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## Main Question or Discussion Point

I have been thinking about a simple thought experiment in classical electromagnetism

that seems to disobey conservation of energy.

I'd be very interested to hear where people think I'm going wrong.

Imagine that we have two oppositely charged objects with charges [itex]+q[/itex], [itex]-q[/itex] and

masses [itex]m[/itex].

Assume that they slid without friction on two vertical poles placed at a

distance [itex]d[/itex] apart.

First we lift them from the ground up to a height [itex]h[/itex] so that each one has a

potential energy [itex]mgh[/itex].

Now we let them drop simultaneously.

The equation of motion of each object (in CGS units for clarity) is given by:

[tex]m a = - mg + \frac{q^2}{d\ c^2} a + \frac{2}{3}\frac{q^2}{c^3} \dot{a}[/tex]

The first term is the weight of the object.

The second term is the force due to the electric field induced by the

acceleration of the other charged object.

The third term is the Abraham-Lorentz radiation reaction force.

The above equation of motion has a constant acceleration solution given by:

[tex]a = \frac{-g}{1 - q^2 / d\ m c^2}[/tex]

The work done by each charged object as it falls a distance [itex]h[/itex] is given by:

W = Force [itex]\times[/itex] distance

[tex]W = \frac{-mg}{1 - q^2 / d\ m c^2} * -h[/tex]

[tex]W = \frac{mgh}{1 - q^2 / d\ m c^2}[/tex]

Thus the energy we get out is

What's wrong with the calculation?

that seems to disobey conservation of energy.

I'd be very interested to hear where people think I'm going wrong.

Imagine that we have two oppositely charged objects with charges [itex]+q[/itex], [itex]-q[/itex] and

masses [itex]m[/itex].

Assume that they slid without friction on two vertical poles placed at a

distance [itex]d[/itex] apart.

First we lift them from the ground up to a height [itex]h[/itex] so that each one has a

potential energy [itex]mgh[/itex].

Now we let them drop simultaneously.

The equation of motion of each object (in CGS units for clarity) is given by:

[tex]m a = - mg + \frac{q^2}{d\ c^2} a + \frac{2}{3}\frac{q^2}{c^3} \dot{a}[/tex]

The first term is the weight of the object.

The second term is the force due to the electric field induced by the

acceleration of the other charged object.

The third term is the Abraham-Lorentz radiation reaction force.

The above equation of motion has a constant acceleration solution given by:

[tex]a = \frac{-g}{1 - q^2 / d\ m c^2}[/tex]

The work done by each charged object as it falls a distance [itex]h[/itex] is given by:

W = Force [itex]\times[/itex] distance

[tex]W = \frac{-mg}{1 - q^2 / d\ m c^2} * -h[/tex]

[tex]W = \frac{mgh}{1 - q^2 / d\ m c^2}[/tex]

Thus the energy we get out is

*more*than the energy that we put into the system in the first place.What's wrong with the calculation?