A coil placed in a time-varying magnetic field dissipates power P. Its number of turns, area enclosed by each turn and wire radius are N, A and r, respectively. A second geometrically similar coil has 2N turns, turn area 2A and wire radius 3r, and is placed in the same time-varying magnetic field. If the power dissipated in the second coil is α√2 P, find α.
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Question

A coil placed in a time-varying magnetic field dissipates power P. Its number of turns, area enclosed by each turn and wire radius are N, A and r, respectively. A second geometrically similar coil has 2N turns, turn area 2A and wire radius 3r, and is placed in the same time-varying magnetic field. If the power dissipated in the second coil is α√2 P, find α.

Options

G1 Physics · Accepted Answer

Induced emf magnitude is ε = NA|dB ÷ dt|. Hence, ε² ∝ N²A². The coil resistance is R = ρℓ ÷ (πr²). For a coil with N turns, ℓ = N(2πR(coil)). Also, A = πR(coil)², so R(coil) = √(A ÷ π). Therefore, ℓ ∝ N√A. Thus, R ∝ N√A ÷ r². Power dissipation is P = ε² ÷ R. Therefore, P ∝ [N²A²] ÷ [N√A ÷ r²] = N × (A raised to the power 3 ÷ 2) × r². For the second coil, P' ÷ P = (2N ÷ N) ×[(2A ÷ A) raised to the power 3 ÷ 2] ×(3r ÷ r)² = 2×(2√2)×9 = 36√2. Given, P' = α√2 P, so α = 36. Correct option: A.