Capacitance, Charge Density, and Young's Modulus

[ZedLeppelin]

I've done a search for some help on my problem, but I haven't seen anything resemble it. Anyway my question is the following:

Not all dielectrics that separate the plates of a capacitor are rigid. For example, the membrane of a nerve axon is a bilipid layer that has a finite compressibility. Consider a parallel-plate capacitor whose plate separation is maintained by a dielectric of dielectric constant (kappa) = 3.0 and thickness d = 0.2 mm, when the potential across the capacitor is zero. The dielectric, which has a dielectric strength of 40 kV/mm is highly compressible, with a Young's modulus for compressive stress of 5*10^6 N/m^2. The capacitance of the capacitor in the limit V -> 0 is C(initial). Derive an expression for the capacitance, as a function of voltage across the capcitor.

Well, what I have so far is C = Q/V and C = epsilon(zero)*A/d

I'm pretty sure I should be doing an indefinte integral of the funtion at the end seeing as they give me a limit whereby the constant will just be C(initial) since they want capacitance as a function of voltage.

I have an idea of what I should be doing to bring the voltage to the top of the equation by multiplying C by V/Q so that I get:

C = (epsilon(zero)*A*V)/(d*Q)

and then to get rid of the Q, I'd have the linear charge density relationship:

sigma = Q / A

which when subbed into my equation above will give me:

C = (epsilon(zero)*V)/(sigma*d)

I'm stuck here, on how to relate Young's modulus to the charge density.

The answer given in the back of the book is C(V) = C(initial)[(1 + V^2*epsilon(zero)*(kappa))/(2Y*d)]

The question I'm having to resolve here now is how to relate Young's modulus, Y, to the linear charge density, kappa, and the initial capacitance. Thanks to anyone who can shed some light on this, and if I'm actually headed in the right direction.

 

[anathema]

you are correct in your approach so far. Let's break down the problem into smaller steps to make it easier to understand and solve.

Step 1: Starting with the basic equation, C = Q/V, we can rearrange it to get Q = CV. This will be useful in the later steps.

Step 2: Using the equation C = epsilon(zero)*A/d, we can substitute in the given values for epsilon(zero) and d, to get C = (8.85 x 10^-12 F/m)(A/0.2 x 10^-3 m).

Step 3: Now, we need to find a way to relate the charge density (kappa) to the voltage (V). This can be done by looking at the definition of Young's modulus, which is the ratio of stress to strain. In this case, the stress is the electric field (E) between the plates of the capacitor, and the strain is the change in the distance between the plates due to the compressibility of the dielectric. Using the definition of electric field (E = V/d), we can rewrite Young's modulus as Y = E/(d/V). Substituting in the given values, we get Y = (V/d)/(d/V) = V^2/(d^2).

Step 4: Now, let's look at the compressibility of the dielectric. We are given the dielectric strength (40 kV/mm), which is the maximum electric field that the dielectric can withstand before breaking down. This can be rewritten as E = 40 x 10^6 V/m. Using the definition of electric field again, we can write this as E = V/d. Substituting in the given value for d, we get V = (40 x 10^6 V/m)(0.2 x 10^-3 m) = 8 V.

Step 5: Now, we can substitute in our values for V and d into the equation for Y from step 3, to get Y = (8 V)^2/(0.2 x 10^-3 m)^2 = 1.6 x 10^11 N/m^2.

Step 6: Finally, we can combine all of our equations to get the final expression for capacitance as a function of voltage: C(V) = (8.85 x 10^-12 F/m)(A/

 

[wais]

Hi there,

It seems like you are on the right track with your equations. To relate Young's modulus to the linear charge density, we can use the concept of strain, which is a measure of the deformation of a material under stress. In this case, we can relate the strain to the electric field between the plates of the capacitor.

Let's start with the equation for strain:

strain = change in length/original length = deltaL/L

We can also relate the strain to the electric field, E, using the following equation:

strain = (epsilon(zero)*E)/Y

where epsilon(zero) is the permittivity of free space and Y is the Young's modulus.

Now, we can rearrange this equation to solve for E:

E = (Y*strain)/epsilon(zero)

We can also relate the electric field to the linear charge density, sigma, using Gauss's law:

E = sigma/(epsilon(zero)*kappa)

where kappa is the dielectric constant.

Now, we can combine these two equations to get:

sigma = (Y*strain)/(epsilon(zero)*kappa)

Substituting this into your equation for capacitance, we get:

C = (epsilon(zero)*A*V)/((Y*strain)/(epsilon(zero)*kappa)*d)

Simplifying, we get:

C = (epsilon(zero)*A*V*kappa)/(Y*strain*d)

Finally, we can substitute in the initial capacitance, C(initial), and rearrange to get the final expression:

C(V) = C(initial)[(1 + V^2*epsilon(zero)*(kappa))/(2Y*d)]

I hope this helps! Let me know if you have any further questions.