A ceiling fan having 3 blades of length 80 cm each is rotating with an angular velocity of 1200 rpm. The magnetic field of earth in that region is 0.5 G and angle of dip is 30°. The emf induced across the blades is N π × 10⁻⁵ V. The value of N is _________.

Solution

Physics sir · 4 steps

The fan turns in a horizontal plane, so only the vertical part of Earth's field cuts the blades.

1. Which field counts. The blades sweep a horizontal plane, so only the vertical component of Earth's field cuts them. With dip δ = 30°, B_V = B sin δ = 0.5 × 1/2 = 0.25 G = 0.25 × 10⁻⁴ T.
2. EMF of one blade. A rod rotating about one end gives an emf of 1/2Bω L² between the centre and the tip. All three blades are in parallel between the same two points, so the emf is that of one blade.

   ε = 1/2B_V ω L²

3. Put in numbers. ω = (1200 × 2π)/60 = 40π rad s⁻¹ and L² = (0.8)² = 0.64 m². So ε = 1/2 × 0.25 × 10⁻⁴ × 40π × 0.64 = 3.2π × 10⁻⁴ V.
4. So. ε = 32π × 10⁻⁵ V, so N = 32.

The common slip: 64. Using the full field of 0.5 G instead of its vertical component, or dropping the 1/2 in 1/2Bω L², doubles the emf and gives 64.

The idea to keep. For anything rotating in a horizontal plane, use B_V = B sin δ. For a vertical plane facing the field, use B_H = B cos δ. The number of blades does not change the emf.

Answer from the NTA final answer key. Solution by PrepWiser.