Does the superposition of electric fields not hold for moving charges?

[Hiero]

If a single point charge is moving, then the component of the electric field normal to the motion is larger (by the gamma factor) than if the charge were stationary.

Now consider an infinite line of charges (with a small separation, the same between each charge). If the whole line is moving along, then by length contraction the linear density (and hence electric field) is increased by gamma. But by the first paragraph, each individual charge should have its field increased by gamma. By this reasoning we should expect the field to be larger by gamma squared, as compared to the frame which sees the line stationary.

On the one hand I know this is silly, because length contraction of a line of charges could be how you derive the first paragraph in the first place. On the other hand though this really bothers me; why should the field of a single charge be different if it's alone or in a line?

 

[mfb]

Where exactly do you see a problem?

Superposition always works.

 

[Dale]

I agree with @mfb. The superposition principle still holds

 

[Orodruin]

because length contraction of a line of charges could be how you derive the first paragraph in the first place

No, it could not.

Consider a line charge ##\rho = \rho_0## in the rest frame of the charges. This gives a 4-current ##J \propto (\rho_0,0)##. Transforming this current to a system moving at velocty ##v## relative to the first gives the 4-current ##J' \propto \gamma \rho_0 (1,-v)##. The ##\gamma## here is exactly the length contraction of the distance between the charges, but the 4-current includes not only the charge density but also the additional current. This gives you an EM-source with a stationary charge and a stationary current, leading exactly to the factor of ##\gamma## in the electric field.

The point is that you are artificially adding another factor of ##\gamma## on top not caring about how the electric field of the point charge changes in the frame where the charges are moving. The component orthogonal to the direction of motion is given by
$$
E_\perp' = \frac{\gamma kQ}{\gamma^2(x'+vt')^2 + y'^2},
$$
What you seem to be neglecting is the factor of ##\gamma^2## in the denominator. Let's take the situation at time ##t' = 0## which leads to the ##vt'## term disappearing and use a spacing such that ##x_n = \delta n## in the rest frame of the charges. By length contraction, you then obtain ##x'_n = \delta n/\gamma##. The superposition of all of the charges then leads to
$$
E_\perp' = \sum_{n=-\infty}^\infty \frac{\gamma kQ}{\gamma^2 (\delta n/\gamma)^2 + y^2}
= \gamma kQ \sum_{n=-\infty}^\infty \frac{1}{(\delta n)^2 + y^2} = \gamma E_\perp.
$$
No contradiction anywhere.

 

[jartsa]

If a single point charge is moving, then the component of the electric field normal to the motion is larger (by the gamma factor) than if the charge were stationary.

Now consider an infinite line of charges (with a small separation, the same between each charge). If the whole line is moving along, then by length contraction the linear density (and hence electric field) is increased by gamma. But by the first paragraph, each individual charge should have its field increased by gamma. By this reasoning we should expect the field to be larger by gamma squared, as compared to the frame which sees the line stationary.

On the one hand I know this is silly, because length contraction of a line of charges could be how you derive the first paragraph in the first place. On the other hand though this really bothers me; why should the field of a single charge be different if it's alone or in a line?

Let's consider a line charges that starts moving, without any length contraction of the line occurring. Like electrons in a wire.

A test charge next to the line feels some extra force from the closest part of the line, and somewhat reduced force from the other parts of the line. Net force is unchanged. So net field is unchanged. If we think about field lines and how motion causes them to turn ... well then we get confused. Somehow that turning does not change the net field, when the charges form a line.

 

[Hiero]

Ah, I see. The rule is that at any point in spacetime the normal field is larger by gamma in the moving-charge frame than in the rest frame of the charge. This is not the same as saying the normal field is larger by gamma as compared to an identical-except-at-rest distribution of charge in the same frame, as I was mistakenly imagining.

No, it could not.

The way I saw it derived was by a plane of charge moving (along the plane). Is this an improper derivation, or is there some key difference about a plane vs. a line?

 

[Orodruin]

The way I saw it derived was by a plane of charge moving (along the plane). Is this an improper derivation, or is there some key difference about a plane vs. a line?

I might have written that part to hastily and later forgot to remove it. However, I think it is much simpler to just Lorentz transform the field tensor right away. It will give you the result right off the bat as you know that the field has to be ##kQ/r^2## in the radial direction in the rest frame of the charge.

 

[Hiero]

However, I think it is much simpler to just Lorentz transform the field tensor right away.

Not if you've never seen a field tensor! I'll learn these more elegant formulations in due time though.

Thanks for your help sir, your first reply hit the nail on the head.