A circular conducting loop of resistance R is placed in a constant, uniform magnetic field B perpendicular to its plane. The radius of the loop is made to increase uniformly with time t as r = c t , where c is a positive constant. What is the ratio of the total heat dissipated in the loop during the time interval t=0 to t=t₀ compared to the heat dissipated during the interval t=t₀ to t=2t₀ ?

Step-by-step solution

The area of the loop at any time t is A = r^2 = (ct)^2 = c^2 t^2 . The magnetic flux passing through the loop is = B A = B c^2 t^2 . The induced emf is: = - d dt = -B c^2 d dt (t^2) = -2B c^2 t The electrical power dissipated in the loop is: P = ^2 R = 4B^2 ^2 c^4 R t^2 The total heat H dissipated over a time interval is the integral of power with respect to time: H = P dt . Heat dissipated from t=0 to t=t₀ : H₁ = ₀^ t₀ 4B^2 ^2 c^4 R t^2 dt = 4B^2 ^2 c^4 R [ t^3 3 ]₀^ t₀ = 4B^2 ^2 c^4 3R t₀^3 Heat dissipated from t