Question

A conducting loop of finite resistance lies on the \(x-y\) plane. There is a constant magnetic field in the \(y\)-direction. The area of the loop varies with time \(t\) as \[ A=A_0(1+\sin t) \] The figure that correctly indicates the qualitative behaviour of the power dissipated in the loop as a function of time is:

• A. Increasing curve
• B. Repeated positive humps touching zero periodically
• C. V-shaped curve
• D. Constant power

Hint:

Power dissipated in a resistor is always positive. If \(P\propto \cos^2 t\), the graph never goes below the time axis. Flux depends on area when magnetic field is constant. Square functions produce repeated positive humps.

Answer 1

The Correct Option is B

Step 1: Set up the flux.
The field $B$ is constant and lies in the $y$-direction; the loop lies in the $x$-$y$ plane with area $A = A_0(1 + \sin t)$. The flux is $\Phi = B A$.
\[ \Phi = B A_0 (1 + \sin t) \]
Step 2: Induced emf by Faraday's law.
\[ \varepsilon = -\frac{d\Phi}{dt} = -B A_0 \cos t \]
Step 3: Power dissipated.
A loop of resistance $R$ dissipates $P = \varepsilon^2/R$.
\[ P = \frac{B^2 A_0^2}{R}\cos^2 t \]
Step 4: Sign of the power.
Because $P$ depends on $\cos^2 t$, it is never negative.
Step 5: Where it vanishes.
Whenever $\cos t = 0$, the power touches zero. Between those instants it rises to a maximum and falls again.
Step 6: Shape of the graph.
The plot is a sequence of identical positive humps that touch the time axis periodically, which is option (B).
\[ \boxed{\text{Repeated positive humps touching zero periodically}} \]