A ceiling fan having 3 blades of length $80\ \text{cm}$ each is rotating with an angular velocity of $1200\ \text{rpm}$. The magnetic field of earth in that region is $0.5\ \text{G}$ and the angle of dip is $30^\circ$. The emf induced across the blades is $N\pi\times10^{-5}\ \text{V}$. The value of $N$ is ______.
Numerical value type. Enter your answer.
Answer: 32
The blades rotate in a horizontal plane, so only the vertical component of the field matters:
$$B_V = 0.5\times10^{-4}\sin30^\circ = 0.25\times10^{-4}\ \text{T}$$
$\omega = \dfrac{1200\times2\pi}{60} = 40\pi\ \text{rad s}^{-1}$. Between the centre and the tip of a blade:
$$\varepsilon = \tfrac12B_V\omega L^2 = \tfrac12(0.25\times10^{-4})(40\pi)(0.8)^2 = 32\pi\times10^{-5}\ \text{V}$$
$N = 32$.
Solution by Sreeraj P, M.Sc Physics