Question: pulling apart capacitor plates

A parallel-plate capacitor with plate area 2.0 cm^2, air-gap .5mm, is connected to a 12V battery, if the air-gap is pulled to .75mm, how much work is required?

calculated Q=4.2E-11 C, V(d=0.75)=18V, delta PE=Q*delta V = 2.52E-10 J, but the answer is 1.3E-10 J, which could be a half of my answer. PE/2? why? Would you please explain this to me? Thanks

Ok, so the thing you are changing when you pull a parallel plate capacitor apart is the capacitance (it's ability to store energy).

The capacitance depends on distance like:

C=εo*(A/d)

so clearly the capacitance will decrease if the distance between the plates is increased. Let's consider the capacitance before (C), and after (C'):

C=εo*(A/d)
C'=εo*(A/d')

Ok. So now let's consider how this will effect the energy stored in the capacitor. This can be written in 3 different ways (by substituting Q=CV into the first):

PE=(1/2)QV=(1/2)CV²=(1/2)Q²/C

Which of these should we use? Well, since the plates are connected to a 12 V battery, the voltage will be the same before and after moving the plates. Note that the charge may change so we can't use the expressions with Q in them because we don't know how Q is changing (ok, yes, we could figure it out, but it would be equivalent to using the second expression, so let's not bother).

All right, so let's put it together:

W=-ΔPE=-((1/2)C'V²-(1/2)CV²)
W=-((1/2)εo*(A/d')V²-(1/2)εo*(A/d)V²)
W=-εo*A*V²/2((1/d')-(1/d))

Plugging in the numbers given in the equation above for the work done, I get: 8.49e-11J (using 12V). You also write 18V in your question, with that number I get an answer: 1.91e-10J. Sorry to muddy the waters with more answers... the method should be correct. (maybe you can recheck which numbers the question uses?)

If you haven't already, I'd suggest taking a look at the similar example (though with constant charge not voltage) posted on the main page: Ch. 17, #50.

Hope that helps. Feel free to ask questions by posting comments.

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At this point in the year, everything is fair game for you, so any question you have is fair game for me. Feel free to ask me anything you have covered in the course (or even things from last semester that might come up).

On that note, a piece of wisdom from my 3rd year mechanics professor:

You don't have problems, you have exercises! ~Dr. Melvin Calkin