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- Homework Statement
- I have to find the capacitance of a paralell plate capacitor which is filled with a dielectric ##\epsilon(r)## and has circular plates of Radius ##R_0##. The lower plate carries a charge ##-Q## and is situated at ##z=0## and the upper plate carries a charge ##Q## situated at ##z=d##. Boundary effects can be neglected and ##d<<R_0##. This cylindrical system is paralell to the ##z##-axis.

- Relevant Equations
- $$\epsilon(r)=\epsilon_0+\Delta\epsilon\frac{r}{R_0}$$ where $$r=\sqrt{x^2+y^2}$$

Hey guys! I'm having trouble with the solution that I arrived at.

Through boundary conditions I'm able to determine ##\vec{D}## as $$\vec{D}=-\frac{4Q}{R_0^2}\hat{e_z}$$ (In CGS units)

Trough that I'm able to get the electric field as $$\vec{E}=-\frac{1}{\epsilon(r)}\frac{4Q}{R_0^2}\hat{e_z}$$

Now I can integrate to get the potential difference between ##z=0## and ##z=d##: $$U=\int_0^d-\frac{1}{\epsilon_0+\Delta\epsilon\frac{\sqrt{x^2+y^2}}{R_0}}\frac{4Q}{R_0^2} dz$$

Finally I get the capacitance as: $$C=\frac{(\epsilon_0+\Delta\epsilon\frac{\sqrt{x^2+y^2}}{R_0})R_0^2}{4d}$$

I wondered if that solution can be correct as I always thought about the capacitance ##C## as a constant. Here it dependents on the radius of cylinder. Would I need to integrate the capacitance over ##\phi## from ##0\to2\pi## and ##r## from ##0\to R## (so in cylindrical coordinates) to get the right solution or is it fine as it stands?

I would appreciate any thoughts and comments on this problem! Thank you!

Through boundary conditions I'm able to determine ##\vec{D}## as $$\vec{D}=-\frac{4Q}{R_0^2}\hat{e_z}$$ (In CGS units)

Trough that I'm able to get the electric field as $$\vec{E}=-\frac{1}{\epsilon(r)}\frac{4Q}{R_0^2}\hat{e_z}$$

Now I can integrate to get the potential difference between ##z=0## and ##z=d##: $$U=\int_0^d-\frac{1}{\epsilon_0+\Delta\epsilon\frac{\sqrt{x^2+y^2}}{R_0}}\frac{4Q}{R_0^2} dz$$

Finally I get the capacitance as: $$C=\frac{(\epsilon_0+\Delta\epsilon\frac{\sqrt{x^2+y^2}}{R_0})R_0^2}{4d}$$

I wondered if that solution can be correct as I always thought about the capacitance ##C## as a constant. Here it dependents on the radius of cylinder. Would I need to integrate the capacitance over ##\phi## from ##0\to2\pi## and ##r## from ##0\to R## (so in cylindrical coordinates) to get the right solution or is it fine as it stands?

I would appreciate any thoughts and comments on this problem! Thank you!