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Question: A circular coil of mean radius of \(7{\text{ }}cm\) and having \(4000\) turns is rotated at the rate...

A circular coil of mean radius of 7 cm7{\text{ }}cm and having 40004000 turns is rotated at the rate of 18001800 revolutions per minute in the earth’s magnetic field (B=0.5 gaussB = 0.5{\text{ }}gauss), the peak value of emf induced in coil will be?
A. 1.158 V1.158{\text{ }}V
B. 0.58 V0.58{\text{ }}V
C. 0.29 V0.29{\text{ }}V
D. 5.8 V5.8{\text{ }}V

Explanation

Solution

In the given question we have to find the peak value of the emf induced in the coil. We have to convert the magnetic field in the SI unit, as the answers are mentioned in the SI unit. Then, we have to find the area of the circular coil and the angular velocity in sec. Then by substituting the values of all the variables in the relation between emf induced, magnetic field, area, number of turns and angular velocity, we will find the solution.

Formula Used:
The maximum value of induced emf which is the peak emf in the coil with the help of the formula,
e=BAnωcos⁡ωte = BAn\omega \cos \omega t
The variables are defined as,
e=e = maximum induced emf
B=B = magnetic field due to the coil
A=A = Area of the coil
n=n = number of turns in the coil
ω=\omega = angular velocity

Complete step by step answer:
The Earth’s magnetic field,
B=0.5 gauss ⇒B=0.5×10−4 Wbm2B = 0.5{\text{ }}gauss \\\ \Rightarrow B= 0.5 \times {10^{ - 4}}{\text{ }}\dfrac{{Wb}}{{{m^2}}}
The radius rr of the coil=7 cm=0.07 m = 7{\text{ }}cm = 0.07{\text{ }}m
Thus, the area AA of the coil=π(7×10−2)2=154×10−4 m2 = \pi {\left( {7 \times {{10}^{ - 2}}} \right)^2} = 154 \times {10^{ - 4}}{\text{ }}{m^2}
The rate of rotation is,
f=1800 min⁡−1 ⇒f=180060 sec⁡−1 ⇒f=30 sec⁡−1f = 1800{\text{ }}{\min ^{ - 1}} \\\ \Rightarrow f = \dfrac{{1800}}{{60}}{\text{ }}{\sec ^{ - 1}} \\\ \Rightarrow f = 30{\text{ }}{\sec ^{ - 1}}

Thus, the angular velocity,
ω=2πf ⇒ω=60π rad.s−1\omega = 2\pi f \\\ \Rightarrow \omega = 60\pi {\text{ }}rad.{s^{ - 1}}
The number of turns of the coil is n=4000n = 4000. Now, we have to find the maximum value of induced emf which is the peak emf in the coil with the help of the formula,
e=BAnωcos⁡ωte = BAn\omega \cos \omega t
The value is at peak when cos⁡ωt=1\cos \omega t = 1. Substituting all the values in the equation we get,
e=0.5×10−4×154×10−4×4000×60π ∴e=0.58 Ve = 0.5 \times {10^{ - 4}} \times 154 \times {10^{ - 4}} \times 4000 \times 60\pi \\\ \therefore e = 0.58\,V
So, the peak value of emf induced in the coil is 0.58 V0.58{\text{ }}V.

Therefore, the correct option is B.

Note: It must be noted that the maximum emf induced in a coil is the peak value of emf when the cosine angle associated with it is at its highest value which is 11. We must convert all the values into a standard unit as different units provide a different value which cannot be considered as standard, also the answer is in SI unit, so we must convert them to SI units for the ease of calculation.