# Work to Move parallel capacitor Plates

1. ### Typhon4ever

43
Hi, in my book there are two capacitor plates distance x apart (they are kept connected to a constant voltage source). These plates are moved apart to distance 3x. In order to find the work done to move the capacitors apart (using W=F dl=QE dl), my book takes the charge Q of one plate and electric field felt on that one plate and integrates it from x to 3x. Why can't we take the take the charge and electric field felt on both plates and integrate half the distance of x to 3x?

### Staff: Mentor

OK, but Q and E are not constant as the distance changes.

Who says you can't?

3. ### Typhon4ever

43
Okay, how would I set up the integration of both plates? My book says the Q on one plate is $\frac{\epsilon_0 A V}{L}$ and the electric field felt on that one plate is $\frac{V}{2L}$ how would I change these to be both plates?

### Staff: Mentor

The only thing that would change would be your variable of integration. Since you want to move each plate half the distance, for each plate the position y would relate to the distance between them by L = 2y. Then just integrate from y = x/2 to y = 3x/2. You'll need to multiply your answer by 2, of course.

Compare that to just moving one plate (like in your book). In that case y would be the total distance, so L = y. And you'd integrate from y = x to y = 3x. (And not multiply by 2.)

Each method should give the same answer for the work done.

5. ### Typhon4ever

43
Oh, I see. You have to focus on one plate. There is no possible way to calculate work done on both plates at once? Is that because the Q between them both is 0?

### Staff: Mentor

I'm not sure what you mean. You are calculating the work done on both plates--it's the same on both so that's why you multiply by 2.

7. ### Typhon4ever

43
What I meant is that is there a way to calculate the total work done without multiplying by two? Or can we only calculate the work on one plate and have to double it.

### Staff: Mentor

Well, you can always do the integration twice! Once for each side.

9. ### Typhon4ever

43
Isn't that the same thing as multiplying by 2?

### Staff: Mentor

Yep.

But since the two plates experience the same force and movement, why not take advantage of symmetry?

43
Thanks!