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 4.0Ω 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.



Given:


Radius of cylindrical region


Length of wire =80cm


Resistance of the wire


Rate of increase of magnetic field =0.01T/s


Area of loop inside magnetic field =area of semicircle of radius 0.1m



We know that,


Flux (ϕ) of magnetic field (B) through the loop of cross section area A in the magnetic field is given by




Since magnetic field is perpendicular to the loop the flux becomes



Rate of change of magnetic field wrt. time is given by



(since area of cross section in magnetic field does not change with time, A remains constant)


Now,


by faraday’s law of electromagnetic induction


…(i)


Where


ϵ =emf produced


ϕ =flux of magnetic field


using eqn.(i) the emf induced in the loop is given by



Hence the current through the loop (i) of resistance R is



Putting the values of A, R and in the above eqn. the magnitude of current is



Therefore current induced in the frame is


1