Potential due to Point Charge — Definition

Definition

Imagine you have a tiny, imaginary positive charge, let's call it a 'test charge', that you want to move around in space. Now, if there's another charge, say a big positive charge, sitting somewhere, it creates an 'electric field' around it.

This field is like an invisible force-field that pushes away any other positive charges. If you try to bring your tiny positive test charge closer to this big positive charge, you'll have to push against the electric field's repulsive force.

This means you're doing 'work'.

Electric potential is essentially a measure of how much 'potential energy' a unit positive test charge would have if it were placed at a particular point in an electric field. Think of it like a hill. If you push a ball up a hill, it gains potential energy. The higher the point on the hill, the more potential energy it has. Similarly, in an electric field, some points are 'higher potential' and some are 'lower potential'.

Specifically, for a single, isolated point charge (a very small charge concentrated at a single point), the electric potential at any point around it depends on two things: the magnitude and sign of the source charge, and how far away the point is from that charge. The formula for this is $V = \frac{kQ}{r}$, where $Q$ is the source charge, $r$ is the distance from the charge to the point where you're measuring the potential, and $k$ is a constant ($k = \frac{1}{4piepsilon_0}$).

If the source charge $Q$ is positive, the potential $V$ will be positive. This means you have to do positive work to bring a positive test charge from infinitely far away (where potential is considered zero) to that point.

If $Q$ is negative, the potential $V$ will be negative. In this case, the electric field itself would pull the positive test charge towards the negative source charge, meaning the field does positive work, and you, as an external agent, would do negative work (or the field would do work on its own) to bring it in, resulting in a negative potential.

It's crucial to remember that potential is a scalar quantity, meaning it only has magnitude and sign, not direction, which simplifies calculations when dealing with multiple charges.