Hello(adsbygoogle = window.adsbygoogle || []).push({});

Here's a situation I would like to discuss...

Suppose we have an ideal parallel plate capacitor which is filled with air (or vacuum) with permittivity = 1. Now if a dielectric slab of width equal to the space (d) between the plates is inserted into the space between the plates, it will be pulled in with a force, provided there is some charge on the capacitor plates. There are two possibilities:

(a) Constant Potential Difference across the capacitor plates imposed by an ideal emf source which is kept connected at all times to the capacitor. (Charge will change as the extent of the slab inside the capacitor changes)

(b) Constant charge on the capacitor plates, maintained by removing the emf source after the capacitor is charged (and only then is the dielectric brought in).

I know how to compute the force on the dielectric in both cases. In case (a) the force turns out to be independent of the length of the slab inside the dielectric and in case (b) it is a function of this length. The problem is to "prove" that in case (a) if the dielectric is displaced slightly from its equilibrium position (inside the capacitor) it will perform simple harmonic motion.

I want to prove it from first principles by showing that there is a restoring force on the slab which tends to bring it back to its position of stable equilibrium and for small displacements satisfies a relation of the form,

[tex]m\frac{d^2x}{dt^2} + kx = 0[/tex]

where m = mass of slab, k = effective "spring constant", x = displacement from stable equilibrium position.

The only forces on the slab are its weight and the pulling force of the capacitor plates (due to induced charge). Neglecting fringing effects and assuming that the slab has no vertical displacement, the only force that should enter an equation of the form above is the pulling force.

But it is independent of x so it can't result in simple harmonic motion. What is wrong here?

I would be very grateful if someone could offer suggestions or ideas regarding this problem.

Thanks and cheers,

Vivek

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Simple Harmonic Motion of a Dielectric Slab in a Parallel Plate Capacitor

**Physics Forums | Science Articles, Homework Help, Discussion**