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Question
A uniform magnetic field B exists in a cylindrical region of radius 10 cm as shown in figure. A uniform wire of length 80 cm and resistance 4.0 Ω is bent into a square frame and is placed with one side along a diameter of the cylindrical region. If the magnetic field increases at a constant rate of 0.010 T/s, find the current induced in the frame.
Solution
The magnetic field lines pass through coil abcd only in the part above the cylindrical region.
Radius of the cylindrical region, r = 10 cm
Resistance of the coil, R = 4 Ω
The rate of change of the magnetic field in the cylindrical region is constant and is given by
\[\frac{dB}{dt} = 0 . 010 T/s\]
The change in the magnetic flux is given by
\[\frac{d\phi}{dt} = \frac{dB}{dt}A\]
The induced emf is given by
\[e = \frac{d\phi}{dt} = \frac{dB}{dt} \times A = 0 . 01\left( \frac{\pi \times r^2}{2} \right)\]
\[ = \frac{0 . 01 \times 3 . 14 \times 0 . 01}{2}\]
\[ = \frac{3 . 14}{2} \times {10}^{- 4} = 1 . 57 \times {10}^{- 4} V\]
The current in the coil is given by
\[i = \frac{e}{R} = \frac{1 . 57 \times {10}^{- 4}}{4}\]
\[ = 0 . 39 \times {10}^{- 4} \]
\[ = 3 . 9 \times {10}^{- 5} A\]
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