MHT CET · Physics · Electromagnetic Induction

A toroid is a long coil of wire wound over a circular core. If 'r' and 'R' are the radii
of the coil and toroid respectively, the coefficient of self-induction of the toroid is
(The magnetic field in it is uniform and \(\mathrm{R}>>\mathrm{r}\) )
\(\left(\mathrm{N}=\right.\) number of turns of the coil and \(\mu_{0}=\) permeability of free space)

A \(\frac{2 \mu_{0} \mathrm{r}^{2}}{\mathrm{~N}^{2} \mathrm{R}}\)
B \(\frac{\mu_{0} \mathrm{~N}^{2} \mathrm{R}^{2}}{2 \mathrm{r}}\)
C \(\frac{\mu_{0} \mathrm{~N}^{2} \mathrm{R}^{2}}{2 \mathrm{R}}\)
D \(\frac{\mu_{0} \mathrm{R}}{2 \mathrm{~N}^{2} \mathrm{r}^{2}}\)

Verified Solution

Answer & Solution

Correct Answer

(C) \(\frac{\mu_{0} \mathrm{~N}^{2} \mathrm{R}^{2}}{2 \mathrm{R}}\)

Step-by-step Solution

Detailed explanation

\(\mathrm{L}=\frac{\phi}{\mathrm{I}}, \phi=\mathrm{NAB}\)
\(\mathrm{B}=\mu_{0} \mathrm{nI}\)
where \(n=\frac{N}{2 \pi R}\)
\(\therefore \phi=\mathrm{N} \pi \mathrm{r}^{2}\left(\mu_{0} \frac{\mathrm{N}}{2 \pi \mathrm{R}} \mathrm{I}\right)\)
\(\phi=\frac{\mu_{0} N^{2} r^{2} I}{2 R}\)
\(\mathrm{L}=\frac{\phi}{\mathrm{I}}=\frac{\mu_{0} \mathrm{~N}^{2} \mathrm{r}^{2}}{2 \mathrm{R}}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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