Question

A capacitor consists of two concentric cylinders. The inner cylinder has a radius of 0.001 m and the outer cylinder a radius of 0.0011 m. The length of the capacitor is 1 m. If centered on the z-axis, the region 0 < ∅ < π has a dielectric constant of 2 and the region π < ∅ < 2π a dielectric constant of 4. Find the capacitance. Ignore fringing fields.

Answer 1

Answer:

The capacitance is 1.75 nF

Explanation:

From the question we are given that

    The inner radius is $r_{in} = 0.001$

     The outer radius is $r_{out} = 0.0011 \ m$

    Length of the capacitor is $L = 1m$

    The dielectric constant is $Di = 2 \ for \ 0 < \phi < \pi$

   The dielectric constant is  $Di_2 = 4 \ for \ \pi < \phi < 2\pi$

Generally the capacitance of a capacitor can be mathematically represented as

                $C = \frac{\pi \epsilon_0 Di_1 L}{ln\frac{r_{out}}{r_{in}} } + \frac{\pi \epsilon_0 Di_2L}{ln\frac{r_{out}}{r_{in}} }$

                   $= \frac{\pi \epsilon_0 L (Di_1 + Di_2)}{ln\frac{r_{out}}{r_{in}} }$

                  $= \frac{(3.142)(8.85*10^{-12})(1)(2+4)}{ln\frac{0.0011}{0.001} }$

                  $=1.75*10^{-9} F$

                  $1.75nF$