Question

Figure shows a long straight wire carrying current $l$ and a square conducting wire loop of side $I$, at a distance '$a$' from current wire. Both the current wire and loop are in the plane of paper.Find the flux of magnetic field of current wire,passing through the loop.

Answer 1

I is the current carrying in the wire and L is the side of the square loop. The loop is placed at a distance a from the wire. Then the magnetic field is given by :

$B=\frac{{\mu }_{0}I}{2\pi x}$

From the definition of flux

$d\varphi =B.ds$

$=\frac{{\mu }_{0}I}{2\pi x}.adx$

$so,\varphi =\frac{{\mu }_{0}aI}{2\pi }\underset{a}{\overset{a+L}{\int }}\frac{1}{x}$

$\varphi =\frac{{\mu }_{0}Ia}{2\pi }\ln \left(\frac{a+L}{a}\right)$

Therefore, the flux will be, $\varphi =\frac{{\mu }_{0}Ia}{2\pi }\ln \left(\frac{a+L}{a}\right)$.