Question

A direct current $I$ flows in a long straight conductor whose cross section has the form of a thin half-ring of radius $R$. The same current flows in the opposite direction along a thin conductor located on the axis of the first conductor. If the magnetic induction force between the given conductors is $F=\frac{{\mu }_0{I}^x}{{\pi }^{2}R}$. Find x.

Answer 1

Let us first find magnetic field at $O$ due to ${}^{\prime }{1}^{\prime }$
$dB=\frac{{\mu }_{0}dI}{2\pi R}$
where
$dI=\frac{IRd\theta }{\pi R}=\frac{I}{\pi }d\theta$
$\Rightarrow dB=\frac{{\mu }_0}{2\pi R}\frac{1}{\pi }d\theta$
Net field will be along $ON$
$B={\int }_{-\pi /2}^{\pi /2}dB\cos \theta ={\int }_{-\pi /2}^{\pi /2}\frac{{\mu }_{0}I}{2{\pi }^{2}R}\cos \theta d\theta =\frac{{\mu }_{0}I}{{\pi }^{2}R}$
$F=BIl=\frac{{\mu }_0{I}^2}{2{\pi }^{2}R}l\Rightarrow \frac{F}{l}=\frac{{\mu }_0{I}^2}{{\pi }^{2}R}$