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Step-by-Step Solution
Step 1: Understand the Physical Situation
We have a rectangular coil of 100 turns, each with an area of 14 \times 10^{-2}\,\text{m}^2 , rotating in a uniform magnetic field of 3\,\text{T} . The coil is rotating at 360\,\text{rev/min} about an axis perpendicular to the magnetic field. We need to find the maximum value of the induced emf.
Step 2: Recall the Relevant Formula
From Faraday’s law of electromagnetic induction, when a coil rotates in a uniform magnetic field, the instantaneous emf is given by:
e(t) = N\,B\,A\,\omega \,\sin(\omega t)
where:
N is the number of turns.
B is the magnetic field.
A is the area of the coil.
\omega is the angular speed (in rad/s).
The maximum value of the emf, E_{\max} , occurs when \sin(\omega t) = 1 , so:
E_{\max} = N\,B\,A\,\omega \,.
Step 3: Convert Rotational Speed to Angular Speed
The coil rotates at 360\,\text{rev/min} . First, convert revolutions per minute to revolutions per second:
360\,\text{rev/min} = \frac{360}{60} \,\text{rev/s} = 6\,\text{rev/s} \,.
Since 1\,\text{rev} = 2\pi\,\text{rad} , the angular speed \omega is:
\omega = 6 \times 2\pi = 12\pi \,\text{rad/s} \,.
Given \pi = \frac{22}{7} , we may use that for any numerical calculation.
Step 4: Substitute Known Values
Now substitute N = 100 , B = 3\,\text{T} , A = 14 \times 10^{-2}\,\text{m}^2 (which is 14/100 = 0.14\,\text{m}^2 ), and \omega = 12\pi\,\text{rad/s} into the formula for E_{\max} :
E_{\max} = N \times B \times A \times \omega \,.
E_{\max} = 100 \times 3 \times (14 \times 10^{-2}) \times 12\pi \,.
Step 5: Perform the Numerical Calculation
Compute step by step:
Compute N \times B \times A = 100 \times 3 \times (14 \times 10^{-2}) = 100 \times 3 \times 0.14 = 42 \,.
Compute \omega = 12\pi = 12 \times \frac{22}{7} = \frac{264}{7} \,.
Hence, E_{\max} = 42 \times \frac{264}{7} = 42 \times \frac{264}{7} \,.
42 \times \frac{264}{7} = 42 \times \frac{264}{7}
= \frac{42 \times 264}{7} \,.
Since 42/7 = 6 , we get 6 \times 264 = 1584 \,.
Therefore, the maximum value of the emf is:
E_{\max} = 1584\,\text{V}
