# B Equation for the frequency of light from an accelerating charge

#### ealbers

Summary
Need the equation for the frequency of light emitted from a accelerating charge.
Say I have a electron in space, its accelerating along say the x axis at 10 meters per sec^2, what frequency of light does it emit?
Thanks!

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Staff Emeritus
There is no such equation. It's not a single frequency. The equations that do exist, which are complicated, really cannot be explained at the B level.

#### ealbers

Really?
Can't they measure this? Say take a vacuum tube and shoot electrons one at a time down it and measure the waves given off? A old tv with an electron gun has accelerating electrons, does it emit waves?

#### Ibix

Of course you can measure the electromagnetic field of an accelerating charge. I doubt it's simple to do because you want to ignore whatever powerful field is accelerating the charge, but it could be done.

However, taking a look at the equations for the electric and magnetic fields of an accelerating charge (see equation 32 in this PDF) and the diagram of the electric field and Poynting vector (diagram at top of p6 in the above) I suspect that the frequency spectrum is continuous, broad, and time-varying. So I'd be surprised if there's "a frequency", so much as a time-and-distance dependent power spectrum.

Why do you want to know?

#### ealbers

I was reading about the equivalence principal and was curious how it handles a electron in the accelerating rocket vs one on the earth....seems the one in the rocket should radiate a frequency, just wondering what frequency the earth one would give off. I THOUGHT it would be some kind of simple linear relationship between the charge and the acceleration q and a say, but apparently its a bit more complicated.

#### Ibix

That's complicated. The short answer is that the equivalence principle only applies locally (the formal statement is that second derivatives of the metric can be made to vanish at one event, and are negligible in a small volume around it). However, the electromagnetic field of a charge fills all of space (in principle), so we don't necessarily expect the equivalence principle to apply here.

I have to admit that the formal discussion around this went over my head the last time it came up. A search of the relativity forum may well turn up the thread if you want to see.

Staff Emeritus
Can't they measure this?
I never said they couldn't. I said that there is not a single frequency and that the expression is complicated - too complicated for B-level.

Staff Emeritus
just wondering what frequency the earth one would give off
It gives off no radiation, as it is not accelerated. (Well; actually it emits a very small amount of radiation with a frequency of 1/24 h because the earth is spinning)

#### A.T.

Say I have a electron in space, its accelerating along say the x axis at 10 meters per sec^2, what frequency of light does it emit?
See this old thread:

In short: A charge with constant acceleration doesn't radiate, neither in a rocket nor at rest in a gravitational field.

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