Question

In figure, a square loop $PQRS$ of side $a$ and resistance $r$ is placed near an infinitely long wire carrying a constant current $I$. The sides $PQ$ and $RS$ are parallel to the wire. The wire and the loop are in the same plane. The loop is rotated by ${180}^o$ about an axis parallel to the long wire and passing through the mid-points of the sides $QR$ and $PS$. The total amount of charge which passes through any point of the loop during rotation is

• A. $\frac{{\mu }_{0}Ia}{2\pi r}\ln 2$
• B. $\frac{{\mu }_{0}Ia}{\pi r}\ln 2$
• C. $\frac{{\mu }_{0}I{a}^2}{2\pi r}\ln 2$
• D. cannot be found because time of rotation is not given
• E. answer required

Answer 1

The correct option is B $\frac{{\mu }_{0}Ia}{\pi r}\ln 2$

Magnetic flux density at a distance x from infinitely long current carrying wire = $B=\frac{\mu I}{2\pi x}$

flux linking the loop initially = $\varphi ={\int }_a^{2a}Badx={\int }_a^{2a}\frac{\mu I}{2\pi x}adx={\left|\frac{\mu Ia}{2\pi }lnx\right|}_a^{2a}=\frac{\mu Ia}{2\pi }ln2$

Upon rotating the loop, total change in flux is $△\varphi =2\times \frac{\mu Ia}{2\pi }ln2=\frac{\mu Ia}{\pi }ln2$

emf induced in circuit = $\frac{△\varphi }{△t}$

total charge that has moved = $i△t=\frac{emf}{r}△t=\frac{1}{r}\frac{△\varphi }{△t}△t=\frac{\mu Ia}{\pi }ln2\frac{1}{r}$