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Question
A circular coil of radius 10 cm, 500 turns and resistance 200 Ω is placed with its plane perpendicular to the horizontal component of the Earth's magnetic field. It is rotated about its vertical diameter through 180° in 0.25 s. Estimate the magnitude of the emf and current induced in the coil. (Horizontal component of the Earth's magnetic field at the place is 3.0 ✕ 10−5 T).
Solution
Horizontal component of the Earth's magnetic field, B = 3 × 10−5 T
Radius of the coil, r = 10 cm = 0.1 m
Number of turns, N = 500
Resistance of the coil, R = 200 Ω
Angular speed of rotation of coil,
The Emf induced in the coil is a function of time and is given by
\[\omega = \frac{d\theta}{dt} = \frac{\pi}{0 . 25}\]
\[ \Rightarrow \omega = \frac{3 . 14}{0 . 25} = 12 . 56 \text { rad/s }\]
Area of the coil,
\[A = \pi r^2 \]
\[ \Rightarrow A = 3 . 14 \times (0 . 1 )^2 = 0 . 0314 m^2 \]
\[e = NAB\omega\sin\omega t\]
Maximum emf induced in the coil is given as,
\[e = NAB\omega\]
\[ \Rightarrow e = 500 \times 0 . 0314 \times 3 \times {10}^{- 5} \times 12 . 56\]
\[ \Rightarrow e = 5 . 91 \times {10}^{- 3} V\]
Maximum induced current in the coil is given as,
\[i = \frac{e}{R}\]
\[ \Rightarrow i = \frac{5 . 91 \times {10}^{- 3}}{200} = 2 . 95 \times {10}^{- 5} A\]
Disclaimer: As it is not mentioned in the question, that whether instantaneous or maximum value of the induced emf and current in the coil are to be calculated. Thus, the maximum values of induced emf and current are calculated. The instantaneous values of the induced emf and current are zero for the given instant.
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