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(D) The coefficient of self-induction $L$ is defined as $L = \frac{\phi}{I}$,where $\phi$ is the total magnetic flux linked with the coil and $I$ is t

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$\frac{\mu_0 N^2 r^2}{2 R}$

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(D) The coefficient of self-induction $L$ is defined as $L = \frac{\phi}{I}$,where $\phi$ is the total magnetic flux linked with the coil and $I$ is the current flowing through it.For a toroid with $N$ turns,the total flux $\phi$ is given by $\phi = N \cdot A \cdot B$,where $A$ is the cross-sectional area of the toroid and $B$ is the magnetic field inside it.The cross-sectional area $A = \pi r^2$.The magnetic field inside a toroid is $B = \mu_0 n I$,where $n = \frac{N}{2 \pi R}$ is the number of turns per unit length.Substituting these into the flux equation:$\phi = N (\pi r^2) (\mu_0 \frac{N}{2 \pi R} I) = \frac{\mu_0 N^2 r^2 I}{2 R}$.Finally,the self-induction $L = \frac{\phi}{I} = \frac{\mu_0 N^2 r^2}{2 R}$.

$A$ toroid is a long coil of wire ($N$ turns) wound over a circular core. The coefficient of self-induction of the toroid is [The magnetic field in it is uniform and $R >> r$,where $r=$ radius of wire,$R=$ radius of coil] ($\mu_0=$ permeability of free space).

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