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Question:

A conducting metal circular-wire-loop of radius r is placed perpendicular to a magnetic field which varies with time as B=B₀e⁻ᵗ⁄τ, where B₀ 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

π²r⁴B₀⁴/τR

π²r⁴B₀²R/τ

π²r⁴B₀²/τR

π²r⁴B₀/τR

Solution:

Flux through the loop, Φ=→B.→S
Φ=B₀πr²e⁻ᵗ⁄τ
ε=-dΦ/dt=B₀πr²/τe⁻ᵗ⁄τ
Heat Q=∫∞₀ ε²/R dt
Q=∫∞₀B₀²π²r⁴e⁻²ᵗ⁄τ/Rτ² dt
=[B₀²π²r⁴e⁻²ᵗ⁄τ/2Rτ]∞₀=π²r⁴B₀²/2τR