No magnetic field from a static charge - Is there a simple physical argument to show why?

Question

For a charge moving in an electric field $\vec E$, its equation of motion is given by the electric part of the Lorentz force $$\frac d {dt}\gamma m \vec v = e\vec E$$This comes from the conservation of relativistic energy in a static electric field. But a magnetic field would still make this conservation law true since the magnetic force is always orthogonal to the velocity of the charge and therefore doesn't change its energy.

Is there a simply physical argument that shows why a static charge doesn't create a magnetic field?

Answer 1

The electric and magnetic fields form one object, the electromagnetic field tensor. This tensor represents an oriented plane at each point in space(time). An easy way to visualize this is to think of the magnetic field vector. Instead of the vector, think about the plane to which that vector is normal. This is the fundamental nature of the EM field.

The electric field is like this, too, except the planes have one direction along the time axis and one direction along a spatial axis.

When a point charge moves through space and time, it traces out a plane. One of the plane's directions is the direction it moves--a stationary charge "moves" through time. The other direction is based on the direction between it and an observer.

Since a charge at rest only moves through time, it sweeps out planes that have at least one timelike direction. This means its EM field contains only electric--not magnetic--components.

Answer 2

As a guess, I'd say for a static charge, the conservation of momentum requires the direction of the Lorentz force to be independent of the direction of the other moving charge's velocity. This then implies a magnetic field can't be created by a static electric charge. I haven't got a clue how to prove this though lol.

Answer 3

A static charge only produces an electric field; cf. Maxwell's equations, Gauss' law.

Answer 4

The fact is, magnetism is nothing more than electrostatics combined with relativistic motion. Other definitions fall into two classes:

1. Really advanced speculation in the form of theory-of-everything-ness that I won't go into here.

2. Empirical observations from circa the year 1900, where people thought "huh, isn't it nifty that electromagnetism turns out to be relativistically invariant?" If you really believe that the magnetic field is some magical, independent entity that has to be tested in every conceivable configuration, then sure, it's possible that a static charge (or a flux capacitor or a unicorn) will produce a magnetic field.

But, these days we understand that there is only electrostatic attraction/repulsion between charges. A particle's acceleration can always be seen to be due solely to electrostatic effects if you transform into its rest frame. Thus you shouldn't be looking at any energy-conserving rule to derive the Lorentz force. You should start with relativistic mechanics and $F \propto Qq/r^2$, and all of Maxwell's/Lorentz's laws will follow.