Question

I have built a cavity resonator consisting of two concentric conducting cylinders having radii of 20 cm and 27 cm and an overall length of 36 cm. The space between the cylinders is filled with powdered titanium dioxide, with a dielectric constant of 78. In the following, you may ignore the effects that occur at the ends of the cylinders. (a) Find the capacitance of this device. Note that this requires that you find the electrostatic potential of the device, using a dummy charge. (b) The device was not designed to be a capacitor, but rather to produce radio waves. Discuss the ways in which the presence of capacitance might affect this device.

Answer 1

a) In this problem to calculate the capacitance one needs to calculte electric field every where due to cylindrical symmetry and electrostatic potential.

Applying Gauss's law to the above cylindrical capacitor problem

$E = \frac{\lambda }{2\pi \varepsilon _0r}------1$

$\Delta V = \frac{\lambda }{2\pi \varepsilon _0}\ln (\frac{b}{a})------2$

Where $\lambda = Q/L$

The capacitance is given by

$C = \frac{Q}{\left | V \right |}------3$

Substituting the above all known in the third equation gives $C = \frac{2\pi \varepsilon _0L}{\ln (b/a)}$

Where b = 27 cm, a = 20 cm, L = 36 cm. Capacitor is filled with tio2 so instead of epsilon _0 one has to put relative permitivitty. $\varepsilon _r = \varepsilon _s / \varepsilon _0$ where $\varepsilon _r$ is 78 and $\varepsilon _0$ = 8.854x10^-12 F/m

Substituting above all gives.

$C = \frac{2 \times3.14\times78\times8.854\times10^{-12}\times0.36}{\ln \frac{0.27}{0.20}}$

C=5.206 x 10^-9 Farad