Capacitor with sinusoidal voltage placed near a plate

[unscientific]

Homework Statement

An electrode with a varying voltage V = V₀ + V_ACe^jωt is placed near a plate. The current output from the electrode is then measured. We want to find the relation between the current output and the amplitude of vibration. We can assume V_AC is small relative to V₀.

Homework Equations

Idea:

1. Constant charge in plate -> Constant force exerted -> separation changes -> capacitance changes
2. Apply sinusoidal voltage->phase change of force for every ∏ rad -> oscillation
3. these leads to changes in output current detected in electrode.

The Attempt at a Solution

V = V₀ + V_ACe(jωt)
Q = CV
I = V*dC/dt + C*dV/dt
I = jCV_ACωe^jωt + V*dC/dt


C = ε₀A/d
dC/dt = - (ε₀A/d²) * d(d)/dt

I = jCV_ACωe^jωt + V*(ε₀A/d²) * d(d)/dt

To find: d(d)/dt

F = Q²/(2ε₀A)
m*dv/dt = Q²/(2ε₀A)

v = Q²/(2mε₀A) * t

therefore, the current is:

I = jCV_ACωe^jωt + Q³/(2ε₀Am) * (1/d) * t

So the current depends both on time and inversely to displacement?

 

[unscientific]

bumpp. My supervisor believes the current is proportional to the displacement though.

 

[unscientific]

It's application is on a torsional oscillator driven capacitively.

 

[rude man]

If this is a driven system then there must be some sort of torsional spring restraining element between the plates pulling them apart. Because the plates are always drawn together by the applied voltage irrespective of its polarity. What is the torsional spring constant? What is d when the torsion torque is zero?

OR: are you sure this is not just a displacement sensor? In other words, is d not the independent variable? In which case force is not a consideration.

 

[unscientific]

If this is a driven system then there must be some sort of torsional spring restraining element between the plates pulling them apart. Because the plates are always drawn together by the applied voltage irrespective of its polarity. What is the torsional spring constant? What is d when the torsion torque is zero?

OR: are you sure this is not just a displacement sensor? In other words, is d not the independent variable? In which case force is not a consideration.

The plate is in fact a torsional oscillator, driven capacitively by the electrodes. So a sinusoidal voltage is applied to the electrode which attracts the oscillator at a certain frequency ω. The current depends on the capacitance, which in turn depends on the distance between the electrode and the oscillator. I want to find the relationship between the current observed and the distance between them.

 

[rude man]

I want to find the relationship between the current observed and the distance between them.

So why are you pursuing force on the plates? As I suspected, d is the independent variable. You already did a lot of analysis & came up with (incl. sign correction)
I = jCVACωejωt - V*(ε0A/d2) * d(d)/dt

I suggest replacing d with h to avoid confusion.

So that's I = jCV_acωe^jωt - V*(ε0A/h²) * dh/dt

Now, replacing C with ε₀A/h and V with V₀ + V_ace^jwt you get

I = jwε₀AV_ac/h e^jwt - (V₀ + V_ace^jwt)ε₀A (1/h²) dh/dt.

And that is as far as you can go without knowing what h(t) is.
Of course, replace each term with its real part to get the actual dependence of I on h.

Note that if dh/dt = 0 you get the standard relation between current and voltage of a fixed capacitor.

 

[unscientific]

So why are you pursuing force on the plates? As I suspected, d is the independent variable. You already did a lot of analysis & came up with
I = jCVACωejωt + V*(ε0A/d2) * d(d)/dt

I suggest replacing d with h to avoid confusion.

So that's I = jCV_acωe^jωt + V*(ε0A/h²) * dh/dt

Now, replacing C with ε₀A/h and V with V₀ + V_ace^jwt you get

I = jwε₀A/h e^jwt - (V₀ + V_ace^jwt)ε₀A (1/h²) dh/dt.

And that is as far as you can go without knowing what h(t) is.
Of course, replace each term with its real part to get the actual dependence of I on h.

Note that if dh/dt = 0 you get the standard relation between current and voltage of a fixed capacitor.

So I thought. I'm not sure how he got it as linearly related to d; I'll have to look at his working when I go back to the lab and i'll post it here.

 

[rude man]

So I thought. I'm not sure how he got it as linearly related to d; I'll have to look at his working when I go back to the lab and i'll post it here.

Thanks. Only thing I can think of is if dh/dt is small, but even then I goes as 1/h, not h.

I suspect we don't have the complete picture. Could you post the question as originally posed? Thanks for posting the answer also, wish more OP's would do that.