Expert: Daniel Mazur - 11/10/2009

Question
Let me translate my question from my book (if I can do so):
We have the capacitor with capacitance 13.5 pF it charged with a battery with the potential V=12.5 v , then we pick up the battery and put the dielectric with  K=6,5   between the capacitor.
It asked how much the potential energy of capacitor before and after putting the dielectric is?
Ui=1/2(CV2)=1055pJ
Uf= Ui/k=162 pJ
W= Uf - Ui=893 pJ
This reduction of energy means that the energy appeard when we put the dielectric there and the Capacitor gives forth to dielectric and works on dielectric about 893 pJ.
If diectric could swing between the capacitor and if there was no fiction , the dielectric with constant mechanical energy (893 pJ) swinged.
This system energy transfered to kinetic energy of dielectric and potential energy of electrical field oscillationally.
But I want to know the frequency of dielectric when it swings (maybe from HOOK LAW or potential energy or with ω=√(k/m)
one of the experts tell me To find U as a function of x, calculate the energy of a capacitor that is partially filled.
but I'm not  sure it's right
C1=έ0A/X for empty part ,and C2=έ0A/k(L-X ) for dielectric
We can suppose that these 2 parts are like 2 parallel  capacitors so the equal capacitor is: C=c1+c2
Also the V of them is similar to the total V.
We have the C and I think I should calculate V as a function of x but I don’t know how.
And if its ok U=1/2(cv2)?
was that right?
thanks


Answer
Hi,

[Q]We have the capacitor with capacitance 13.5 pF it charged with a battery with the potential V=12.5 v , then we pick up the battery and put the dielectric with  K=6,5   between the capacitor.[/Q]
I see you mean that you place the dielectric between the capacitor PLATES. But it is not clear to me, if by "pick up the battery" you actually mean disconnecting it from the capacitor. As you may know, a connected and disconnected battery make two completely different problems.

Then (forgive my own notation E for energy) Ei = Ci*V^2/2 =1055 pJ as you wrote. Assuming that K = eps_r, the relative permittivity of the dielectric, then the Cf = K*Ci. This is, where the solution forks, but only the one with removal of the battery gives you an oscillating dielectric solution:
If you do disconnect the battery from the capacitor, then you work with it in a constant-charge mode, where E = Q^2/(2*C). As the charge Q = Ci*V = 169pC, the Ef = (Ci*V)^2/(2*Cf) = Ci^2*V^2/(2*K*Ci) = Ci*V^2/(2*K) = Ei/K = 162 pJ. The energy difference is indeed W = 893pJ and without friction the dielectric will oscillate (or swing, if you like it better).

There are two things you would need to proceed. One you have spotted: What is the V as a function of x or the energy as a function of x? Let's say the size of the capacitor's plates is L in the direction of motion of the dielectric. Then you know that Q = const = Cx*Vx = [C1(x)+C2(x)]*Vx .
For parallel capacitor C1(x) = eps0*x*H/d and C2(x) = K*eps0*(L-x)*H/d . You made a mistake here. Like you, I use x=0 for full penetration and x=L for dielectric completely outside the capacitor. Then of course
Cx=(eps0*H/d)*(x+K*(L-x))
and Vx is calculated from
Vx=Q/Cx= V*Ci/Cx = V*(eps0*H*L/d)/Cx = V*L/(x+K*(L-x)).

The energy of the system is then described as E(x) = (1/2)*Q^2/Cx = V^2*Ci^2/(2*Cx) = (eps0*H*V^2*L^2)/(2*d*(x+K*(L-x))). From this functional dependence we can already see, that this is NOT going to be a harmonic oscillator. A harmonic one needs to take the form of E =(1/2)*k*x^2 . Our result is not even naturally an even function of x, which means that for it to function properly as a symmetric potential well, we need to implant absolute values of x thus:
E(x) = (eps0*H*V^2*L^2)/(2*d*(|x|+K*(L-|x|)))

This describes a completely anharmonic oscillator and it is impossible to calculate the frequency exactly. You can try the Taylor expansion of this function about x=0, but as the co-efficient of the |x|^1 term of the expansion is non-zero, it is eps0*H*V^2*(1-K)/(2*d*K^2), I think you cannot hope even for an approximate solution.

Secondly, you would need to know the dielectric's inertial mass! Even if you convinced anyone that the oscillations are nearly harmonic and you got the stiffness 'k', you also need to put in the mass to actually extract the frequency.

I am sorry that your problem has no nice solution.

Cheers,
Daniel