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

The capacitance is 1.75 nF

Step-by-step 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 = (\pi \epsilon_0 Di_1 L)/(ln(r_(out))/(r_(in)) ) + (\pi \epsilon_0 Di_2L)/(ln(r_(out))/(r_(in)) )`

`= (\pi \epsilon_0 L (Di_1 + Di_2))/(ln(r_(out))/(r_(in)) )`

`= ((3.142)(8.85*10^(-12))(1)(2+4))/(ln(0.0011)/(0.001) )`

`=1.75*10^(-9) F`

`1.75nF`