A conducting loop of finite resistance lies on the x - y plane. There is a constant magnetic field in the z direction. The area of the loop varies with time t , as A = A 0 ( 1 + sin t ) in appropria

Question

A conducting loop of finite resistance lies on the x - y plane. There is a constant magnetic field in the z direction. The area of the loop varies with time t , as A = A 0 ​ ( 1 + sin t ) in appropriate units. The figure that correctly indicates the qualitative behaviour of the power P dissipated in the loop as a function of time is :NEET 2026 (Re-NEET)

TestHub · Accepted Answer

The magnetic flux passing through the loop is given by:
$Φ = B \cdot A$

Given that the area varies with time as $A = A_{0} (1 + sin t)$ and the magnetic field $B$ is constant, the flux is:
$Φ = B A_{0} (1 + sin t)$

According to Faraday's law of electromagnetic induction, the induced emf $ε$ is:
$ε = - \frac{d Φ}{d t} = - B A_{0} \frac{d}{d t} (1 + sin t) = - B A_{0} cos t$

The power $P$ dissipated in the loop of resistance $R$ is:
$P = \frac{ε ^{2}}{R} = \frac{( - B A _{0} cos t ) ^{2}}{R} = \frac{B ^{2} A _{0}^{2}}{R} cos^{2} t$

From this expression, we can see that the power $P$ varies as $cos^{2} t$ .
At $t = 0$ , $P = \frac{B ^{2} A _{0}^{2}}{R} cos^{2} (0) = \frac{B ^{2} A _{0}^{2}}{R}$ (which is a maximum value).
At $t = \frac{π}{2}$ , $P = 0$ .

The graph of $P$ versus $t$ will start from a maximum positive value, decrease to zero, and then increase again, oscillating between $0$ and a maximum value. This behavior is correctly represented by the graph in option (3).

Answer:

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Options:

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