Why electric potential energy at a point is the same as amount of work done to bring the charge to that point from infinity?

Question

Let's consider a positive charge at a point in the space. Now, if we want to bring a another positive charge to a point within the electric field of the first charge, then we have to do some work against the electrostatic force of the first charge and that work is stored as potential energy in the system.

But if we think there is a negative charge nearby, then we don't have to perform any work to bring the positive charge from infinity to that point in the electric field because here the work is done by the system which loses its potential energy. So, by definition, the potential energy of the system is zero, when the distance between two charges is infinity, and when we have brought the negative charge to that point in the electric field of the first charge, the potential energy is negative. But how can energy be negative?

In the case of earth we use this formula to determine potential energy of a particle:

$$\mathrm{P.E.}=mgh$$

Because we know that if we want to send the particle $h$ units up from earth, we have to do some work and that work is stored as potential energy.

Now, for the charges, if we hold the negative charge at earth and the positive charge as the particle then we have to perform some work to take away the positive from the negative and that work should be stored as potential energy. Then we got the maximum potential energy to be infinity and not zero, and the potential energy would never be negative in any point.

Answer 1

But how can energy be negative?

Your question is not entirely clear to me, but is it correctly understood that you essentially are asking why electric potential energy can be negative?

The answer is that the value of potential energy doesn't matter. Only the difference matters. If you compare the amount of stored potential energy with another amount of potential energy, then there might be stored more or less. If more, then positive; if less, then negative. The sign only tells you how much energy that is stored compared to or relative to some reference level that has been chosen more or less randomly.

And the reference level in case of electric charges is usually chosen as the potential energy stored when the charges are infinitely far apart. We don't know what the energy level is for such a situation, so we just call it zero.

• If two positive charges are infinitely far apart, we call the potential energy level zero. Move them closer together, and the potential energy of this two-particle-system changes. We can call this change positive, why not. If you release them, then the charges will move away from each other again, back to the original energy level of zero.

• If a positive and a negative charge are infinitely far apart, we still call the potential energy level zero. Move them closer together, and the potential energy again changes. But this time it changes "differently". Because if you let go, then they don't move farther apart again; they rather move even closer together. It is sort of opposite to the above situation. So, we'll call this potential energy level negative instead.

Negative energy is therefore nothing special. It just means "less than whichever we called zero". And when an energy level is smaller in one situation than in another situation, then it just means that the system wants to be in that former situation - it wants to be in a state of lowest possible energy.