Question

Magnetic flux during time interval $\tau$ varies through a stationary loop of resistance $R$, as ${\varphi }_B=at(\tau -t)$. Find the amount of heat generated during that time. Neglect the inductance of the loop.

• A. $\frac{a^2{\tau }^3}{R}$
• B. $\frac{a^2{\tau }^2}{2R}$
• C. $\frac{a^2{\tau }^3}{3R}$
• D. $\frac{a^2{\tau }^3}{4R}$

Answer 1

Answer: (C)

$\frac{a^2{\tau }^3}{3R}$

$i=\frac{d\varphi }{dt}\frac{1}{R}=\frac{a(\tau -2t)}{R}$

Heat produced $H={\int }_0^{\tau }{i}^{2}Rdt$

$H={\int }_0^{\tau }\frac{a^2(\tau -2t{)}^2}{R}dt=\frac{a^2{\tau }^3}{3R}$