Question

A long, cylindrical solenoid with 140 turns per centimeter has a radius of 1.5 cm. (a) Neglecting end effects, what is the self-inductance per centimeter of the solenoid? (b) If the current through the solenoid changes at the rate 8 A/s, what is the voltage induced per centimeter of the solenoid?

Answer 1

a )

The magnetic field of a solenoid is

`B = \mu n I`

where n is the total number of turns per length,
`n = N/l`.

This can be derived from Ampere's law, but usually it is given in questions.

The flux through a single loop in the solenoid is
`B\pi r^2`, and the total flux through solenoid is

`\Phi = (\mu_0 n I) (\pi r^2)(nl)`

where
`l` is the total length of the solenoid. Here, the third parenthesis is the total number of turns in the solenoid.

The self inductance of the solenoid is

`L = (\Phi)/(I) = ((\mu_0 n I)(\pi r^2)(nl))/(I) = \mu_0 n^2\pi r^2 l`

`L = \mu_0 (140)^2 \pi (1.5)^2 l = 44100\mu_0\pi l`

`L = 44100* \mu_0\pi l`

`L = 44100 (4 * 3.14 * 10^(-7)) * 3.14 * l\\L = 0.17 l`

b)

By Faraday's law, the voltage induced in a solenoid is

`|\varepsilon| = L(dI)/(dt).`

`(dI)/(dt)` is given in the question as 8 A/s.

Hence,
`|\varepsilon| = 0.17* l * 8`

The voltage induced per centimeter of the solenoid is

`|\varepsilon| = (0.17 * l * 8)/(l)`

`|\varepsilon| = 1.36V/cm.`

Step-by-step explanation:

If the exact numerical answer is needed, we should use the following values:

`\mu_0 = 4\pi * 10^(-7)`

`\pi = 3.14`

The reason we are using the notation
`|\varepsilon|` is that the Faraday's Law indicates the direction of the current as well. However, the question does not ask the direction of the current, so the correct way use the Faraday's law is to use absolute value of the induced voltage.