Magnetism and radiation as a consequence of SR

[jainabhs]

Please refer the doc attached.
The document is available at http://physics.weber.edu/schroeder/mrr/MRRhandout.pdf

It explains magnetism and radiation as a consequence of SR.
In the section for magnetism, it is depicted that, with respect to test charge Q, the distance between the -q charges gets contracted. That is fine. But why do they say that distance between +q charges has increased. Why is that?
According to me, only the distance between -q charges should decrease and the distance between +q charges should remain constant as they are not in motion w.r.t test charge Q. And length contraction for -q charges is sufficient to explain charge imbalance in the conductor,in turn, applying attraction to test charge Q

Please explain...thanks in anticipation

 

[pervect]

Here's how you work out the charge distribution. Assume an ideal wire, so that its potential is zero. A real wire will have some small resistance which complicates the analysis, but it can be safely ignored by setting the resistance to zero and eliminating this complicating factor.

Solve for the electric field given the boundary conditions that the potential of the wire is zero. This is very easy - the electric field is the gradient of the potential, and the potential can be taken to be zero everywhere (on and off the wire). Note that this makes an additional assumption - that the wire started out with no net charge in order for us to say it had a zero potential.

So now that we've established that the electric field around a perfectly conducting uncharged wire is zero regardles of whether it's carrying currnet, you can find the charge densities and how they transform by Gauss law. The correct solution for charge densities will have no electric field in the laboratory frame. Thus in the laboratory frame, the two charge densities must be equal. This leads to the results described in the paper when viewed in the rest frame of the wire.

 

[Entropia]

Thank you for your question. The document you have provided is a handout by Dr. Daniel Schroeder from Weber State University, which discusses the effects of special relativity (SR) on magnetism and radiation. In order to understand why the distance between the +q charges increases in the presence of a test charge, it is important to understand the concept of length contraction in SR.

Length contraction is a phenomenon that occurs when an object moves at high speeds relative to an observer. According to SR, the length of an object in the direction of its motion will appear shorter to an observer in a different frame of reference. This effect is a consequence of the fact that the speed of light is constant for all observers, regardless of their relative motion.

Now, let us consider the scenario described in the document where a test charge Q is moving with a constant velocity v relative to a conductor with two stationary charges, +q and -q. As Q moves closer to the conductor, the distance between the -q charges appears to decrease due to length contraction. This is because the -q charges are moving towards Q and are therefore affected by length contraction. However, the +q charges are stationary and not affected by length contraction. So why does the document state that the distance between the +q charges increases?

The answer lies in the fact that the distance between the +q charges increases from the perspective of the test charge Q. In other words, the distance between the +q charges appears to increase for Q because it is moving towards them at a high speed. This is a consequence of length contraction and can be explained using the Lorentz transformation equations in SR.

In conclusion, the document is correct in stating that the distance between the +q charges increases from the perspective of the test charge Q due to length contraction. This effect, along with the length contraction of the -q charges, results in a net charge imbalance in the conductor and leads to the observed attraction between the test charge Q and the conductor. I hope this helps to clarify your confusion.