Question

In an ac generator, if a coil of \( N \) turns and area \( A \) is rotated at \( \nu \) revolutions per second in a uniform magnetic field \( B \), then the motional emf produced is equal to \[ (t = 0 s, the coil is perpendicular to the field) \]

• A. \( NBA (2\pi \nu) \sin(2\pi \nu t) \)
• B. \( NBA^2 (2\pi \nu) \sin(2\pi \nu t) \)
• C. \( N^2 B^2 A^2 (2\pi \nu) \sin(2\pi \nu t) \)
• D. \( NBA (4\pi \nu) \sin(2\pi \nu t) \)

Hint:

In ac generators, the emf produced is sinusoidal and depends on the number of turns, area of the coil, magnetic field strength, and the frequency of rotation.

Answer 1

The Correct Option is A

The emf induced in a rotating coil in a magnetic field is given by the formula: \[ \mathcal{E} = NBA (2\pi \nu) \sin(2\pi \nu t) \] Where: - \( N \) is the number of turns of the coil - \( B \) is the magnetic field strength - \( A \) is the area of the coil - \( \nu \) is the frequency of rotation - \( t \) is the time At \( t = 0 \), the coil is perpendicular to the field, so the maximum induced emf occurs at this time. The term \( (2\pi \nu) \) accounts for the angular frequency of the rotation, and the sine function represents the variation of the induced emf over time. Thus, the motional emf produced is \( \boxed{NBA (2\pi \nu) \sin(2\pi \nu t)} \).

Answer 2

Step 1: Understand the principle of an AC generator.
An AC generator works on the principle of electromagnetic induction, where an emf is induced in a rotating coil placed in a uniform magnetic field.

Step 2: Expression for magnetic flux.
At any time \( t \), the magnetic flux through the coil is:
\[ \phi = NBA \cos(\theta) = NBA \cos(2\pi \nu t) \] where:
- \( N \) is the number of turns,
- \( B \) is the magnetic field,
- \( A \) is the area of the coil,
- \( \nu \) is the frequency in revolutions per second,
- \( \theta = 2\pi \nu t \) is the angle rotated at time \( t \).

Step 3: Use Faraday’s law of electromagnetic induction.
\[ \text{emf} = -\frac{d\phi}{dt} = -\frac{d}{dt}(NBA \cos(2\pi \nu t)) \] \[ = NBA \cdot 2\pi \nu \sin(2\pi \nu t) \]

Step 4: Sign convention.
The negative sign in Faraday’s law indicates direction of induced emf (Lenz's law), but for magnitude:
\[ \text{emf} = NBA (2\pi \nu) \sin(2\pi \nu t) \]

Step 5: Conclusion.
The motional emf produced is \( \boxed{NBA (2\pi \nu) \sin(2\pi \nu t)} \)