Consider a long solenoid of length $I$ and radius $r$. If $n$ is the number of turns per unit length

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Consider a long solenoid of length $I$ and radius $r$. If $n$ is the number of turns per unit length and $\mu_{0}$ is the permeability of free space, the inductance of the solenoid is :

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Sol. [Key Concept: Self-inductance of a long solenoid] Self inductance of solenoid$$\begin{matrix} & L = \frac{N\phi}{i} \ & L = \frac{NBA}{i} \ & L = \frac{N\mu_{0}niA}{i}\left\lbrack n = \frac{N}{I} ightbrack \ & L = N\mu_{0}nA \ & L = \left( \frac{N}{I} ight)\mu_{0}nAl \ & L = \mu_{0}n^{2}Al \ & L = \mu_{0}n^{2}\pi r^{2}l\ \left\lbrack \because A = \pi r^{2} ightbrack \end{matrix}$$

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