A plane loop is shaped in the form as shown in figure with radii a= 20 cm and b= 10 cm and placed in a uniform time varying magnetic field ${B = B}_{0}$ $\sin (ω t)$ where $B_{0} {=10m T and ω=10rads}^{-1}$ . Find the amplitude of the induced current, in ampere, in the loop if its resistance per unit length ${λ=50×10}^{-3} {Ωm}^{-1} .$ The inductance of the loop is negligible

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Detailed Solution

Instantaneous flux through loop is

$\begin{aligned} ϕ = π a^{2} B \cos 8^{0} + π b^{2} B \cos 180^{\circ} \\ \Rightarrow ϕ = π (a^{2} - b^{2}) B \\ \Rightarrow ϕ = π (a^{2} - b^{2}) B_{0} \sin (ω t) \end{aligned}$

Differentiating W. r. t. time

$\begin{aligned} | \frac{d ϕ}{d t} | = π (a^{2} - b^{2}) B_{0} ω \cos (ω t) \\ \Rightarrow I = \frac{π (a^{2} - b^{2}) B_{0} ω Cos (ω t)}{R} \end{aligned}$

$I_{M a x} = I_{0} = \frac{π (a^{2} - b^{2}) B_{0} ω}{2 π (a + b) λ}, λ =$ resistance per unit length

$\begin{aligned} I_{Max} = I_{0} = \frac{1}{2 λ} (a - b) B_{0} ω \\ I_{Max} = I_{0} = \frac{1}{2} (20 - 10) \times 10^{- 2} \times 10 \times 10^{- 3} \times \frac{100}{50 \times 10^{- 3}} \\ I_{Mac} = I_{0} = 1 A \end{aligned}$