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

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

G1 Physics · Accepted Answer

Φ=∫0^a βtr²(2πrdr)=πβta⁴/2. Hence |ε|=πβa⁴/2 and P=ε²/R=π²β²a⁸/(4R).