Current density question

[Anisotropic Galaxy]

http://higgs.phys.washington.edu/phy121b/HW/light speed changes.pdf

Above is what I need to do. So essentially, I have to show that in the new reference frame, that the new magnetic force will counter the old electric force. So then I have to use the equation of attraction due to a wire, and then some lorentz contraction. buit then i think I'm lost...Thanks!

 

[lightgrav]

So, in the new frame, are the + charges closer together or farther apart?
Are the - charges closer together or farther apart?

The "old" Electric Force was zero, as measured in frame S.
The electric Force in frame S' is not zero. (recall Gauss to find E).
The magnetic Force due to the (moving) + charges in frame S' is not zero.
(recall Ampere to find B).
Do they cancel?

 

[kyle8921]

I would approach this question by first understanding the basic concepts involved. Current density refers to the amount of electric current per unit area, and it is represented by the symbol J. In the given scenario, we are dealing with a wire carrying electric current and its interaction with a moving charge. This involves the principles of electromagnetism, specifically the Lorentz force law which describes the force acting on a charged particle in an electric and magnetic field.

In order to show that in the new reference frame, the new magnetic force will counter the old electric force, we need to consider the effects of the relative motion between the wire and the charge. In the new reference frame, the wire will appear to be moving and therefore, its length will be contracted according to the Lorentz contraction formula. This means that the wire will appear shorter in the new reference frame, and as a result, the current density will also change.

Next, we can use the equation for the magnetic force on a moving charge, which is given by F = qvB, where q is the charge, v is the velocity of the charge, and B is the magnetic field. In this case, the charge is moving perpendicular to the wire, so we can use the right-hand rule to determine the direction of the force. Since the charge is moving in the same direction as the wire, the force will be in the opposite direction.

Now, we need to consider the electric force on the charge due to the wire. This can be calculated using Coulomb's law, which states that the force between two charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. In this case, the charge on the wire will be distributed over its length, so we need to use the current density (J) to calculate the total charge on the wire.

Finally, by comparing the equations for the magnetic and electric forces, we can see that they are equal in magnitude but opposite in direction. This means that the new magnetic force will indeed counter the old electric force, as desired.

In conclusion, by understanding the principles of electromagnetism and applying the concepts of current density and Lorentz force, we can show that in the new reference frame, the new magnetic force will counter the old electric force. This demonstrates the importance of considering relative motion and frame of reference in understanding physical phenomena.