A toroid is a long coil of wire wound over a circular core. If 'r' and 'R' are the radii of the coil and…

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)
  1. $\frac{2 \mu_{0} \mathrm{r}^{2}}{\mathrm{~N}^{2} \mathrm{R}}$
  2. $\frac{\mu_{0} \mathrm{~N}^{2} \mathrm{R}^{2}}{2 \mathrm{r}}$
  3. $\frac{\mu_{0} \mathrm{~N}^{2} \mathrm{R}^{2}}{2 \mathrm{R}}$
  4. $\frac{\mu_{0} \mathrm{R}}{2 \mathrm{~N}^{2} \mathrm{r}^{2}}$

Solution

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

Asked in: MHT CET 2020 (20 Oct Shift 1)

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