A circular conducting ring of radius a and resistance R lies in a perpendicular field B(r, t)=βtr² throughout its area. Determine the Joule-heating rate.
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Question

A circular conducting ring of radius a and resistance R lies in a perpendicular field B(r,t)=βtr² throughout its area. Determine the Joule-heating rate.

G1 Physics · Accepted Answer

Φ=∫0^a βtr²2πrdr=πβta⁴/2. Therefore ε=πβa⁴/2, I=ε/R and P=I²R=ε²/R=π²β²a⁸/(4R).