Question Regarding The Movement of Charges

Question

We know that two electrons repel each other since they have like charges, which means they move in opposite directions. But how can they move if they exert equal and opposite charges, aren't the forces balanced which means there is no movement?

Answer 1

As a motivational example, consider your air-table puck experiment from highschool. If you have a pair of pucks at rest, and this one pushes that one, then the two of them will move off in opposite directions. If they are equal mass then they move with equal speeds. The total momentum is therefore zero, since this one is exactly opposite that one.

The thing that lets you easily calculate this is the center of momentum. This is the reference frame where the total momentum adds to zero. In this frame, when you have collisions and forces and so on, the total momentum remains zero.

So two for electrons. This one provides a force on that one. That one provides a force on this one.

The forces are equal and opposite. That is, the change in momentum of this one is the same size and opp. direction as that one.

The center of momentum does not change its motion. If you start in the center of momentum frame then the total momentum starts zero and stays zero. But each electron does move.

Answer 2

Let $-q_1$ be placed at x = 0. Let $-q_2$ be placed at x = 1.

Charge 1 exerts a force on charge 2 via Coulomb’s law. The only force exerted on charge 2 comes from the electric field of charge 1. I.e. the only force on charge 2 is a repelling force due to charge 1. Because the repelling force is the only charge exerted on charge 2 and no other force is present to balance it, charge 2 accelerates and moves due to the force.