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I'm curious, how would a charged particle, let's say an electron, move in a simple uniform electric field?

My first guess would be that it would follow Newton's second law of motion and move with a constant acceleration:

[tex]$ \dot{x}=\frac{Eq}{m} $[/tex]

where E is the fields intensity, q the charge, and m it's mass.

However, an accelerated charge should generate electromagnetic waves.

These have energy and so the motion wouldn't accelerate as fast.

I found that the energy radiated by these should be equal to:

(Larmor formula) for norelativistic speeds.

So I would assume that the work done by the electric field on the charge in a fraction of time:

[tex]$ P_{el} = q\cdot E \cdot \dot{x} $[/tex]

would be used to a) increase the kinetic energy of the particle b) used for the radiation energy:

[tex]\[P_{el}=\frac{d}{dt}(\frac{1}{2}m\dot{x}^2) + P_{rad}\][/tex]

That leads to the differential equation:

[tex]$ A \cdot \dot{x} = m \cdot \dot{x} \cdot \ddot{x} + B \cdot \ddot{x}^2 $[/tex]

where

[tex]$ A=Eq ; B=\frac{{q^2}}{{6\cdot\pi\eps_0\ c^3}} $[/tex]

(just to make it look simpler).

I have now idea how to solve this.

Do you think my reasoning is right? If so, can anybody find a solution?

How is the motion of a charged particle in a uniform electric field truly described?

I'd think this is quite a simple question, but I couldn't find any simple answer to it on the Internet.

I appreciate any help.

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# A Charged Particle in a Uniform Electric Field

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