Question

The relation between the total magnetic flux linked with a coil of resistance \(48\Omega\) and time is \(\phi=5-24t\). If there is no heat loss and the heat capacity of the coil is \(5\text{ JK}^{-1}\), then the rise in temperature of the coil during \(t=3\text{ s}\) to \(t=5\text{ s}\) is:

• A. \(9.6\text{ K}\)
• B. \(2.4\text{ K}\)
• C. \(4.8\text{ K}\)
• D. \(3.6\text{ K}\)

Hint:

When magnetic flux varies linearly with time, induced emf remains constant because \(E=d\phi/dt\) is constant.

Answer 1

The Correct Option is C

Concept: Induced emf is \[ E=\left|\frac{d\phi}{dt}\right| \] Heat generated in time \(t\) is \[ H=\frac{E^2}{R}\,t \] Rise in temperature: \[ \Delta T=\frac{H}{C} \]

Step 1: Find induced emf. \[ \phi=5-24t \] \[ E=\left|\frac{d\phi}{dt}\right| \] \[ =24\text{ V} \]

Step 2: Calculate heat generated from 3 s to 5 s. Time interval \[ \Delta t=2\text{ s} \] \[ H = \frac{24^2}{48}\times2 \] \[ = 12\times2 \] \[ =24\text{ J} \]

Step 3: Find rise in temperature. \[ C=5\text{ JK}^{-1} \] \[ \Delta T=\frac{24}{5} \] \[ =4.8\text{ K} \] \[ \boxed{4.8\text{ K}} \]