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Step-by-Step Solution
Step 1: Convert the magnetic field from gauss to tesla
Earth’s magnetic field in the problem is given as 0.5 G. Recall that 1 G = 10^{-4} T, so:
B = 0.5 \times 10^{-4}\,\text{T} = 5 \times 10^{-5}\,\text{T}\,.
Step 2: Determine the relevant (vertical) component of the magnetic field
We are given the angle of dip as 30^\circ . The vertical component of the magnetic field B_v is:
B_v = B \sin(\theta) = \bigl(5 \times 10^{-5}\bigr) \sin(30^\circ)\,.
Since \sin(30^\circ) = \tfrac{1}{2}, this becomes:
B_v = \bigl(5 \times 10^{-5}\bigr) \times \tfrac{1}{2} = 2.5 \times 10^{-5}\,\text{T}\,.
Step 3: Convert the rotational speed to angular velocity
The ceiling fan rotates at 1200 rpm (revolutions per minute). Converting to revolutions per second (rps):
1200\,\text{rpm} = \dfrac{1200}{60} = 20\,\text{rps}\,.
Then, the angular velocity \omega in radians per second is:
\omega = 2\pi \times \text{(rps)} = 2\pi \times 20 = 40\pi \,\text{rad/s}\,.
Step 4: Express the blade length in meters
Each blade length is 80\,\text{cm} , so:
\ell = 0.80\,\text{m}\,.
Step 5: Use the formula for induced emf in a rotating rod
For a rod rotating in a perpendicular magnetic field with angular velocity \omega , the induced emf \varepsilon is:
\varepsilon = \dfrac{1}{2} B_v \,\omega\, \ell^2\,.
Substituting values:
\varepsilon = \dfrac{1}{2} \times \bigl(2.5 \times 10^{-5}\bigr)\times \bigl(40\pi\bigr)\times (0.80)^2\,.
(0.80)^2 = 0.64\,.
So, B_v \,\omega \,\ell^2 = \bigl(2.5 \times 10^{-5}\bigr) \times 40\pi \times 0.64\,.
Numerically first: 2.5 \times 0.64 = 1.6\,.
Then multiplying by \tfrac{1}{2} : \tfrac{1}{2} \times 1.6 = 0.8\,.
Next, multiply by 40\pi : 0.8 \times 40\pi = 32\pi\,.
Introducing the factor 10^{-5} from earlier, we get
\varepsilon = 32\pi \times 10^{-5}\,\text{V}\,.
Step 6: Identify the value of N
Given that the induced emf is expressed as N\pi \times 10^{-5}\,\text{V}, we compare it with our result 32\pi \times 10^{-5}\,\text{V}.
Hence, N = 32\,.
