A mass m of copper of density d and resistivity ρ is drawn into wire of radius a and bent into one circular loop of radius r, placed perpendicular to a changing field B. The induced current is [m/(xπρd)](dB/dt). Find x.
Physics · Mixed Chapter
JEE Mains

Question

A mass m of copper of density d and resistivity ρ is drawn into wire of radius a and bent into one circular loop of radius r, placed perpendicular to a changing field B. The induced current is [m/(xπρd)](dB/dt). Find x.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — Express wire volume in terms of mass: For copper density d and wire radius a, mass m = density×volume: m/d = πa²×(2πr) = 2π²ra². Thus a² = m/(2π²rd). Step 2 — Resistance of circular wire: Wire length = 2πr and cross-section = πa². R = ρ(2πr)/(πa²) = 2ρr/a². Step 3 — Induced emf: The circular loop encloses area πr². For a perpendicular changing field, |emf| = πr²|dB/dt|. Step 4 — Induced current: I = emf/R = [πr²|dB/dt|]/[2ρr/a²] = [πra²/(2ρ)]|dB/dt| = [m/(4πρd)]|dB/dt|. Step 5 — Compar…