Question

Two concentric circular coils, one of small radius $r$. and the other of large radius $r_{2}$, such that $r_{1}<<\,r_{2}$, are placed coaxially with centres coinciding. The mutual inductance of the arrangement is

• A. $\frac{\mu_{0} \pi r_{1}^{2}}{r_{2}}$
• B. $\frac{\mu_{0} \pi r_{1}^{2}}{2 r_{2}}$
• C. $\frac{\mu_{0} \pi r_{2}^{2}}{r_{1}}$
• D. $\frac{\mu_{0} \pi r_{2}^{2}}{2 r_{1}}$

Answer 1

Answer: (B)

Let a current $I_{2}$ flow through the outer circular coil. The field at the center of the coil is $B_{2}=\frac{\mu_{0} I_{2}}{2 r_{2}} .$ Since the other co-axially placed coil has a very small radius, $B_{2}$ may be considered constant over its cross-sectional area.
Hence, $\phi=\pi r_{1}^{2} B_{2}$
$\phi=\frac{\mu_{0} \pi r_{1}^{2} I_{2}}{2 r_{2}}=M_{12} \cdot I_{2}$
$ \Rightarrow M_{12}=\frac{\mu_{0} \pi r_{1}^{2}}{2 r_{2}}$