# Analogy between E and B fields

Hi,

I was simply wondering this for some time now... I am constantly seeing similarities between electric (E) fields and magnetic (B) fields.

A few examples, Coulomb's law and Biot-Savart's law:
$$dE = \frac{1}{4 \pi \epsilon_0} \frac{dQ}{r^2}$$
$$dB = \frac{\mu_0}{4 \pi} \frac{I dl}{r^2}$$

They are nearly the same... Especially the constants $\mu_0$ and $epsilon_0$, they always seem to be taking eachother's place, where $epsilon_0$ is always replaced with $(\mu_0)^{-1}$ (not just in this example but everything I have ever come across, like in EM-waves etc...)

Also in the case of time-varrying E or B fields they seem to be related, you can't only look at one without considering the other anymore (maxwell equations etc)...

I have heard / read a bit about this and it seems to be that E and B fields are essentially the same, in quantum mechanics, or in (general?) relativity (or both? I dunno..)...
Is this true?

If anyone has some information on this that would be great, I'm very interested in this...
Thanks!

EDIT
I don't know if it's the same with you guys but for some reason I cannot see the latex images in my post... Seems to be something wrong...

The electric and magnetic fields are unified in special relativity.

It is possible to derive the Biot-Savart law using a thought experiment with Coloumb's law and a moving observer. When we use the Lorentz transformations of special relativity to transform coordinates, the Biot-Savart law is the result.

Is this transformation difficult to actually calculate? Could you perhaps show an example?

G01
Homework Helper
Gold Member
Is this transformation difficult to actually calculate? Could you perhaps show an example?

The transformation is not too difficult, but it could take some time to reproduce it here on the forum. I suggest trying google first, or checking out Chapter 12 of Griffth's "Introduction to Electrodynamics" if available.

For instance, here is a talk given by Daniel Shroeder from Weber State on this topic:

http://physics.weber.edu/schroeder/mrr/MRRtalk.html

Last edited:
Thanks that seems like an interesting read!

The transformation is not too difficult, but it could take some time to reproduce it here on the forum. I suggest trying google first, or checking out Chapter 12 of Griffth's "Introduction to Electrodynamics" if available.

For instance, here is a talk given by Daniel Shroeder from Weber State on this topic:

http://physics.weber.edu/schroeder/mrr/MRRtalk.html

I don't happen to have a Griffith. I do, however understand the vacum gauge symmetry of E and B. Does Griffith discuss any non-vacuum symmetry? just curious...

Electricity and magnetism are two extremes of the same phenomenon (electromagnetism). If you were to approach this quantum mechanically, with the addition of a few things, Maxwell's equations will show them to be practically identical. So then why is there a magnetic force and electric force...actually...I don't know and I have not gone into quantum mechanics...but it has something to do with charges moving, so I would assume magnetic fields occur due to moving Electric Fields...and if electric fields move, they are not always constant in a fixed frame of reference...and so magnetic fields occur essentially when there is a changing electric field...and electric fields occur with changing magnetic fields, which is apparently symmetric.

I don't happen to have a Griffith. I do, however understand the vacum gauge symmetry of E and B. Does Griffith discuss any non-vacuum symmetry? just curious...

Yes, we are not talking about the vacuum guage symmetry, but rather a process of deriving the magnetic field of a infinite line of current using the lorentz transformations and coloumb's law.

Yes, we are not talking about the vacuum guage symmetry, but rather a process of deriving the magnetic field of a infinite line of current using the lorentz transformations and coloumb's law.

Yes, of course. My mistake. Checking my notes, the gauge doesn't enter into it, apparently. It seems however, that if
$$dE = \frac{1}{4 \pi \epsilon_0} \frac{dQ}{r^2}$$
is expressed in a coordinate independent form using
$$d*F=-*J$$ where J is a 1-from,
$$dB = \frac{\mu_0}{4 \pi} \frac{I dl}{r^2}$$
will be included in the same equation, where the Lorentz transform is implicit. I can't simply tell by looking as these two equations if that would be the case.

Another one

There's Another similarity between the Gauss' And Amperean laws

Integral(closed surface) E.ds = q(enclosed)/Epsilon nought

Integral(closed loop) B.dl= I(enclosed)/Mu nought

I know, there are loads, and it's obvious from the maxwell equations that time-varying E and B fields are physically connected. However it is not directly obvious that they would be the same thing (if you look at it in the right way)...