Question

A circular loop of radius $0.3\, cm$ lies parallel to a much bigger circular loop of radius $20\, cm$. The centre of the small loop is on the axis of the bigger loop. The distance between their centres is $15\, cm$. If a current of $2.0 \,A$ flows through the smaller loop, then the flux linked with bigger loop is

• A. $6.6 \times 10^{-9}$ weber
• B. $9.1 \times 10^{-11}$ weber
• C. $6 \times 10^{-11}$ weber
• D. $3.3 \times 10^{-11}$ weber

Answer 1

Answer: (B)

As field due to current loop $1$ at an axial point
$B_{1}=\frac{\mu_{0}I_{1}R^{2}}{2\left(d^{2}+R^{2}\right)^{3 /2}}$
Flux linked with smaller loop $2$ due to $B_{1}$ is
$\phi_{2}=B_{1}A_{2}=\frac{\mu_{0}I_{1}R^{2}}{2\left(d^{2}+R^{2}\right)^{3/ 2}}\pi r^{2}$
The coefficient of mutual inductance between the loops is
$M=\frac{\phi_{2}}{I_{1}}=\frac{\mu_{0}R^{2}\pi r^{2}}{2\left(d^{2}+R^{2}\right)^{3 / 2}}$
Flux linked with bigger loop $1$ is
$\phi_{1}=MI_{2}=\frac{\mu_{0}R^{2}\pi r^{2}l^{2}}{2\left(d^{2}+R^{2}\right)^{3 / 2}}$
Substituting the given values, we get
$\phi_{1}=\frac{4\pi\times10^{-7}\times\left(20\times10^{-2}\right)^{2}\times\pi\times\left(0.3\times10^{-2}\right)^2\times2}{2\left[\left(15\times10^{-2}\right)^{2}+\left(20\times10^{-2}\right)^{2}\right]^{3/ 2}}$
$\phi_{1}=9.1\times10^{-11}$ weber