A point charge +q rests halfway between two steady streams of positive charge of equal charge per unit length λ, moving opposite directions and each at c/3 relative to the point charge. With equal electric forces on the point charge, it would remain at rest. Consider the situation from a frame moving right at c/3.
a) Find the charge per unit length of each stream in this frame.
b) Calculate the electric force and the magnetic force on the point charge in this frame, and explain why they must be related the way they are.
2. The attempt at a solution
I know that the charge density of each moving line of charge will change due to relativistic length contraction. I also managed to show that the new charge density is simply the original charge density multiplied by the Lorentz factor:
L=L0/γ, where L is the contracted length
λ0=Q/L0, λ=Q/L, where lamda is the new charge density
Solve each for Q, set them equal, and rearrange to obtain
λ/λ0=L0/L=γ, ∴ λ=γλ0
But here's where I'm stuck. I know I now need to find the Lorentz factor γ for each moving line of charge. I also know that γ=(1-v2/c2)-1/2. In this formula, I need to know v, which I thought would be computed using a Lorentz transformation for velocity, namely u'=(u-v)/(1-uv/c2).
I do have the final answer to part a, which is given as λ(√8)/3 and 5λ(√8)/12 in the textbook. However, I can't produce these values, because when I insert the values that I thought would be correct into the velocity transformation equation for u and v (namely c/3 for each), one u' would be equal to 0, yielding a Lorentz factor of 1, and the other u' would be incorrect as well.
I think I understand the method behind solving the problem (although I could be wrong), but it seems like I'm having trouble conceptually trying to figure out the velocities relative to one another.
Any help would be greatly appreciated, thanks!