# Slightly confused about potential energy in an electrostatic field

## Main Question or Discussion Point

Hi everyone, I don't know how I manage it but I've gone and confused myself about something which I was fairly confident about right before an exam... again!

So here's the deal.

Let's imagine two particles A and B which have charge Q and q respectively and are separated by a distance r (I know, imaginative right?).
Now, if A is held in place and B is released then I can calculate its maximum velocity through these steps:

1) Get the EPE of B $\frac{k_{e}Qq}{r}$
2) Assume all of the EPE becomes KE and so $\frac{m_{B} v_{max}^2}{2} = \frac{k_{e}Qq}{r}$
3) Rearrange for v and job done

But whilst I've been happily doing this for a while, I realised that A is losing EPE as well. If A is being held stationary and is losing EPE, but only B is gaining KE then surely the KE gain of B ought to be the sum of the EPE loss of A and B otherwise energy conservation is violated.
But as all the calculations I've ever done have never shown this to be the truth I am doubting this thought.
So, I had a quick discussion with someone and we came to the conclusion that maybe we had a more fundamental misunderstanding of EPE and that in reality the system has an EPE given by equation 1. When B is allowed to move from A, the EPE of the system decreases to zero as the KE of B increases by the same amount.

So, anyone want to jump in and help me out?!

Cheers,
James

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Doc Al
Mentor
Don't think of the EPE as belonging to either A or B. Instead, think of it as belonging to the system of A and B together.

Lovely, that's what I was expecting to hear. Thanks a lot

Hello, James.

To expand a bit on Doc Al's answer.

Strictly according to Coulomb's Law the force between the (any two) charges is

$$F = \frac{1}{{4\pi {\varepsilon _0}}}\frac{{{Q_1}{Q_2}}}{{{r^2}}}$$

This force requires that there be two charges for it to exist and is regarded as the fundamental law of electricity.

This law provides the link between electricity and mechanics.

Very often we find it convenient to consider one charge fixed (as you have done with A) and to consider that a generates an 'electric field'. Then we often call the other charge (your B) a 'test charge'.
We then attribute potential energy to B as it moves about in the 'field' of A. Again as you have done.

go well