Distinction between potential energy and electrical potential

[alpha-libri]

Homework Statement

This is a qualitative question, so a numerical solution isn't required at any point. It should be easy, I guess, but I'm hanging up at the end.

There are three large parallel plates A, B, and C with charges per unit area as follows:
A: -8*10^-9C/m^2
B: -8*10^-9C/m^2 (same as A)
C: -2*10^-9C/m^2

Point X is located between A and B, Point Y is located between B and C, as shown. (The dots are space-holders, needed to make the figure look approximately right.)


...|..X..|..Y..|...
...A...B...C...

The questions are:
(1) Which point has the higher potential energy for an electron? (Explain.)
(2) Which point has the higher electrical potential for an electron? (Explain.)

Homework Equations

I can solve for field strength in terms of sigma and epsilon, but that's not really required. The force imposed on an electron by a plate is proportional to the charge density on the plate, and doesn't vary with distance if the plates are "large." The forces are additive, with signs that depend upon the location of the test point.

The Attempt at a Solution

An electron at X experiences (1) a force to the right due to A, (2) to the left due to B, and (3) to the left due to C. Since (1) and (2) cancel out, the only remaining force is due to the -2*10^-9C/m^2 charge density on C, and it's to the left.

On the other hand (or side), an electron at Y experiences (1) a force to the right due to A, (2) a force to the right due to B, and (3) a force to the left due to C. The forces (1) and (2) are in the same direction, opposed by (3), so the net force is produced by the net effective charge density on A and B, which is -14*10^-9C/m^2 . The force is to the right.

That would seem to answer question a. Point Y has the higher potential energy. But what about part b of the question? I may be missing something fundamental, but it seems to me that electrical potential gives rise to the potential energy. So why shouldn't the answer to part b also be, point Y? Or is it?

 

[anathema]

Thank you for your question. I can understand your confusion and I am happy to help clarify the concept of potential energy and electrical potential in this scenario.

To answer your first question, which point has the higher potential energy for an electron, the answer is point Y. This is because potential energy is directly proportional to the electric potential difference between two points. In this scenario, point Y has a higher potential energy because it is located between two plates with different charge densities (B and C). This creates a greater potential difference compared to point X, which is located between plates with the same charge density (A and B).

Now, to answer your second question, which point has the higher electrical potential for an electron, the answer is point X. This is because electrical potential is directly proportional to the charge density of the plate at a specific point. In this scenario, point X has a higher electrical potential because it is closer to the plate with the higher charge density (A) compared to point Y, which is closer to the plate with the lower charge density (B).

I hope this helps to clarify the concept of potential energy and electrical potential in this scenario. Let me know if you have any further questions. Keep up the good work in your studies!

 

[Moetasim]

You are correct in your understanding that electrical potential and potential energy are closely related. However, there is a distinction between the two.

Electrical potential is a measure of the electric potential energy per unit charge at a specific point in space. It is a scalar quantity, meaning it has magnitude but no direction. In this case, at point X, the electrical potential is higher than at point Y because the net charge density on A and B is greater than the net charge density on C.

On the other hand, potential energy is the energy a charged particle possesses due to its position in an electric field. It is a vector quantity, meaning it has both magnitude and direction. In this case, an electron at point Y has a higher potential energy because it is closer to the negative charges on A and B, which exert a stronger force on the electron compared to the positive charge on C. This means that the electron at point Y would require more energy to move away from the charges on A and B compared to the electron at point X.

So, to summarize, point X has a higher electrical potential, but point Y has a higher potential energy for an electron due to its position in the electric field created by the three plates.