Question

A circular loop of radius $0.3cm$ lies parallel to a much bigger circular loop of radius $20cm$. The centre of the small loop is on the axis of the bigger loop. The distance between their centres is $15cm$. If a current of $2.0A$ 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
${\varphi }_2=B_1{A}_2=\frac{{\mu }_0{I}_1{R}^2}{2{\left(d^2+R^2\right)}^{3/2}}\pi r^2$
The coefficient of mutual inductance between the loops is
$M=\frac{{\varphi }_2}{I_1}=\frac{{\mu }_0{R}^2\pi r^2}{2{\left(d^2+R^2\right)}^{3/2}}$
Flux linked with bigger loop $1$ is
${\varphi }_1=M{I}_2=\frac{{\mu }_0{R}^2\pi r^2{l}^2}{2{\left(d^2+R^2\right)}^{3/2}}$
Substituting the given values, we get
${\varphi }_1=\frac{4\pi \times 10^{-7}\times {\left(20\times 10^{-2}\right)}^2\times \pi \times {\left(0.3\times 10^{-2}\right)}^2\times 2}{2{\left[{\left(15\times 10^{-2}\right)}^2+{\left(20\times 10^{-2}\right)}^2\right]}^{3/2}}$
${\varphi }_1=9.1\times 10^{-11}$ weber