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.

Answer 1

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
$\frac{dB}{dt}=0.010\mathrm{T}/\mathrm{s}$

The change in the magnetic flux is given by
$\frac{d\mathrm{\varphi }}{dt}=\frac{dB}{dt}A$

The induced emf is given by
$$\begin{gathered} e=\frac{d\mathrm{\varphi }}{dt}=\frac{dB}{dt}\times A=0.01\left(\frac{\mathrm{\pi }\times r^2}{2}\right) \\ =\frac{0.01\times 3.14\times 0.01}{2} \\ =\frac{3.14}{2}\times {10}^{-4}=1.57\times {10}^{-4}\mathrm{V} \end{gathered}$$
The current in the coil is given by
$$\begin{gathered} i=\frac{e}{R}=\frac{1.57\times {10}^{-4}}{4} \\ =0.39\times {10}^{-4} \\ =3.9\times {10}^{-5}\mathrm{A} \end{gathered}$$