Question

A coil having ' \(N\) ' turns and resistance ' \(R \Omega\) ' is connected to a galvanometer of resistance ' \(6 \text{ R } \Omega\) '. The magnetic flux linked with this coil changes from \(\phi_1\) weber to \(\phi_2\) weber in time ' \(t\) ' second. The induced current in the circuit is

• A. \(\frac{N(\phi_2 - \phi_1)}{t}\)
• B. \(\frac{N(\phi_2 - \phi_1)}{7Rt}\)
• C. \(\frac{N(\phi_2 - \phi_1)}{Rt}\)
• D. \(\frac{N(\phi_2 - \phi_1)}{6Rt}\)

Hint:

The induced current depends on the total resistance of the entire closed loop, not just the coil itself.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
Faraday's Law of Induction states that a change in magnetic flux induces an e.m.f. The total resistance of the circuit determines the resulting current via Ohm's Law.
Step 2: Key Formula or Approach:
1) Induced e.m.f. in a coil: \(e = -N \frac{d\Phi}{dt}\). In magnitude, \(|e| = \frac{N(\phi_2 - \phi_1)}{t}\).
2) Induced current: \(i = \frac{e}{R_{total}}\).
Step 3: Detailed Explanation:
The coil and galvanometer are connected in series.
The total resistance of the circuit is:
\[ R_{total} = R_{coil} + R_{galvanometer} = R + 6R = 7R \]
The average induced e.m.f. is:
\[ e = \frac{N(\text{change in flux})}{\text{time}} = \frac{N(\phi_2 - \phi_1)}{t} \]
The induced current is:
\[ i = \frac{e}{R_{total}} \]
\[ i = \frac{N(\phi_2 - \phi_1)}{t} \div (7R) \]
\[ i = \frac{N(\phi_2 - \phi_1)}{7Rt} \]
Step 4: Final Answer:
The induced current is \(\frac{N(\phi_2 - \phi_1)}{7Rt}\).