Question

A toroid is wound over a circular core with total no. of turns equal to $N$. The radius of each turn is $r$ and radius of toroid is $R(>>r)$. The coefficient of self-inductance of the toroid is given by

• A. $L=\frac{{\mu }_{0}N{r}^2}{2R}$
• B. $L=\frac{{\mu }_{0}Nr}{2R}$
• C. $L=\frac{{\mu }_{0}N{r}^2}{R}$
• D. $L=\frac{{\mu }_0{N}^2{r}^2}{2R}$

Answer 1

The correct option is D $L=\frac{{\mu }_0{N}^2{r}^2}{2R}$

$L=\frac{\varphi }{I},\varphi =NAB$
$B={\mu }_{0}nI$
where $n=\frac{N}{2\pi R}$
$\therefore \varphi =N\pi r^2\left({\mu }_0\frac{N}{2\pi R}I\right)$
$\varphi =\frac{{\mu }_0{N}^2{r}^{2}I}{2R}$
$L=\frac{\varphi }{I}=\frac{{\mu }_0{N}^2{r}^2}{2R}$