Force between charges

Question

Let us take two opposite charges are moving away from each other in directly opposite direction ......it's obvious that kinetic energy of both of them will decrease as both of them are attracting each other .So can we say both the particles are doing work on each other?

Remember that the electric field is also an object that can store energy and do work. It would be somewhat more complete (and avoid misunderstandings like "where did the energy go?" later) to say that both charges are doing work on the electric field. — probably_someone, Sep 18 '18 at 13:27

score 3 · Answer 1 · answered Sep 18 '18 at 13:33

Yes you can. $$W=\int\vec F \cdot d\vec x$$

Since the electric force from one charge onto another is directed along a line between the two charges, $\vec F\cdot d\vec x$ is non-zero within any displacement $d\vec x$. Therefore work is being done.

Another way you can see this is thinking about what you have already realized: the kinetic energy of both is decreasing. The relationship between work and change in kinetic energy is given by $$W_{\text {net}}=\Delta K$$

Since the only force acting on one particle is the force from the other, the net work done on one particle is just the work performed by the other particle. Since $\Delta K \neq 0$, there is work being done.

score 0 · Answer 2 · answered Sep 18 '18 at 14:54

So can we say both the particles are doing work on each other?

Yes.
Consider charge 1 as the system.
It has an external Coulomb force acting on it due to charge 2.
As charge 1 moves that external force does work on it and this results in a change in the kinetic energy of charge 1.
The same can be done when charge 2 is chosen to be the system.

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There is another way of considering the situation.

Let the system be the two charges with charge $q_1$ and $q_2$.
If the zero of electric potential energy of the system is when the separation of the charges is infinite then the electric potential energy is $-k_{\rm e}\frac{q_1q_2}{r}$ when the separation of the charges is $r$.
That energy is stored in the electric field.
As the separation of the charges increases the electric potential energy increases (becomes less negative) whilst the kinetic energy of the charges decreases by the same amount.
The change in kinetic energy of each charge depends on its mass relative to the mass of the other charge as momentum of the system has to be conserved assuming that there are no external forces acting on the system.

Within this system there are two internal forces - the attractive force on charge 1 due to charge 2 and the equal in magnitude and opposite in direction attractive force on charge 2 due to charge 1 .
These forces do work on their respective charges and so change the kinetic energy of the charges and hence of the system.

Force between charges

2 Answers2