Need help on electric potential definition

Question

I'm having trouble understanding electric potential. In my book it says "an electric force acts on a charge situated in an electric field." I understand this part. Then it goes on to say "If a charge is moved over a distance against this force, work is done against the electric force and the system gains electric potential energy." I don't understand what it means by "the system gains electric potential energy." What is "the system," and wouldn't it be more correct to say that "the charge gains electric potential energy" (since only the charge is being moved). I'm really confused.

Answer 1

Basically, the system is simply that which is studied in a problem in physics. It refers to that which we want to know more about, in this case the moving electric charge in the presence of the electric field.

Be cautious with the terms 'electric potential' and 'potential energy', since they're two different things. Electric potential is defined as potential energy per unit charge. You should think of potential energy as 'the energy stored within a system', that ultimately will result in a force. Be aware that this is only for a more intuitive understanding of potential energy. A more correct way of defining it would be 'the energy an object has due to its position in a force field' (the electric field in this case). Force can also be defined as a decrease in said potential energy: \begin{equation} \mathbf{F}=-\nabla U \end{equation} Here $\mathbf{F}$ is the force and $U$ is the potential energy.

In electromagnetism, the potential energy of an object in an electric field, as well as the resulting force, is dependent of its charge. Dividing both by the object's charge gives us two things that are independent of the object: the electric field itself and the potential. Both are dependent of position only (and off course, the system that gives rise to them). This division gives us a new relation: \begin{equation} \mathbf{E}=-\nabla V \end{equation} Here $\mathbf{E}$ is the electric field and $V$ is the electric potential. So the electric field can be seen as a decrease in electric potential. To be clear, boldface means the quantity is a vector and therefore has a direction. You can thinks of $-\nabla$ as an operation that gives you the direction in which something decreases most rapidly, like $U$ and $V$.

I hope this helps at least a bit!

Answer 2

You are right that the charge gained potential energy. But this statement is only true because the charge is part of a system. We cannot talk about the electrical potential of a charge unless it is in an electric field - which means that there is "something else" that is essential for our definition of the potential.

We say the "system" (the charge, plus whatever was generating the electric field) has increased in potential energy, since something from outside must have acted on the system - it did work on the charge (a component of the system) in order to move it.

Consider this example:

I have two charges, A and B, that are a distance x apart. I discharge A (that takes a certain amount of energy since the charge on A experiences a force from the charge on B). Now I move B (which takes no energy since A is discharged). Finally, I put the charge back on A. Because B is a different distance, it takes a different amount of energy to charge A than it did to discharge it.

When I have finished, the situation I have is indistinguishable from the situation I would have had if I had moved B and left A alone. In both cases, the potential energy of the system has changed. But in the first case I did no work on B.

This is why it makes more sense to consider the system when you talk about the potential energy - B has no energy except for the fact that it is in the field of A.