[The_X]

I'm reviewing for final exams and I'm having some trouble on one particular problem. I thought some physics/math majors might be able to help me out with this one.

We're supposed to consider a stationary infinite line of electric charge. In the rest frame there is no current or magnetic field associated with this. However, in a moving frame there suddenly appears a B and I. I did the calculations and apparently both of these depend on beta and gamma, but I have absolutely no idea why this happens. The question is basically asking us to explain why there is a current and magnetic field in the moving frame but not in the rest frame.

Anyone?

Anyone?

Bueller?

 

[Arainach]

Stationary electric charges don't generate a magnetic field - moving ones do. Likewise, current is merely moving electric charge. So, in the moving frame of reference, the magnetic field and current appear.

 

[kvant]

Well, it's quite some time since I played with Maxwell's equations, but the thing is that Lorentz transformations do mix electric (E) and magnetic (B) fields together. It can happen that in some frames one of them completely disappears and vice versa. It is the same thing as mixing time and space while changing from one frame to another. If I remember correctly, the four-vector of elmag. potentials (phi,Ax,Ay,Az) transforms in the same way as four-vector of space-time coordinates (t,x,y,z).

More intuitively, static charge induces only electric field, to get non-zero magnetic field you need a current (i.e., moving charge). In your rest frame there is only static charge, in the moving frame you have a current. One doesn't even need relativity to see this efect. It's just that relativity (Lorentz transformations) make it consistent.

I might try to go into some formulas, but it's a bit too late for it today.

 

[trains are bad]

If they are looking for a more basic explanation, think of it this way.

I'm sure you know that a line of stationary charge creates an electric field. Radial, drops off as 1/r, all that.

And, that a current-carrying wire (or line of moving charge) creates a magnetic field described by the Biot-Savart law, right hand rule, iron filings on a piece of paper, compass held next to wire, all that good stuff.

Well, in the moving reference frame, the line of stationary charge looks like a line of moving charge.

 

[circularlogic]

It is easier if you pretend the frame to be the perspective of some point-charge, let's say it's an electron. In the still frame, you can glean from classical physics that the electron will feel no B and I since the electron isn't moving. In relativistic terms, just as trains are bad stated, the electron is stationary relative to the charge line. Now begin to move the frame [read: electron] and in classical physics you now have a moving electron near a charge line, hence B and I are no longer 0.

 

[configure]

As I recall, it goes something like this (essentially what five or so people have said just before me

)

Electromagnetic force is 'Lorentz-invariant', that is, it is the same in every reference frame. In fact, the Lorentz transformations were deduced by Lorentz by looking at electromagnetism. Anyway, so consider the stationary frame of the line of charge. No magnetic field, and only an electric field generated by the charges.

Now, consider an inertial frame moving with respect to the line of charge (with a velocity along the line, that is). In that frame, Lorentz contraction plays havoc with the line of charge - its not the same, is it? Now there's more charge per unit length. And so, to compensate, a magnetic field springs up; sort of like its doing the book-keeping.


Frankly, its worth it to just scrap all understanding of EM at some point and just go about it via the Field tensor approach. But then again, that's utterly useless unless you've broken your head on the vector approach.


Anyway, good luck with your final!

 

[The_X]

Thanks guys. I think what Configure said is essentially what I need. Because the charge density changes the field appears... I guess this would just be due to conservation of energy, right? I'll have to see if it mathematically works out by checking up on whether my equations show lambda changing.