A uniform magnetic field B exists in a cylindrical region of radius 10 cm as sown in figure. A uniform wire of length 80 cm and resistance 4.0 $℧$ is bent into a square frame and is placed with one side along a diameter of the cylindrical region. If the magnetic field increase at a constant rate of 0.010 T/s, find the current induced in the frame.

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Solution

Actually induced emf will depend on the orientation of the wireframe wrt to the axis of the cylinder.

The magnetic field is directed along the axis of the cylinder, and if the wireframe is placed with its plane parallel to the side (planar) face of the cylinder, then there is an emf induced in the wire. The flux due to mag field B passes only through a semicircular area of a cross-section of a cylinder.

The flux through the wire frame $ϕ = \frac{- d ϕ}{d t} = - 0.05 π \frac{d B}{d t}$
= $- 50 π$ micro Volts
current = $\frac{e m f}{r e s i s t a n c e} = - 12.5 π$ micro ampere.

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