Question

A metallic rod of length 1 m held along east-west direction is allowed to fall down freely. Given horizontal component of earth’s magnetic field B_H = 3 × 10^-5 T. The emf induced in the rod at an instant t = 2s after it is released is ( Take g = 10 ms^-2 )

• A. 3 × 10^-3 V
• B. 3 × 10^-4 V
• C. 6 × 10^-3 V
• D. 6 × 10^-4 V

Answer 1

The Correct Option is D

Given Information: 
Length of rod, \( l = 1\,m \)
Horizontal component of Earth's magnetic field, \( B_H = 3 \times 10^{-5}\,T \)
Acceleration due to gravity, \( g = 10\,ms^{-2} \)
Time after release, \( t = 2\,s \)

Step-by-Step Explanation:

Step 1: Find velocity (\(v\)) of the rod after time \(t\):

Rod is falling freely under gravity, thus initial velocity \( u = 0 \), so velocity after \( t \) seconds is:

\[ v = u + gt = 0 + 10 \times 2 = 20\,m/s \]

Step 2: Calculate induced emf (\(E\)):

The induced emf in a rod moving perpendicular to a magnetic field is given by:

\[ E = B_H \times l \times v \]

Substitute values into this formula:

\[ E = (3 \times 10^{-5}\,T) \times (1\,m) \times (20\,m/s) \]

Simplify:

\[ E = 6 \times 10^{-4}\,V \]

Final Conclusion:
The induced emf in the rod at \( t = 2\,s \) is \(6 \times 10^{-4}\,V\).

Answer 2

The emf induced in a falling rod is given by the formula: \[ \text{emf} = B \cdot L \cdot v \] Where: - \( B \) is the magnetic field strength, - \( L \) is the length of the rod, - \( v \) is the velocity of the rod at time \( t \). The velocity \( v \) of the rod after time \( t \) due to gravity is given by: \[ v = g \cdot t \] Where: - \( g = 10 \, \text{m/s}^2 \) is the acceleration due to gravity, - \( t = 2 \, \text{s} \) is the time. Substitute \( v = g \cdot t \) into the emf formula: \[ \text{emf} = B \cdot L \cdot (g \cdot t) \] Substitute the given values: \[ \text{emf} = (3 \times 10^{-5} \, \text{T}) \cdot (1 \, \text{m}) \cdot (10 \, \text{m/s}^2 \cdot 2 \, \text{s}) \] \[ \text{emf} = 6 \times 10^{-4} \, \text{V} \] Thus, the emf induced in the rod at \( t = 2 \, \text{s} \) is \( 6 \times 10^{-4} \, \text{V} \).