Find the inductance of a uniformly wound solenoid having N turns and length l. Assume that l is much longer than the radius of the windings and that the core of the solenoid is air.

Question

Find the inductance of a uniformly wound solenoid having N turns and length [latex] \ell ext {. Assume that } \ell [/latex] is much longer than the radius of the windings and that the core of the solenoid is air.

Accepted Answer

We can assume that the interior magnetic field due to the current is uniform and given by Equation 30.17:

[latex] B=\mu_{0} \frac{N}{\ell} I=\mu_{0} n I [/latex] (30.17).

[latex] B=\mu_{0} n I=\mu_{0} \frac{N}{\ell} I [/latex].

where [latex] n=N / \ell [/latex] is the number of turns per unit length. The magnetic flux through each turn is

[latex] \Phi_{B}=B A=\mu_{0} \frac{N A}{\ell} I [/latex].

where A is the cross-sectional area of the solenoid. Using this expression and Equation 32.2, we find that

[latex] L=\frac{N \Phi_{B}}{I} [/latex] (32.2).

[latex] L=\frac{N \Phi_{B}}{I}=\frac{\mu_{0} N^{2} A}{\ell} [/latex] (32.4).

This result shows that L depends on geometry and is proportional to the square of the number of turns. Because [latex] N=n \ell [/latex], we can also express the result in the form

[latex] L=\mu_{0} \frac{(n \ell)^{2}}{\ell} A=\mu_{0} n^{2} A \ell=\mu_{0} n^{2} V [/latex]. (32.5).

[latex] ext { where } V=A \ell ext { is the interior volume of the solenoid. } [/latex]

Question 32.1: Find the inductance of a uniformly wound solenoid having N t...

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