# Determining the sign of Work in Electric Problems

CaneAA
I am having a difficult time determining the sign of work in electrostatic problems. I'm attempting to reason out a general rule based on the given examples in my book and to relate it to work done by gravity--but I'm getting two contradictory observations. I don't know if maybe there are different rules depending on whether the force doing the work is conservative or non-conservative...

For example: Consider an electron moving toward a region of higher potential (+ side; lower potential energy) in an electric field. From what I understand, the work done by the electric force is POSITIVE.
If, however, the electron is being pulled toward the negative plate [there has to be an external non-conservative force doing this, right?], then the work done by the electric force, which is opposing this motion, is NEGATIVE.
So, it would appear that in determining the sign of the work done by an electric force [a conservative force, like gravity], if the particle is moving to a region of lower potential energy (ex. toward + plate), the electric work is positive. And vice versa.

So, this would be the same as the work done by gravity. If an object is dropped (moving to a region of lower potential energy) --> gravity does positive work. And vice versa.

But now, if you want to move an electron toward the negative plate (higher potential energy), you need to apply an external non-conservative force. But the work done by this force (in moving the electron to a region of higher potential energy) is POSITIVE.

So, I guess my main source of confusion is that for one type of force, moving the electron to a region of higher potential energy results in positive work, and for another type of force (i.e. electric force), moving the electron towards a region of higher potential energy, results in negative work.

So, ultimately, my question is: How do you determine the sign of the work done? Are there 2 rules for determining the sign of work depending on whether the force doing the work is conservative or non-conservative--or am I missing a simpler connection between the sign and work?

Thank you!

## Answers and Replies

Homework Helper
Gold Member
$$W = \int_P \vec F \cdot \vec {ds}$$

In other words, the work done by a force is the dot product between the force vector and the displacement vector, over the path.

Put yourself into the mix. Suppose that you are the electric force pushing the electron. Or, conversely, suppose that you are some external force pushing the electron against the electrical force. Or substitute gravity into the above.

Does the object move in the same direction which you are pushing? If so, you are doing positive work. If the object moves in the opposite direction you are pushing, it means something is doing work on you! In that case, you are doing negative work. So, the work that you do is positive if you are the one succeeding in pushing in the object in the direction that you are pushing.

If work was easy, they wouldn't call it "work".

CaneAA
Does the object move in the same direction which you are pushing? If so, you are doing positive work. If the object moves in the opposite direction you are pushing, it means something is doing work on you! In that case, you are doing negative work. So, the work that you do is positive if you are the one succeeding in pushing in the object in the direction that you are pushing.

Thank you!