Preparing for Electricity and Magnetism (advanced, undergraduate level)

[LightningXI]

Hello PF,

I am a few weeks from starting my second year as an undergraduate. This fall I will be taking the Electricity and Magnetism course for physics majors (crosslisted for graduate students). Last spring I took the introductory physics course on electrostatics and magnetostatics. The latter and multivariable calculus are the two prerequisite courses to take before E&M; I took multivariable last fall.

I have emailed the instructor about enrolling into this course, and he considers that E&M will certainly be a challenge due to my relatively short exposure to physics. He advised me the following (we will be using David Griffith's 3rd ed. Introduction to Electrodynamics):

I would study the first chapter on vector analysis. If you have really internalized the way the fundamental theorem of calculus generalized to two and three dimensions, you are in a good position. We'll review this at the beginning of the semester. Then you could study the beginning of chapter 2.

Good news is I have managed to grab a copy of the textbook in the nearest library.

Because I was working over the summer in a physics lab/internship, I have just started to go over my calculus, since I have mainly forgotten some of the details. Good thing is I have retained most of the concepts, but again, not in full detail. I will be engaging in intensive study sessions not only to review, but also to self-study/learn before the start of the semester.

I have noticed that the "surface and volume" integrals and the "Fundamental Theorems (of calculus, of gradients, Green's -- divergences, and of curls -- Stokes')" are used a little differently from the strictly math-oriented multivariable calculus course or proof-based linear algebra course (which I took last spring). For instance, my multivariable course mainly involved parametric equations and vector equations (e.g. int (dQ/dx - dP/dy)) for Green's), whereas Griffith explains it geometrically. I am slightly confused, and I would really like to achieve what my instructor said, and more: "[internalizing] the way the fundamental [theorems] work [in math AND physics]."

I would appreciate any useful mathematics and/or physics resources I could use to prepare over the next few weeks. It does not matter if these resources involve more advanced concepts, just as long as you tell what I should know before going over them. Any study material ranging from recapitulated introductory to advanced levels would be great!
Thank you!

P.S. If this thread does not belong in this section, please move accordingly and my apologies for that inconvenience!
Edit: Please move to HW/Coursework! Sorry for the inconvenience.

 

[Muphrid]

My quick and dirty interpretation of the fundamental theorem of calculus is, "The integral of a function over a boundary is equal to the integral of the derivative over the region that boundary encloses." The 1D fundamental theorem, the divergence theorem, Stokes' theorem---they all follow this basic idea. Ultimately, to make these theorems useful, you'll end up using them at their lowest level--in terms of components and such. Griffiths is probably just introducing them geometrically to give some intuitive grasp of what the theorems say. Physics texts love doing this for the less mathematically inclined. Don't get hung up on it if you already do understand the underlying math.

EM theory tends to be physicists' first real exposure to applications of vector analysis and calculus, and as such I feel it can almost get bogged down in these particulars to the point it obscures the simple nature of the topic. EM theory is more or less the study of Maxwell's equations and the different fields that result from various broad classes of sources (i.e. charges and currents).

For instance,
Case: no current, only time-independent charge density -> time-independent electric fields ("electrostatics")
Case: time-varying charge desnity -> time-varying electric fields ("electrodynamics")
Case: Time-independent current density, no charge density -> time-independent magnetic fields ("magnetostatics")
Case: Time-dependent currents -> time-dependent magnetic fields

This is a simplfied model of the breadth of electromagnetism, but it should give a general idea. In general, an idea of methods to solve differential equations will be very useful: you'll see Green's functions and separation of variables, maybe even spherical harmonics and the general idea of orthogonal functions (perhaps not for an undergrad course, though).

In all seriousness, though, a large part of classical EM theory is just applications of solving PDEs. That's not to say it's easy (it's not), but the concepts introduced in a diff. eq. course are immensely useful for connecting what's going on to stuff you might already know and can work with.

 

[GarageDweller]

I think it's worth your time to learn more about the two vector field operators, how the expressions came to be and their history. This will give you a more physical idea of what they really do.
You will understand they both have great geometric meaning. The divergence at a point is the amount a field is "generated" there, the curl of a field is the stationary circulation about the vector. For example, understanding the divergence this way will make gauss' law of magnetism very intuitive.

 

[Philip Wood]

I learned my vector calculus from a textbook on Electricity and Magnetism, whose origins went back over a hundred years. The book (Abraham and Becker) is long out of print, but I still approve of the way the authors used fluid velocity, v, as their 'field vector' when introducing vector calculus. So flux (the integral of v.dA)was the volume of fluid emerging through a surface. This gave immediate motivation for Stokes's theorem and the divergence theorem. One then felt completely comfortable using them with the more abstract fields such as E and B.

 

[vanhees71]

This is indeed one of the best books ever written on E&M, particularly on the relativistic treatment of moving bodies. E.g., the explanation of the unipolar (homopolar) generator is a masterpiece! It's even somewhat better than my all-time favorites on classical physics which is by Sommerfeld (his Lectures on Theoretical Physics, which are also translated to English).

Fortunately Becker/Sauter (which is a follow-up edition of Abraham/Becker) is not out of print. The English translation is still available as a Dover book:

http://store.doverpublications.com/0486642909.html

 

[LightningXI]

Don't get hung up on it if you already do understand the underlying math.

This is precisely what I will do for now. I think that thinking strictly about the math won't help me progress through the actual physics material that Griffith really wants to talk about.

For instance,
Case: no current, only time-independent charge density -> time-independent electric fields ("electrostatics")
Case: time-varying charge desnity -> time-varying electric fields ("electrodynamics")
Case: Time-independent current density, no charge density -> time-independent magnetic fields ("magnetostatics")
Case: Time-dependent currents -> time-dependent magnetic fields

This was a nice way to put it, and it is just how I really learned the math behind Maxwell's equations in my introductory physics course. The internalizing part is costing me a bit more effort; I'm mainly struggling a bit in the use of the different theorems (despite my understanding of them) as key steps to solving, for instance, a volume element including integral that must be simplified using integration by parts, the chain rule, and/or the product rule. Sometimes things become more complicated for me with curvilinear coordinates.

In all seriousness, though, a large part of classical EM theory is just applications of solving PDEs. That's not to say it's easy (it's not), but the concepts introduced in a diff. eq. course are immensely useful for connecting what's going on to stuff you might already know and can work with.

The good thing is that I will be simultaneously taking a differential equations course, which will range beyond to second order and so on. Thank you, Muphrid.

I think it's worth your time to learn more about the two vector field operators, how the expressions came to be and their history. This will give you a more physical idea of what they really do.

The history would indeed help a bit, but I am not sure how useful it could be. The operators clearly describe aspects of physical phenomena, by definition, if I am not mistaken.

I learned my vector calculus from a textbook on Electricity and Magnetism, whose origins went back over a hundred years. The book (Abraham and Becker) is long out of print, but I still approve of the way the authors used fluid velocity, v, as their 'field vector' when introducing vector calculus. So flux (the integral of v.dA)was the volume of fluid emerging through a surface. This gave immediate motivation for Stokes's theorem and the divergence theorem. One then felt completely comfortable using them with the more abstract fields such as E and B.

Fluid velocity is an interesting and descriptive way to put it, I agree. But personally, I feel more limited by mathematical constraints, when seeing things too generally at times, like saying "fluids" or "surfaces". Because physics problems often attempt to deal with real-world-phenomena, would it be not better to actually know how these "fluids" or fields act differently with respect to the theorems? I would think that people who learn the maths generally could be used to dealing with these vector fields in that generic way, but then, when actually learning physics, they could easily confuse, for instance, the orthogonal relationships (and their effects on each other) between E and B. Are E and B, then, not "abstract" but rather, specific cases with respect to the fundamental theorems of vector/integral calculus?

This is indeed one of the best books ever written on E&M, particularly on the relativistic treatment of moving bodies. E.g., the explanation of the unipolar (homopolar) generator is a masterpiece! It's even somewhat better than my all-time favorites on classical physics which is by Sommerfeld (his Lectures on Theoretical Physics, which are also translated to English).

Fortunately Becker/Sauter (which is a follow-up edition of Abraham/Becker) is not out of print. The English translation is still available as a Dover book:

http://store.doverpublications.com/0486642909.html

Thank you for the resource! I will see if I can find it in a library nearby!

 

[WannabeNewton]

Is this class the advanced EM class one usually takes after intro EM or is it an honors intro EM class? If you have access to the book Electricity and Magnetism by Purcell then by all means check it out. It's an intro EM text usually used in honors classes at almost all unis in the US that have honors physics but you will see that if you did the problems in it a lot of the problems will repeat again in Griffiths. It is a good book to bridge the gap.

 

[Jorriss]

https://www.amazon.com/dp/0393925161/?tag=pfamazon01-20

Vector calculus taught in the context of E&M. It's better if read as a review, so it sounds just right for you.

 

[LightningXI]

Is this class the advanced EM class one usually takes after intro EM or is it an honors intro EM class? If you have access to the book Electricity and Magnetism by Purcell then by all means check it out. It's an intro EM text usually used in honors classes at almost all unis in the US that have honors physics but you will see that if you did the problems in it a lot of the problems will repeat again in Griffiths. It is a good book to bridge the gap.

This class is the advanced EM; there is no honors EM in my school. The course number is 324, which is also a crosslisted 500-level class for graduate students as 524. I will check the book out, thanks!

https://www.amazon.com/dp/0393925161/?tag=pfamazon01-20

Vector calculus taught in the context of E&M. It's better if read as a review, so it sounds just right for you.

Thank you!