A question regarding the concept of potential difference between two points in an electric field, as stated in my 12th grade book

Question

My 12th grade physics book on electrostatics says:

Potential difference between two points in electric field can be defined as work done in displacing a unit positive charge from one point to another against the electric forces.

By this logic, the potential difference between two points in an electric field should always be positive because the work done in moving a unit positive charge against the direction of electric field will always be positive. But potential difference can also be negative, so what exactly is my book describing?

Answer 1

Consider a system where the electric force is due to a negative charge, not a positive charge as is usually assumed.

Now, when it is said that:

... work done in displacing a unit positive charge... against the electric forces

what is meant is that an external force that is equal in magnitude but opposite in direction to the electric force is applied.

Now, suppose, as the figure shows, you move the charge q (which is positive) from A to B. So, the direction of your force is AB. The big charge Q will try instead to make it move towards A. So, it's direction is BA. Now, the force you apply and the direction of the charge's displacement is in the same direction. Therefore, in this case work done by the external force (you) is positive.

Let us now assume the charge moves from B to A. In this case, the electric force is still in the direction BA, and you are applying a force in the direction AB. In this case, the direction of the displacement and the external force are opposite to each other, so the work done is negative.

Answer 2

the potential difference between two points in an electric field should always be positive because the work done in moving a unit positive charge against the direction of electric field will always be positive.

This statement which you said is fundamentally wrong.

Think as if you are holding a charged particle. If the particle applies a force on you to reach from one point to another then the particle has done the work on you. Therefore, work done by external agent is negative and vice versa.

Hence, potential difference can be both negative and positive.

Answer 3

You're right; the definition that you quote is flawed. I prefer this ...

The potential difference of point A relative to point B is the work done per unit test charge by the electric field on a test charge, $q$, going from A to B. So $$\text dV=-\tfrac 1q q\vec E.\text d \vec r=-\vec E.\text d \vec r$$

in which $\text d\vec r$ is the displacement of point B relative to point A and $\text dV$ is the potential of B relative to A.

Apart from the confusion over signs, another thing I don't like about the definition that you quote is its use of another force pushing against the electric force; it strikes me as an unnecessary and messy introduction (though not wrong in itself).