A square loop of side 2 . 0 cm is placed inside a long solenoid that has 50 turns per centimetre and carries a sinusoidally varying current of amplitude 2 . 5 A and angular frequency 700 rad s - 1 . T

Question

A square loop of side 2 . 0 cm is placed inside a long solenoid that has 50 turns per centimetre and carries a sinusoidally varying current of amplitude 2 . 5 A and angular frequency 700 rad s - 1 . The central axes of the loop and solenoid coincide. The amplitude of the emf induced in the loop is x × 10 - 4 V . The value of x is ___________ ( Take, π = 22 7

TestHub · Accepted Answer

It is given that the current is varying sinusoidally. The current can be written as $I = I_{0} \sin ω t$ .
The magnetic field through the solenoid $B = μ_{0} n I$
The flux through the square is $ϕ = μ_{0} n I A$
The emf is
$ε = μ_{0} n A \times \frac{d (I_{0} \sin ω t)}{d t} \Rightarrow ε = μ_{0} n A I_{0} ω \cos ω t$
The amplitude of the emf is,
$ε = μ_{0} n A I_{0} ω = 4 π \times 10^{- 7} \times \frac{50}{10^{- 2}} \times 4 \times 10^{- 4} \times 2.5 \times 700 \Rightarrow ε = 44 \times 10^{- 4} V$

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