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(C) The coefficient of self-induction $L$ is defined as $L = \frac{\phi}{I}$,where $\phi$ is the magnetic flux and $I$ is the current.For a toroid,the

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$\frac{\mu_{0} N^{2} r^{2}}{2 R}$

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(C) The coefficient of self-induction $L$ is defined as $L = \frac{\phi}{I}$,where $\phi$ is the magnetic flux and $I$ is the current.For a toroid,the magnetic field $B$ inside the core is given by $B = \mu_{0} n I$,where $n$ is the number of turns per unit length.Given $n = \frac{N}{2 \pi R}$,we have $B = \mu_{0} \left( \frac{N}{2 \pi R} \right) I$.The magnetic flux $\phi$ through each turn of the coil is $\phi = B \cdot A$,where $A = \pi r^{2}$ is the cross-sectional area of the coil.Total flux linked with $N$ turns is $\Phi = N \phi = N (B \cdot A) = N \left( \mu_{0} \frac{N}{2 \pi R} I \right) (\pi r^{2})$.Simplifying this,we get $\Phi = \frac{\mu_{0} N^{2} r^{2} I}{2 R}$.Therefore,the self-inductance $L = \frac{\Phi}{I} = \frac{\mu_{0} N^{2} r^{2}}{2 R}$.

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