A conducting circular loop of radius r and resistance R is placed in a uniform magnetic field which is perpendicular to the plane of the loop. The magnetic field varies with time t as B = B₀ ( t) . If the average thermal power dissipated in the loop is P , the amplitude of the magnetic field B₀ is given by:

Step-by-step solution

The magnetic flux through the loop is = B A = B₀ r^2 ( t) . By Faraday's law, the induced EMF is = - d dt = -B₀ r^2 ( t) . The instantaneous power dissipated in the loop is P_ inst = ^2 R = B₀^2 ^2 r^4 ^2 ^2( t) R . The average power over a full cycle is obtained by taking the time average of ^2( t) , which is 1 2 . Thus, the average power is P = B₀^2 ^2 r^4 ^2 2R . Rearranging this equation to solve for B₀ gives B₀ = 2RP r^2 . Answer: 2RP r^2