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Let us imagine a free, negatively charged object that is in rest and placed in an elecric field of a point positive charge. The positive charge has a huge mass and cannot move, so we consider only the movement of the negative charge. The negative charge starts falling the center at time t0.

It is straight forward to find a velocity at given time when the charge simply falls toward the center if we consider only the Coulomb force.

But I would like to find the velocity at a given time, when not only the Coulomb force applies, but we consider bremsstrahlung too. Namely - some of the energy of the accelerating charge is lost due to emision of the electromagnetic radiation.

I look for classical electromagnetic solutions, not quantum mechanical.

How could I do that? In case of the acceleration parallel to the velocity the formula for the bremsstrahlung is rather simple, but I still don't know how to start the calculation. Any hints?

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Possibly something along the lines of: $W_{tot} = \Delta K$, $\frac{\mathrm{d}W}{\mathrm{d}t} = P_{Coulomb}+P_{Brem}$, where $W$ is work, and $P_{Coulomb} = \vec{F}_{Coulomb}\cdot\vec{v}$? Actually, you should probably use the appropriate special relativistic generalizations. – G. Paily Jul 29 '14 at 19:17

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