electromagnetism and lab-work go hand in hand.

### Lab 1

We are all familiar with electronics. We were introduced to currents and potentials by adults telling us not to put our fingers in the electric circuits of our home. This is, at least to me, one of the reasons why the theory of electronics can be hard to grasp; we already have a picture of it! This is why it is very important to have an open mind when it comes to this.

The inner resistance in a voltmeter is not really caused by a resistor.

#### Exercise 1,2 and 4

Internal resistance. What a word. Take for example the voltmeter. Does it have a resistor inside it? Could we not just replace this one with an insulator and have perfect measurements? No, sadly this is not the case.

The voltmeter is a complex thing, however, it is a general theorem, called Thevenin’s theorem, which states that any resistive circuit or network, no matter how complex, can be represented as a voltage source in series with a source resistance.

It is this source resistance we are interested in. The complex electronics of the voltmeter sets up what is interpreted as a resistance by the current flowing in the circuit. It is big, but far from infinite, as this would block the voltmeter from working at all. It is therefore necessary to know the value of it, so that we can control that the error in our measurements are not too big. This goes for all measurements, no matter the field.

#### Exercise 4

The conductors you face in early level electromagnetism are \textit{ideal} in the sense that the current flowing through the material do not loose energy on the way in terms of heat.

At room temperatures in the lab, however, the circumstances are not ideal. The Peltier element exploits this concept of heat transfer by the so-called Peltier effect, where two different materials will transfer heat, or energy for that matter, between each other. This transfer is directly proportional to the current.

If we instead look at it the reversed way: We warm up one of the materials, and cool the other one. This way we induce a heat transfer. If the Peltier effect is reversible, this would imply that there is generated a current in the loop between these materials. It works like a battery! This is electromagnetism in the shape of thermodynamics!

#### Exercise 6

Superconducting is a phenomenon where electromagnetism acts really strange. This weirdness is much like  what scientists discovered when studied the movement of atomic particles. This strange behavior is described (at least attempted) by the weird Quantum Mechanics. So prepare to see some quantum effects with your own eyes! (The Quantum perspective of Electromagnetism is called Quantum Electrodynamics.)

### Lab 2

Magnets are fascinating objects. Have you ever asked yourself why we have magnetic fields? They are generated by charges, but so are electric fields. Why do we distinguish between them? To demonstrate this phenomenon, it is crucial to know that they are generated by moving charges. The relation between an electric and magnetic field is what triggered Einstein to work on his theory of special relativity. One interpretation of the theory is that the dynamics of moving objects gets corrections to make sure that the speed of light is not broken. See where I’m going?

If the next part confuses you, do not worry, it’s beyond the scope of this course!

Consider that you are observing a stationary charged rod while standing still. It’s electric field is given by Coulomb’s law, and the force on a test charge $q$ in this field (from the Lorentz force) is $\vec F = q(\vec E + \vec v\times \vec B) = q\vec E$.

However, let’s move to the left with a constant speed $v$. From our point of view it looks like the rod is moving. From theory we now know that a magnetic field is circling around the rod. But we didn’t do anything spectacular? This seems a bit too magical.

However, like the Lorentz transformations of time and position (i.e. $t_v = \gamma t_0$), the Lorentz force is also invariant of whether or not the rod has a velocity from our point of view. This means that the second term $\vec v \times \vec B$ arises as a relativistic correction to the field from the stationary rod! What you are seeing with your own eyes when you are doing this lab, and when you are guided by a compass for that matter, is the manifestation of special relativity!

This is what an hysterisis curve looks like. In this lab you will make one of your own.

#### Exercise 1

Susceptibility, what a word. It is derived from the fact that the permeability $\mu$ of a material is not equal to the one of vacuum $\mu_0$. Working with $\mu$ and $\mu_0$ are a practical way of using the equation $\vec B = \vec B_0 + \mu_0\vec M$. The ratio between the material’s permeability and the vacuum’s, is used as a measure for this difference: $K_m = \frac{\mu}{\mu_0}$. (This quantity is called the relative permeability.) Since this ratio always stays close to 1, and in some interesting cases extremely close to 1, one usually tables the susceptibility $\chi = K_m -1$ (This can also be expressed as $(\mu-\mu_0)/\mu_0$ and can be viewed as a relative difference between the two permeabilities.)

In other words: It’s a convenient way of expressing how a magnetic field will be different in a “magnetic” material ($\vec M \ne 0$) than in a non-magnetic one. Good luck!

#### Exercise 2

This exercise lets you use your knowledge about solenoids to measure the strength of a magnet. This way of measuring, by simply observing the effects the system of interest has on another system, is a grand example of the strength of physical laws. We can say something about the magnet without even touching it!\footnote{Imagine having to put a thermometer into the sun’s core to measure it’s temperature…} All we need is something to relate the behavior of the solenoid with the properties of the magnet, namely Faraday’s law: Beautiful!

#### Exercise 3

Ferromagnets, without the applied field these materials are not magnetic at all. However, when an external field is applied, the magnetic moments of the atoms inside line up with the field making it magnetic. The interesting thing is that it will stay this way unless we apply a magnetic field in the opposite direction to neutralize it. This behavior is called hysteresis. It is analogous to mixing acid and base in water.

Just like in the previous case, once the material is magnetized, it will alter the magnitude of a magnetic field inside it. The most common way of picturing how the magnetization of the material is altered by the application of a magnetic field is by drawing hysteresis loops. How to read these I will leave as an exercise for the lab, but I assure you it’s very insightful.

### Lab 3

The Hall effect was discovered by the American Edwin Hall in 1879, around the same time Maxwell was working on his equations. 50 years earlier Gauss, Faraday and Ampere published their laws, laying the mathematical framework for the electromagnetism. It represents the perfect interplay between magnetic and electric fields, charges, (semi)conductors and potentials.

In the same way as in lab 2, we can determine the magnetic field (without touching the magnet) by predicting the magnetic field’s interaction with the semiconductor using physical laws. This method is called Hall probing, and is a commercially used method. Can you do lab 2 using this method? If so, which of the methods did you prefer?

# Exercises from last year

If you want to do more or other exercises than the ones provided in the weekly problem sets, you may want to check out last years exercises. There has also been published a solution manual for all these exercises:

# Note on the magnetic vector potential

Magnetic dipole (Source: Wikipedia*)

In electrostatics we found it very convenient to introduce the concept of the electric potential. It gave us a straight forward way of calculating electric fields without doing any vector calculations or using any symmetry arguments. Can we introduce something similar for magnetic fields?  It turns out that because magnetic fields are divergence less we can find a vector potential who’s curl gives us the magnetic field! Even though this magnetic vector potential is not as useful as the electrostatic potential in elementary applications, it turns out to be of major importance in electrodynamics as well as classical mechanics and quantum mechanics. It might therefore be a good idea to get familiar with the concept and some of it’s properties already, especially if you are taking a degree in physics. In this note I explain how to find the vector potential, the concept of a gauge transformation and it’s fundamental equations relating it to currents in both electrostatics and electrodynamics. Read more here:

*

# Magnetic fields from solenoids

Finn Ravndal has written a note on the magnetic field from solenoids. In this note Finn explains how to calculate the field at different points in space around and inside the solenoid. This can come in very handy for those of you who still feel a bit unfamiliar with the concept and for those who want to learn a bit more.

Notat om spoler

# Nanoparticles: An after-exam bonus

I guess you are pretty exhausted after the midterm exam this week and a lot of you have probably had other exams to deal with as well. What better time to share a game with you to put your minds at ease for a little while.

The game is called Nanoparticles and was written by me about half a year ago. It is based on Coulomb’s law with a few modifications and Newtonian physics. But instead of having you calculate anything, the game does it all for you in the background. You can download the game for Ubuntu, Linux, Windows and Nokia phones here (Mac and Android versions will be available sometime in the future):