Question

In Fig, there is a conducting loop ABCDEF of resistance $\lambda$ per unit length placed near a long straight current-carrying wire. The dimensions are shown in the figure. The lone wire lies in the plane of the loop. The current in the long wire varies as $I=I_0\left(t\right)$. The mutual inductance of the pair is

• A. $\frac{{\mu }_{0}a}{2\pi }\ln \left(\frac{2a+l}{l}\right)$
• B. $\frac{{\mu }_{0}a}{2\pi }\ln \left(\frac{2a-l}{l}\right)$
• C. $\frac{2{\mu }_{0}a}{2\pi }\ln \left(\frac{a+l}{l}\right)$
• D. $\frac{{\mu }_{0}a}{2\pi }\ln \left(\frac{a+l}{l}\right)$

Answer 1

The correct option is A $\frac{{\mu }_{0}a}{2\pi }\ln \left(\frac{2a+l}{l}\right)$

Magnetic field at a point distance "$x$" from the wire, $B=\frac{\mu I}{2\pi x}$
Total magnetic flux linking the loop, $\varphi ={\int }_l^{l+a}Ba.dx+{\int }_{l+a}^{l+2a}Ba.dx={\int }_l^{l+2a}Ba.dx={\int }_l^{l+2a}\frac{\mu I}{2\pi x}a.dx=\frac{\mu Ia}{2\pi }\ln \left(\frac{l+2a}{l}\right)$
Inductance of the circuit $=L=\frac{flux}{I}=\frac{\mu a}{2\pi }\ln \left(\frac{l+2a}{l}\right)$