Question

A toroid has $1000\text{turns}$. The inner and outer radii of its core are $15\text{cm}$ and $17\text{cm}$ respectively. The magnetic field in the centre of core for a current of $0.985\text{A}$ is $1.97\text{T}$. The value of relative permeability of core is.

Answer 1

Given:
$N=1000\text{turns};r_1=15\text{cm}$
$r_2=17\text{cm};I=0.985\text{A};B=1.97\text{T}$

So, mean radius of toroid will be

$r=\frac{r_1+r_2}{2}=\frac{15+17}{2}$

$\therefore r=16\text{cm}=16\times {10}^{-2}\text{m}$

$\therefore$ No of turns per unit length is

$n=\frac{N}{2\pi r}=\frac{1000}{2\pi \times 16\times {10}^{-2}}$

And we know that, magnetic field at the mean radius due to toroid will be

$B=\mu nI(\because \mu ={\mu }_0{\mu }_r)$

$\Rightarrow \mu =\frac{B}{nI}=\frac{1.97}{0.985}\frac{2\pi \times 16\times {10}^{-2}}{1000}$

$\therefore \mu =64\pi \times {10}^{-5}{\text{TmA}}^{-1}$

So, relative permeability of core will be

${\mu }_r=\frac{\mu }{{\mu }_0}=\frac{64\pi \times {10}^{-5}}{4\pi \times {10}^{-7}}=1600$