Question

Derive an expression for the magnetic induction at a point due to an infinitely long straight conductor carrying current. Write the expression for the magnetic induction when the conductor is placed in a medium of permeability ${}^{\prime }{\mu }^{\prime }$.

Answer 1

$dB=\frac{{\mu }_0.Idlsin\Theta }{4\pi r^2}$

$l=atan\varphi$ $dl=ase{c}^2\varphi d\varphi$

$a=rccos\varphi =r^2=a^{2}se{c}^2\varphi$

thus,

$dB=\frac{{\mu }_0.Icos\varphi }{4\pi a}d\varphi$

$B=\frac{{\mu }_0.I[sin{\Theta }_1+sin{\Theta }_2]}{4\pi a}$

For infinitely long conductor ${\Theta }_1={\Theta }_2={90}^0$

$B=\frac{{\mu }_0.I}{2\pi a}$

it is placed in medium with $\mu$

$B=\frac{\mu I}{2\pi a}$