Magnetic Field of a moving Electron

[Savio]

Hello Everyone!

In School, Approximative Equations for the fields of capacitors and coils are thrown at us Students; without any proof or explanation. Because I'm not really satisfied with that, I decided to try to calculate the real fields on my own.
For the Capacitor, I have Coulombs law to integrate over two plates. But what is the analogous law for Magnetic Force?
What I found about magnetic fields in books and the internet, was only about more complicated forms and not about Point charges. How can it be that something so elementary seems to be not mentioned anywhere?
So my question is:

What is the magnetic force, which a moving electron exerts on another moving electron?

Thanks and best regards
Savio

PS: please excuse my english, I'm from germany ...

 

[Bill_K]

The field of a particle sitting still is the Coulomb field. So write that down and call it E. Now ask how the electromagnetic field changes when you go to another rest frame. You'll find that in a rest frame that moves with velocity v, the magnetic field resulting from that Coulomb field will be B' = - γ v/c x E.

 

[epsilonjon]

So my question is:

What is the magnetic force, which a moving electron exerts on another moving electron?

If you've looked in school at the electric field you'll know that the direction of it (for positive q) is from source to point, with the magnitude proportional to the charge magnitude \left|q\right| and to \frac{1}{r^{2}}.

Experiments show that the magentic field is also proportional to these two things, but the direction is no longer along the line from source point to field point. Instead, \vec{B} is perpendicular to the plane containing this line and the particle's velocity vector \vec{v}. Furthermore, the magnitude of B is proportional to particle's velocity and also to the sine of the angle between the line from point to source and the velocity vector.

Putting all this together we get:

\vec{B}=\frac{\mu_{0}}{4\pi}\frac{q\vec{v}\times\hat{r}}{r^{2}}

where \hat{r} is the unit vector that points from the source point to the field point, and \frac{\mu_{0}}{4\pi} is a proportionality constant.

This diagram shows nicely the way the B field looks:

[PLAIN]http://img12.imageshack.us/img12/316/18823195.jpg

If you then have another charged particle moving in this B field, the force exerted on it by this field is

\vec{F}=q\vec{v}\times\vec{B}.

You can see from all this that you need to be comfortable with vectors and cross products, which is perhaps why the books you are reading are not covering it? How is your math in this area?

Hope I've helped a bit,
Jon.

 

[jtbell]

You'll find the formula for the magnetic field produced by a moving charge near the end of this page:

Fields due to a moving charge

Combine this with the general formula for the force exerted by a magnetic field.

 

[Savio]

Thanks for the answers!

The field of a particle sitting still is the Coulomb field. So write that down and call it E. Now ask how the electromagnetic field changes when you go to another rest frame. You'll find that in a rest frame that moves with velocity v, the magnetic field resulting from that Coulomb field will be B' = - γ v/c x E.

I am not familiar with special relativity. So I'll need the Lorentz transformation? When I have some time, I will try it.

Experiments show that the magentic field is also proportional to these two things, but the direction is no longer along the line from source point to field point. Instead, \\vec{B} is perpendicular to the plane containing this line and the particle's velocity vector \\vec{v} . Furthermore, the magnitude of B is proportional to particle's velocity and also to the sine of the angle between the line from point to source and the velocity vector.

Putting all this together we get:

\\vec{B}=\\frac{\\mu_{0}}{4\\pi}\\frac{q\\vec{v}\\times\\h at{r}}{r^{2}}

where LaTeX Code: \\hat{r} is the unit vector that points from the source point to the field point, and LaTeX Code: \\frac{\\mu_{0}}{4\\pi} is a proportionality constant.

Thank you! This and the diagram are exactly what I searched.

You can see from all this that you need to be comfortable with vectors and cross products, which is perhaps why the books you are reading are not covering it? How is your math in this area?

I know the definition of the cross product, but I have not really worked with it. I think this is a good time to concern myself a bit more with this.

You'll find the formula for the magnetic field produced by a moving charge near the end of this page:

Fields due to a moving charge

Combine this with the general formula for the force exerted by a magnetic field.

This looks interesting, but although I read something about tensors some time ago, I do not really understand them, so I'm afraid I can not follow your link at the moment.

 

[jtbell]

He just uses tensors for the derivation. The final equation is #1539 near the bottom of the page, for which you need only the vector cross-product.