Question

An infinite wire carrying current $i=i_o$ sin $\omega t$ is kept along the axis of a toroid of mean radius a. The cross sectional area of the toroid is $A(<<a^2)$. Resistance of toroid is $R$, the mean power dissipated P in the toroid is $n\times {10}^{-2}\text{W},$ find $n$.
Number of turns in toroid $=N,i_o=10A,\omega =100rad/s,N=1000,A=1c{m}^2,a=10cm,R=1\mu \Omega$.

Answer 1

Magnetic field $B=\frac{{\mu }_{0}i}{2\pi a}$
magnetic flux$\varphi =\frac{{\mu }_{o}i}{2\pi a}\times A\times N$
emf generated is
$\epsilon =\frac{{\mu }_0{i}_0\omega AN}{2\pi a}\cos \omega t$
As mean value of ${\cos }^2\theta$ is $0.5$
$P={\left(\frac{{\mu }_0{i}_0\omega AN}{2\pi a}\right)}^2\times \frac{1}{2}\times \frac{1}{R}=\frac{2\times {10}^{-8}}{R}=2\times {10}^{-2}$