First slide

Inductance

Question

What is the mutual inductance of a two-loop system as shown with centre separation l

Difficult

A

$\frac{μ_{0} {πa}^{4}}{8 l^{3}}$
B

$\frac{μ_{0} {πa}^{4}}{4 l^{3}}$
C

$\frac{μ_{0} {πa}^{4}}{6 l^{3}}$
D

$\frac{μ_{0} {πa}^{4}}{2 l^{3}}$

Solution

Magnetic field at the location of coil (2) produced due to coil (1)
$B_{1} = \frac{μ_{o}}{4 π} . \frac{2 M}{l^{3}}$
Flux linked with coil (2)
$ϕ_{2} = B_{1} A_{2} = \frac{μ_{o}}{4 π} \frac{2 i ({πa}^{2})}{l^{3}} \times ({πa}^{2})$
Also $ϕ_{2} = Mi \Rightarrow M = \frac{μ_{o} {πa}^{4}}{2 l^{3}}$

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