Question

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 the angle of dip is \( 30^\circ \). The emf induced across the blades is \( N \pi \times 10^{-5} \, \text{V} \). The value of \( N \) is \( \_\_\_\_\_ \).

Answer 1

Correct Answer: 32

Step 1. Effective Vertical Magnetic Field Component Calculation:

Given:

\( B = 0.5 \, \text{G} = 0.5 \times 10^{-4} \, \text{T} \)

The vertical component of the magnetic field, \( B_v \), is calculated using the dip angle \( \delta = 30^\circ \):

\( B_v = B \sin \delta = 0.5 \times 10^{-4} \times \sin 30^\circ = 0.5 \times 10^{-4} \times \frac{1}{2} = \frac{1}{4} \times 10^{-4} \, \text{T} \)

Step 2. Angular Velocity Conversion (rpm to rad/s):

Angular velocity \( \omega \) in rad/s is determined by:

\( \omega = 2 \pi \times f = 2 \pi \times \frac{1200}{60} = 2 \pi \times 20 = 40 \pi \, \text{rad/s} \)

Step 3. Rotation Radius Determination:

Each blade has a length of \( \ell = 80 \, \text{cm} = 0.8 \, \text{m} \). The effective radius \( r \) of rotation is:

\( r = 0.8 \, \text{m} \)

Step 4. Induced Electromotive Force (emf) Calculation:

The emf \( \varepsilon \) induced across the blade tips is:

\( \varepsilon = \frac{1}{2} B_v \omega r^2 \)

Substituting the values:

\( \varepsilon = \frac{1}{2} \times \frac{1}{4} \times 10^{-4} \times 40 \pi \times (0.8)^2 \)

Simplification yields:

\( \varepsilon = \frac{1}{2} \times \frac{1}{4} \times 10^{-4} \times 40 \pi \times 0.64 = 32 \pi \times 10^{-5} \, \text{V} \)

Step 5. Value of \( N \) Conclusion:

Comparing \( 32 \pi \times 10^{-5} \, \text{V} \) with the form \( N \pi \times 10^{-5} \, \text{V} \), we find:

\( N = 32 \)