Determining relative amounts of work done

[RoboNerd]

Homework Statement

Homework Equations

Work = - (integral of) (E dot dl)

The Attempt at a Solution

Hi,

I know that the right answer is D zero, but I fail to understand why. I said that the answer was A as I have different charges, and I thought that depending on how I approach each individual charge by taking either the left-hand path or the right-hand path that my integral of E dot dl would be different resulting in different amounts of work done.

However, the solutions say that the amount of work done is the same regardless of path taken.
Could anyone please explain to me why this is the case (with the electric field being conservative)?

Thanks in advance for being willing to help.

 

[kuruman]

One of the definitions of a conservative force is that the work done by it is independent of the path taken. So the answer to your question is "by definition". Gravity is another conservative force. The work done by gravity when you move from one point to another depends on the height difference, not on how you go from one point to another. With electrostatics replace "height difference" with "potential difference". It's the same idea.

For this question, can you write expressions for the electric potential at points R and S? If these expressions are the same, then the potential difference is zero and the work done by the electric force will also be zero.

 

[haruspex]

the right answer is D zero

Do you mean C zero?

 

[jbriggs444]

I thought that depending on how I approach each individual charge by taking either the left-hand path or the right-hand path that my integral of E dot dl would be different resulting in different amounts of work done.

The four lines labelled with length d are not intended to denote paths. They are intended to denote distances. You cannot integrate E dot dl over any of those lines because they all end up on point charges. The integrals all diverge (i.e. they are all infinite).

If you ignore the fact that the integrals are not well defined, it is clear by symmetry that the work done over a path that approaches the +Q charge from R is equal and opposite to that done on a path that recedes from the +Q charge toward S. They add to zero. Similarly for the work done on the paths approaching and receding from the +2Q charge.

 

[kuruman]

The problem is asking for the work "required to move a charge -Q from point R to point S". One can chose a path that does not bring charge -Q near the given charges, in which case all integrals are well-defined. The work done by the electrical force along any path from R to S depends only on the end points because the electrical force is conservative. This work is the negative of the change in potential energy. The latter is equal to (-Q)ΔV so the question is what is ΔV between points R and S? If it's zero, there is no need to go further. If it's not zero, the work required to move charge -Q (presumably by an external agent) is the negative of the work done by the electrical force, i.e. -(-Q)ΔV = +QΔV.

 

[RoboNerd]

Do you mean C zero?

Yes, I actually did. :-) Thanks for catching that

 

[RoboNerd]

The four lines labelled with length d are not intended to denote paths. They are intended to denote distances. You cannot integrate E dot dl over any of those lines because they all end up on point charges. The integrals all diverge (i.e. they are all infinite).

That is a very good point. I did not even think of that. Thanks for bringing that to my attention!

Thank you for bringing up very interesting points to my attention. I have determined that the potential differences are zero, and thus, no work is done.

Thank you again everyone for the help!