Find the mutual inductance in the arrangement, when a small circular loop of wire of radius $R$ is placed inside a large square loop of wire of side $L (L ≫ R)$ . The loops are coplanar and their centres coincide :

Question

Aicharya · Accepted Answer

The magnetic field due to the rectangular loop at its centre is:

$B = 4 \times \frac{μ_{0} i}{4 π (\frac{L}{2})} [\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}] = \frac{2 \sqrt{2} μ_{0} i}{π L}$

Therefore, the flux through the circular loop $ϕ = π R^{2} \times B = \frac{2 \sqrt{2} μ_{0} R^{2} i}{L}$

If M is the mutual inductance, then

$M \frac{d i}{d t} = \frac{d ϕ}{d t} = \frac{2 \sqrt{2} μ_{0} R^{2}}{L} \frac{d i}{d t}$

$\Rightarrow M = \frac{2 \sqrt{2} μ_{0} R^{2}}{L}$

Find the mutual inductance in the arrangement, when a small circular loop of wire of radius R is placed…

Solution