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In our intro to electrodynamics course, we were told that the interaction between two charges depends on:

➢Magnitude of the charges

➢Separation distance between the charges

➢Velocities of the charges

➢Acceleration of the source charge

➢At a past instant of time! (Electromagnetic radiation travels at the speed of light-Relativistic effects)

The first two are obvious, but how do the next few influence the interactions? Especially, the fifth one (That was the exact thing he wrote.)

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What your instructor was summarising is the Lienard-Weichert potentials.

These are formulae derived from Maxwell’s equations and special relativity that can be used to calculate the EM field due to a charge in general motion. As you can see, the potentials depend on the charge’s velocity, so the field, which is built from the time and spatial derivatives of potential, depends on velocity and acceleration.

Specifically, the reason for the need to use a past instant in time (i.e. the retarded potentials) is because of causality. Signals cannot travel faster than the speed of light, so a particle cannot instantaneously affect another one that is in a different location. Instead, to avoid “spooky action at a distance”, the particle’s position at a previous time is relevant.

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You get the magnitude of the charges and the separation distance. Consider one charge the source and the other the target. Practically we're talking about the source at the prior time and the target at the later time. Usually they talk like it's source at later time and target at later time, and then they have to go back and figure out where the source was at the prior time. There's no value in bothering with that.

We are computing the electric and magnetic fields at the later time at the target position. For that we don't need the target velocity. We only need target velocity to decide how the force affects the target. Which is of course the point of the exercise....

For your convenience, you might as well make the frame be the one where the target is stationary. Then you don't have to consider its velocity and you don't have to consider magnetic fields, which will not affect it. Unless for some reason that frame is not convenient.

You can divide up the source velocity into two parts. There's the velocity inline with the straight-line vector between the source and target. And there's the velocity normal to that vector. Those behave completely differently.

Imagine the source is moving directly toward the target at velocity v.

Integrate the force over a time interval of 1 unit, ignoring inverse square effects. In 1 time unit, the source will get v distance closer. So the force that leaves the target after 1 time unit will take only 1-v time units to arrive. That force is squeezed into a smaller distance, it isn't 1 distance unit but less. If you calculate the density of force along that distance it comes out to $\frac{1}{1-v}$. And if all that force arrives in $1-v$ time units, then the amount of force that arrives in 1 time unit is $1+v$ times as much.

So we get a factor of $\frac{1+v}{1-v}$ compared to a source that isn't moving at all. This is easy and obvious and there's nothing mysterious about it.

When the source moves sideways instead, the factor is $(1-v^2)\sqrt{1+v^2}$.

I can explain the $\sqrt{1+v^2}$. The original equation includes the sum of two vectors, n and -v. n is a unit vector, and this time -v is perpendicular to n. So their sum has length $\sqrt{1+v^2}$.

The other part is harder. I can imagine part of it.

We imagine that the force travels straight from the source to the target. But because the source is moving sideways, the force acts at an angle, not along the direction of movement. I can imagine that a part of the force is lost. If v is like the sine, then the part that isn't lost would be the cosine, $\sqrt{1-v^2}$. (This doesn't explain why v would be the sine when we started out with a ratio of 1:v.)

Something like that happens with light polarization. After polarizing to an angle, the remaining irradiance is $\cos^2$. But this is force, not irradiance, and even if the logic works (for force whose direction is rotated instead of polarization around that direction), still I only get $\sqrt{1-v^2}$ instead of $1-v^2$.

Magnetic fields are a property of the frame. In a frame where the source is stationary there is no magnetic field. In a frame where the target is stationary, the magnetic field has no effect on it. It's only in frames where both target and source move, and only in particular relative directions, that the magnetic field is relevant.

I'm not ready to talk much about acceleration, but notice that the magnitude factors that depend only on velocity also affect the acceleration terms.

We think of radiation as waves with a wavelength. Notice that the equation gives a radiation term whenever the source accelerates, whether there is any periodicity to it or not. Waves are just something that happens, that can give big effects.

If electrons move in a sine wave, in a radio tower, then you get radio waves that are strongest perpendicular to the tower and fade out to nothing directly above and below the tower. Linear polarization.

If electrons move in a circle, they create linear-polarized waves in the plane of the circle, circularly-polarized waves perpendicular to the circle, and elliptically polarized waves of radiation in between.

Anything that happens with radiation can be split up into a sum of sine waves, because Fourier transforms let you divide anything that happens into a sum of sine waves.

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