Along straight wire carries a current $I$. A rectangular loop of sides $a$ and $b$ is placed in the same plane such that the side of length $a$ is parallel to the wire at a distance $d$ from it. Calculate the magnetic flux $\Phi$ through the loop.

Solution

1. The magnetic field $B$ at a distance $x$ from along straight wire carrying current $I$ is given by $B=\frac{{\mu }_{0}I}{2\pi x}$.
2. The magnetic flux $\Phi$ through a differential strip of width $dx$ and length $a$ at distance $x$ is $$\begin{array}{rl}d\Phi & =B\cdot a\cdot dx\\ & =\frac{{\mu }_{0}Ia}{2\pi x}dx\end{array}$$.
3. Integrate $d\Phi$ from $x=d$ to $x=d+b$ to find the total flux: $\Phi ={\int }_d^{d+b}\frac{{\mu }_{0}Ia}{2\pi x}dx$.
4. Evaluating the integral yields $$\begin{array}{rl}\Phi & =\frac{{\mu }_{0}Ia}{2\pi }[\ln x{]}_d^{d+b}\\ & =\frac{{\mu }_{0}Ia}{2\pi }\ln (\frac{d+b}{d})\end{array}$$.
5. Simplifying the logarithmic term results in $\Phi =\frac{{\mu }_{0}Ia}{2\pi }\ln (1+\frac{b}{d})$.
6. Hence the answer is (A).