Question

Consider a long wire carrying a time varying current $i=kt(k>0)$. A circular loop of radius $a$ is placed with its center at a distance $d$ from the wire $(a<<d)$. The magnetic flux through the loop is

• A. $\frac{{\mu }_0{a}^{2}kt}{d}$
• B. $\frac{{\mu }_0{a}^{2}kt}{2d}$
• C. $\frac{{\mu }_0\pi a^{2}kt}{2d}$
• D. $\frac{{\mu }_0\pi a^{2}kt}{d}$

Answer 1

The correct option is B $\frac{{\mu }_0{a}^{2}kt}{2d}$

Since, current in the wire is continuously increasing, we can conclude that the magnetic field due to this wire in the region, is also increasing with time.

Since, $a<<d$, we can assume the magnetic field to be constant throughout the loop.

Magnetic field $B$ due to wire at the location of loop is given by,

$B=\frac{{\mu }_{0}i}{2\pi d}$

Directed perpendicular to the plane of paper and inwards.

$\therefore$ flux through the circular loop is,

$\varphi =\frac{{\mu }_{0}i}{2\pi d}\times \pi a^2$

$\varphi =\frac{{\mu }_0{a}^{2}kt}{2d}$

Hence, option ($b$) is the correct answer.