First slide

Inductance

Question

A small square loop of wire of side / is placed inside a large square loop of wire of side L (L > l). The loop are coplanar and their center coincide. The mutual inductance of the system is proportional to

Moderate

A

l/L
B

l²/L
C

L/l
D

L²/l

Solution

Magnetic field produced due to large loop
$B = 4 \times \frac{μ_{0}}{4 π} \frac{i}{\frac{L}{2}} (\sin 45^{0} + \cos 45^{0}) = \frac{μ_{0}}{4 π} \frac{8 \sqrt{2} i}{L}$
Flux linked with smaller loop
$\begin{array}{l} φ = B (l^{2}) = \frac{μ_{0}}{4 π} \frac{8 \sqrt{2} i l^{2}}{L} \\ ∴ φ = M i \\ \Rightarrow M = \frac{φ}{i} = \frac{μ_{0}}{4 π} \cdot \frac{8 \sqrt{2} l^{2}}{L} \\ \Rightarrow M \propto \frac{l^{2}}{L} \end{array}$

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