Parallel plate capacitor with dielectric in a gravitational field.

[adwodon]

Homework Statement

A square parallel plate capacitor with vertical plates of area A
and distance d, charged with a constant charge Q and is completely
filled with a dielectric material the same dimension as the gap between
the plates, with dielectric constant k and mass m. Assuming
the dielectric is a solid block of material that can move inside the
capacitor with no friction, what would be the equilibrium condition
in the presence of gravitational field? What would be the period of
small oscillations around this equilibrium point?

Homework Equations

C=(k\epsilon₀A)/d

The Attempt at a Solution

Honestly I don't know where to begin with this question. Is it suggesting that the dielectric would oscillate from side to side between the plates? How? I am assuming I've just stared at the problem too long and I am missing something obvious, so if someone could just nudge me in the right direction by pointing out what this question is asking it would help a lot.

 

[Chi Meson]

Vertical plates! There will be two forces in balance, one the weight of the dialectric, the other an electrostatic attraction between plates and the dialectric.

 

[connor415]

Why is there electrostatic attraction toward the dielectric, it has no charge! Adwodon I am pretty sure youre on my course btw. UCL?

 

[adwodon]

Hey
Sorry Chi Meson I forgot to thank you it was a case of me staring at it so long I completely ignored the vertical part, it took a while but I figured it out before our original due date (tuesday), we got an extension though as barely anyone could do all 3 questions (this being the first and easiest).

Connor yes I am at UCL.

The dielectric has no overall charge, but the electrons will move towards the positive plate of the capacitor so you get something like this:

http://upload.wikimedia.org/wikiped...x-Capacitor_schematic_with_dielectric.svg.png

So the attraction is only between the edge of the dielectric.
When a dielectric is fully inserted this force will cancel itself out, but if there is a gap it will pull it in (ie if the dielectric starts to fall out it will be pulled back in)

If you want some help imagine the dielectric is horizontal for now, push it into the dielectric by a distance x

Capacitance of the part filled with dielectric will be:

C1=(e0KLx)/d

Part filled with air:

C2=(e0L(L-x))/d

as the volage across the two parts is the same
C=C1+C2

As charge is constant:

U= (-Q^2)/2C

F= -dU/dx

Thats how you figure out the force the plates put on the dielectric, then just imagine the plates were vertical. As for the small oscillations, just see what happens when the dielectric is pushed a small distance past equilibrium (y, where y<<x).

If you're still having trouble I am easy to spot, I am the guy with the arm covered in tattoos. Although I am pretty sure I've nailed this one I haven't touched the rest of this problem sheet though. Too busy with other work.