# Three Plate Capacitor and Force

1. Sep 29, 2012

### schaefera

1. The problem statement, all variables and given/known data
This is basically question 3.18 in Purcell. You have a plate being suspended into the middle of two other plates, and then you need to determine the force downward on the hanging plate given that the separation between it and either side of the other conductor is s; the plates have length b; the amount of the inside dangling plate between the other two is height y.

2. Relevant equations
F= (1/2)Q^2 * d/dx(1/C)

3. The attempt at a solution
I'm trying to say that this thing acts as two capacitors side by side, but only for the distance y by which it is dangling down. Since U= (1/2)CV^2, the total energy of the system is 2*(1/2)*C*V^2, and C=(by)/(4*pi*s). So I get the total energy as a function of capacitance; I differentiate the energy to get the force on it, and in the end I find that:

F= -bV^2/(4*pi*s).

Is this right?
Can the field between the plates be determined as 4*pi*sigma, or is it 2*pi*sigma since the center plate really has 2*pi*sigma amount of charge per area on each side?

2. Sep 29, 2012

### TSny

That's what I get also. Doesn't guarantee it's correct of course.
I'm not following the statement "...the center plate really has 2*pi*sigma amount of charge per area on each side". Suppose sigma is the charge density on each of the outer plates. What would be the charge density on each face of the inner (center) plate?

Last edited: Sep 29, 2012
3. Sep 30, 2012

### rude man

You haven't given the potential difference between the two outer plates. You also nebulously describe the geometrical setup. If s is the disatance between the cente and either outer plate, what is y again??

It would be much preferable if you quoted the problem verbatim.

4. Sep 30, 2012

### TSny

Here's the setup. As you can see, the two outer plates are actually part of one bent sheet. The problem refers to Eq. 27 of the text which is $F = \frac{Q^2}{2}\frac{d}{dx}(\frac{1}{C})$.

#### Attached Files:

• ###### Prob 3-18 Purcell.jpg
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5. Sep 30, 2012

### rude man

Oh, how a picture is worth a thousand words!

Hint: invoke virtual work! i.e. F*Δy = Δenergy in the E fields.

6. Sep 30, 2012

### schaefera

I did exactly that. It would appear we already found the solution, but thanks!