How can voltage between unlike charges decrease with proximity?

[Brian_D]

Homework Statement

This is an elementary question about voltage and Coulomb's law, not a homework problem per se.

Relevant Equations

$F=(K*qa*qB)/d^2$
$electric~potential~difference=(Fd)/q'$

The first equation is Coulomb's law and the second is an equation for electric potential difference (voltage), where work done on the charge is Fd, that is, the force applied to the charge times the distance over which the force is applied. And where q' is the test charge. What confuses me is a diagram and statement in one of my physics textbooks, Paul Zitzewitz, Glencoe Physics: Principles and Problems, p. 490; the relevant diagram and statement are attached.

The book says regarding part (a) of the diagram, that electric potential is smaller when two unlike charges are closer together. I am not following this. Yes, the second equation says that as the distance decreases, the potential decreases. But how is this compatible with Coulomb's law? The latter says that the force INCREASES with proximity, so how can the potential decrease with proximity?

 

[PeroK]

I guess you should do the maths, as they say. Force isn't voltage.

 

[Brian_D]

The math doesn't make sense to me, which is why I'm asking the question. Each equation makes sense in itself, but I don't understand how the two of them are consistent. Voltage is not force, but it seems related to force, namely the work needed to move a test charge. Since unlike charges attract, it takes positive work to separate them, but the further apart they get, the LESS work it takes, right? This is consistent with Coulomb's law. Doesn't this mean the further apart the charges, the LESS voltage between them? But the textbook is saying the exact opposite. My problem is not with the math, but with the concepts.

 

[PeroK]

You really ought to be able to resolve this for yourself. There is no conceptual difficulty if you do the calculations.

Each charge must do more work on the test charge over the greater distance. Whatever work it does over a short distance is does that and more over the longer distance.

Also, look at the force against distance profile for each charge in each case.

Or, set one charge to zero and see what happens.

There are many ways to see this.

 

[Brian_D]

You said, "Whatever work it does over a short distance it does that and more over the longer distance." Thank you, this answers my question. I had not been thinking about the work as cumulative, but now I see that it is.