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Number2Pencil
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Homework Statement
An electrostatic speaker is constructed using two conductive plates (stators) and an electrostatically charged diaphragm in the middle which vibrates. One of the conductors is grounded, and the other has an amplified voltage applied to it to drive the force.
A uniform charge density, ps (C/m^2) is maintained on the diaphragm. The stator conductors are separated by distance d.
Find the symbol equation for Pressure, P, as a function of Voltage, V(t). Use only the symbols Q, ps, A, E, V (where V=V(t)), d, and F.
Homework Equations
[tex]P=\frac{F}{A}[/tex]
[tex]F=Q(E+v \times B)[/tex]
[tex]Q=\rho_s A[/tex]
[tex]V = -\int{E \cdot dl}[/tex]
The Attempt at a Solution
I am having a hard time conceptualizing how to find voltage or electric field. Here is what I've done:
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METHOD 1:
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Since there are no magnetic fields in this problem...
[tex]F=QE[/tex]
Since all electric fields are parallel (angle = 0)
[tex]V = -\int{Edl}[/tex]
[tex]V = -Ed[/tex]
[tex]E = -\frac{V}{d}[/tex]
So now I have Q, and I have E, let's solve for F:
[tex]F = - \rho_s A \frac{V}{d}[/tex]
Solving for pressure:
[tex]P = \frac{F}{A} = -\rho_s \frac{V}{d}[/tex]
But alas, it says this is incorrect. I am guessing that the E-Field at the plate is more complicated than just -V/d
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--METHOD 2: Point Charges
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So next I thought, "If the charges are uniformly distributed, maybe I can look at just single point charges on the
conductors."
Coulombs Law:
[tex]F = \frac{k Q_{plate} Q}{r^2}[/tex]
We can leave Q as Q since that is a symbol in the answer, but we can use Gauss' Law to come up with a relationship
between electric fields and charge:
[tex]Q = \epsilon_0 E A[/tex]
I took the liberty of getting the results from someone doing a parallel plate capacitor example.
Similarily:
[tex]E = -\frac{V}{d}[/tex]
Giving us:
[tex]Q_{plate} = -\frac{\epsilon_0 V A}{d}[/tex]
and we know that the distance between the positive plate and the diaphragm is d/2, putting all this together:
[tex]F = \frac{4 k \epsilon_0 V A Q}{d^3}[/tex]
Change to pressure by dividing by Area:
[tex]P = \frac{F}{A} = \frac{4 k \epsilon_0 V Q}{d^3}[/tex]
Since our gap medium is air, we know that k is:
[tex]k = \frac{1}{4 \pi \epsilon_0}[/tex]
Plugging and canceling out common terms, I wind up with this answer:
[tex]P = -\frac{VQ}{\pi d^3}[/tex]
But the answer guide said it was incorrect. At this point I will just admit that I am unclear on how to approach this
problem...can someone do some nudging or hand-holding and explain the flaws in my shoddy physics?