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

Concept:

• Magnetic flux through a loop is \[ \Phi=BA \] when magnetic field is constant.

• Induced emf is given by Faraday's law \[ \varepsilon=-\frac{d\Phi}{dt} \]

• Power dissipated in the loop is \[ P=\frac{\varepsilon^2}{R} \]

• Since power depends on the square of emf, it is always non-negative.

Step 1: Write the magnetic flux through the loop
Since magnetic field is constant, \[ \Phi=BA \] Substituting \[ A=A_0(1+\sin t) \] gives \[ \Phi=BA_0(1+\sin t) \]

Step 2: Calculate the induced emf
Using Faraday's law, \[ \varepsilon = -\frac{d\Phi}{dt} \] \[ \varepsilon = -BA_0\cos t \]

Step 3: Determine the power dissipated
\[ P = \frac{\varepsilon^2}{R} \] \[ P = \frac{B^2A_0^2}{R} \cos^2 t \]

Step 4: Study the nature of the graph
Since \[ P\propto \cos^2 t \] the power is always positive. Also, \[ P=0 \] whenever \[ \cos t=0 \] Thus the graph consists of repeated positive arches touching the time axis periodically.

Step 5: Select the correct graph
The graph corresponding to \[ P\propto \cos^2 t \] is Option (B). \[ \boxed{\text{Option (B)}} \]