# 3 repelling parallel line charges

Suppose I had 3 identical parallel line charges parallel to the x-axis of infinite length in the same geometric plane.

The first and third line charges are equidistant from the second.

Now suppose I have a series of observers that have different velocities which vary only along the x-axis (i.e. differences in the x-component of velocity only).

For our first observer, the "center of charge" is stationary, and for the rest of the observers the "center of charge" is moving at some velocity directed along the x-axis. The first observer would observe no component of the electric field directed along the x-axis.

However, the others would observe such electric fields directed along the x-axis.

How do we ensure there is no acceleration of the second line charge despite the electric field exerted on it in the x-direction? Is its momentum affected due to the varying "electric potential" therefore "mass" in some compensating way, ensuring no acceleration, as in (mv)' - m'v = mv' = 0?

One should consider how a neutral wire with an electric current can be represented by the superposition of two line charges with equal and opposite linear charge densities while having different velocities. Notice that in the bottom half of the illustration below labeled "Test charge frame:" (in red) it shows an electric field with a component parallel to the wire that exists because the positive charges contribute to it and yet the negative charges do not negate that contribution. From http://physics.weber.edu/schroeder/mrr/mrrtalk.html

I've included information about this because some may doubt that electric field components in the x-direction are seen by observers having different component velocities in the x-direction than those of the line charges.

• I don't understand exactly the situation you're trying to explain (if you could, try to describe it better or include some figures), but keep in mind that you need Lorentz Transformations and Special Relativity if you want to change the reference frame when electromagnetic fields are present. Apr 14 '19 at 18:40
• What is the distance of wire 1 and 3? Is the same as 1 and 2? Are they in one plane? What is the speed of the first and other observers? Why did you state "others would observe such electric fields directed along the x-axis"? Apr 14 '19 at 20:10
• @DanielTuzes, Let's have the lines exist in the XY plane for example. Let the y positions of these lines be -d, 0, and +d respectively, where d is a function of time as specfied in the inertial frame observer who sees the least possible system momentum where line 2 is stationary. Apr 14 '19 at 20:21
• @DanielTuzes, The observers I specified I believe were sufficent to illustrate problem I encountered as to the electric fields that would or would not be produced at the x-axis as a function of the observer. The length contraction differences between these frames are confined to the x-direction for members of the set of observers as there is no transverse length contraction. The first observer sees no x-component of the velocity of the line charges. Each of the other observers will see the three lines moving in the x-direction at some common rate. Apr 14 '19 at 20:31
• @ErickShock, Per your suggestion, I added an illustration figure with some additional information about the problem. Hopefully, I have stated the problem clearly enough that proceeding to apply Lorentz Transformations and Special Relativity will make sense. Also, I am seeking an intuition for the result that I should expect from the application of these. Apr 14 '19 at 22:50

As I said in my comment, you need special relativity to transform the components of the $$\mathbf E$$ and $$\mathbf B$$ fields. They transform according to \begin{align} \mathbf E'_{||} &= \mathbf E_{||} \\ \mathbf B'_{||} &= \mathbf B_{||} \\ \mathbf E'_{\perp} &= \gamma (\mathbf E_\perp + \mathbf v \times \mathbf B_\perp) \\ \mathbf B'_{\perp} &= \gamma \left ( \mathbf B_\perp -\frac{1}{c^2} \mathbf v \times\mathbf E_\perp \right ) \end{align} where the primed quantities denote the (total) fields at a frame moving with velocity $$\mathbf v$$ relative to your original frame ($$\mathbf v$$ is the velocity of any observer), $$\gamma$$ is the Lorentz factor and the subscript $$||$$ denotes the component parallel to $$\mathbf v$$ and $$\perp$$ the perpendicular component. As you can see, the parallel components remain the same, so no acceleration in the $$x$$-direction will be measured by any observer. The cool stuff happens in the second figure, where a magnetic field is created whose force exactly counterbalances the electric force in order to don't accelerate the charge.
• @Kevin Marinas First, no. The parallel components of the field doesn't change when you alter the $x$ component of their velocity. And yes, the second image is not right. SR demands that both "lines" contract, increasing their charge density. But since they contract in the same way, the net effect keeps the wire neutral. Apr 16 '19 at 3:37