A conducting metal circular-wire-loop of radius r is placed perpendicular to a magnetic field which varies with time as B = B 0 e - t τ , where B 0 and τ are constants at time t = 0 . If the resistance of the loop is R , then the heat generated in the loop after a long time t → ∞ is

Question

A conducting metal circular-wire-loop of radius r is placed perpendicular to a magnetic field which varies with time as B = B 0 e - t τ , where B 0 and τ are constants at time t = 0 . If the resistance of the loop is R , then the heat generated in the loop after a long time t → ∞ is, π 2 r 4 B 0 4 2 τ R, π 2 r 4 B 0 2 2 τ R, π 2 r 4 B 0 2 R τ, π 2 r 4 B 0 2 τ R

MARKS App · Accepted Answer

Here, B = B 0 e - t τ Area of the circular loop, A = π r 2 Flux linked with the loop at any time t , ϕ = B A = π r 2 B 0 e - t τ Emf induced in the loop, ε = - d ϕ d t = π r 2 B 0 1 τ e - t τ Net heat generated in the loop ∫ 0 ∞ ε 2 R d t = π 2 r 4 B 0 2 τ 2 R ∫ 0 ∞ e - 2 t τ d t = π 2 r 4 B 0 2 τ 2 R × 1 - 2 τ × e - 2 t τ 0 ∞ = - π 2 r 4 B 0 2 2 τ 2 R × τ 0 - 1 = π 2 r 4 B 0 2 2 τ R

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