Inductance per unit length near the middle of | vedclass

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Question

(A) The inductance $L$ of a long solenoid is given by the formula:$L = \frac{\mu_0 N^2 A}{l}$Where $N$ is the total number of turns,$A$ is the cross-s

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$\mu_0 \pi \left(\frac{nd}{2}\right)^2$

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(A) The inductance $L$ of a long solenoid is given by the formula:$L = \frac{\mu_0 N^2 A}{l}$Where $N$ is the total number of turns,$A$ is the cross-sectional area,and $l$ is the length.Since the cross-sectional area $A = \pi r^2 = \pi (d/2)^2 = \frac{\pi d^2}{4}$,we substitute this into the equation:$L = \mu_0 \left(\frac{N}{l}ight)^2 	imes l 	imes \frac{\pi d^2}{4}$Given that $n = N/l$ is the number of turns per unit length,the inductance per unit length is:$\frac{L}{l} = \mu_0 n^2 \left(\frac{\pi d^2}{4}ight)$$\frac{L}{l} = \mu_0 \pi \left(\frac{nd}{2}ight)^2$

Inductance per unit length near the middle of a long solenoid is ($\mu_0=$ permeability of free space,$n=$ number of turns per unit length,$d=$ the diameter of the solenoid).

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