Question

A conductor of length 1 metre moves at right angles to a uniform magnetic field of flux density \(2\ \text{Wb/m}^2\) with a velocity of 40 metre/second. Calculate the induced e.m.f. when the conductor moves at an angle of 30 degree to the direction of the field.

• A. 50 V
• B. 30 V
• C. 40 V
• D. 15 V

Hint:

Motional e.m.f. is maximized when the conductor moves perpendicular to the magnetic field (\(\theta = 90^\circ\), \(\sin 90^\circ = 1\)).
When moving at an angle of 30°, the induced e.m.f. is exactly half of this maximum value (\(\sin 30^\circ = 0.5\)).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
When a straight conductor moves through a magnetic field, it cuts the magnetic flux lines, inducing an electromotive force (e.m.f.) across its ends.
This is known as motional or dynamically induced e.m.f.
Key Formula or Approach:
The dynamically induced e.m.f. (\(e\)) in a straight conductor moving through a magnetic field is given by:
\[ e = B \cdot l \cdot v \cdot \sin\theta \]
where:
- \(B\) is the magnetic flux density in Tesla (\(\text{T}\) or \(\text{Wb/m}^2\)).
- \(l\) is the active length of the conductor in meters (\(\text{m}\)).
- \(v\) is the velocity of the conductor in meters per second (\(\text{m/s}\)).
- \(\theta\) is the angle between the direction of motion (velocity vector) and the magnetic field lines.

Step 2: Detailed Explanation:
Let us identify the given parameters from the problem:
- Active length of the conductor, \(l = 1\ \text{m}\).
- Magnetic flux density, \(B = 2\ \text{Wb/m}^2\).
- Velocity of motion, \(v = 40\ \text{m/s}\).
- Angle between the motion and the magnetic field, \(\theta = 30^\circ\).
(Note: The phrase "moves at right angles" in the first sentence describes the initial orientation of the conductor relative to the field lines, but the actual motion to be calculated is at \(\theta = 30^\circ\)).
Substitute these values into the induced e.m.f. formula:
\[ e = B \cdot l \cdot v \cdot \sin(30^\circ) \]
Since \(\sin(30^\circ) = 0.5\):
\[ e = 2 \times 1 \times 40 \times 0.5 \]
\[ e = 80 \times 0.5 = 40\ \text{V} \]

Step 3: Final Answer:
The induced e.m.f. in the conductor is 40 V.