Question

A coil of 'n' turns and resistance $R \Omega$ is connected in series with a resistance $R/2$. The combination is moved for time 't' second through magnetic flux $\phi_1$ to $\phi_2$. The induced current in the circuit is ______.

• A. $\frac{n(\phi_1 - \phi_2)}{3Rt}$
• B. $\frac{2n(\phi_1 - \phi_2)}{3Rt}$
• C. $\frac{2n(\phi_1 - \phi_2)}{Rt}$
• D. $\frac{n(\phi_1 - \phi_2)}{Rt}$

Hint:

Always double-check if the question provides just the coil's resistance or an additional external resistance. A classic trap is calculating the current using only the coil's internal resistance $R$, ignoring the external series component.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We have a coil undergoing a change in magnetic flux, which induces an electromotive force (EMF) according to Faraday's Law. We then need to calculate the induced current using Ohm's Law across the total resistance of the circuit.

Step 2: Key Formula or Approach:
1. Faraday's Law of Induction: The induced EMF is $e = -n \frac{\Delta \phi}{\Delta t}$.
2. Total Resistance: $R_{eq} = R_{coil} + R_{external}$.
3. Ohm's Law: $I = \frac{e}{R_{eq}}$.

Step 3: Detailed Explanation:
First, calculate the total resistance of the series circuit:
$$R_{eq} = R + \frac{R}{2} = \frac{3R}{2}$$
Next, determine the magnitude of the induced EMF ($e$) over time $t$:
$$e = n \left| \frac{\phi_2 - \phi_1}{t} \right| = \frac{n(\phi_1 - \phi_2)}{t}$$
(Taking the magnitude or absolute change depending on standard convention).
Finally, calculate the induced current ($I$):
$$I = \frac{e}{R_{eq}} = \frac{\frac{n(\phi_1 - \phi_2)}{t}}{\frac{3R}{2}}$$
$$I = \frac{2n(\phi_1 - \phi_2)}{3Rt}$$

Step 4: Final Answer:
The induced current is $\frac{2n(\phi_1 - \phi_2)}{3Rt}$, matching option (b).