# Analogy between E and B fields

1. Jun 15, 2008

### Nick89

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...

2. Jun 15, 2008

### Crosson

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.

3. Jun 15, 2008

### Nick89

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

4. Jun 15, 2008

### G01

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: Jun 15, 2008
5. Jun 15, 2008

### Nick89

Thanks that seems like an interesting read!

6. Jun 17, 2008

### Phrak

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...

7. Jun 17, 2008

### Gear300

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.

8. Jun 18, 2008

### Crosson

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.

9. Jun 18, 2008

### Phrak

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.

10. Jun 19, 2008

### Physicspsycho

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

11. Jun 19, 2008

### Nick89

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)...