First slide

Inductance

Question

Two single-turn circular loops of wire have radii R and r, with R >> r. The loops lie in the same plane and are concentric. The mutual inductance of the pair is (approximately)

Moderate

A

$\frac{μ_{0} π r^{2}}{R}$
B

$\frac{2 μ_{0} π r^{2}}{R}$
C

$\frac{μ_{0} π r^{2}}{2 R}$
D

$\frac{3 μ_{0} π r^{2}}{2 R}$

Solution

A current i in the large loop of radius .R produces a magnetic field of magnitude $B = \frac{μ_{0} i}{2 R}$ at its center. Since the radius 2R of the small loop r <.< R, we may treat the flux through the small loop as being approximately $Φ_{B} = B A \cos 0^{\circ} = \frac{μ_{0} i}{2 R} A = \frac{μ_{0} π r^{2} i}{2 R}$ . The mutual inductance of the loops is then
$M = \frac{Φ_{B}}{i} = \frac{μ_{0} π r^{2}}{2 R}$

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