Question

a uniform magnetic field B exists in the cylindrical region of radius 10 cm as .A uniform wire of length 80 cm and resistance 4.0 Ω is bent into a squre frame and is placed with one side along a diameter of the cylindrical region. If the magnetic field increases at a constant rate of 0.01 T/s,Find the current induced in the frame.

Answer 1

Actually induced emf will depend on the orientation of the wire frame wrt to the axis of the cylinder.  It is not clear to me from the above description how the wire frame is placed.

It is also not clear if the magnetic field is along the   axis of cylinder or along a direction perpendicular to the axis.

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

         The flux through the wire frame = Φ = B * (π * 0.1^2 / 2)
           emf = - dΦ/dt = - 0.005 π dB/dt = - 0.005  π * 0.01 = - 50 π micro Volts
           current = emf / resistance = - 12.5 π micro Amperes.


2.  If the magnetic field is along the axis of cylinder and the wire frame is inserted inside the cylindrical region, then the flux through the wire frame is zero.  Hence there is no current.

3. If the wire frame is inserted in the cylinder and the magnetic field is perpendicular to the axis of the cylinder, then there is an induced emf depending on the angle between the magnetic field direction and the plane of wire frame.

   if the angle between the magnetic field B and the normal to the wire frame is θ, then
         emf = - dΦ/dt =  - (0.2^2 Cos θ) * dB/dt  = - 0.04 Cos θ * 0.01  volts
                 = - 400 Cos θ micro volts.
       current = -100 Cos θ  micro ampere,  as  resistance = 4 ohms.