Magnetic flux linked with a stationary loop of resistance $$R$$ varies with respect to time during the time period $$T$$ as follows:
$$\phi = at\left( {T - t} \right)$$
The amount of heat generated in the loop during that time (inductance of the coil is negligible) is

Solution :
Given that $$\phi = at\left( {T - t} \right)$$
Induced emf, $$E = \frac{{d\phi }}{{dt}} = \frac{d}{{dt}}\left[ {at\left( {T - t} \right)} \right]$$
$$ = at\left( {0 - 1} \right) + a\left( {T - t} \right) = a\left( {T - 2t} \right)$$
So, induced emf is also a function of time.
$$\therefore $$ Heat generated in time $$T$$ is
$$H\int\limits_0^T {\frac{{{E^2}}}{R}} dt = \frac{{{a^2}}}{R}\int\limits_0^T {{{\left( {T - 2t} \right)}^2}} dt = \frac{{{a^2}{T^3}}}{{3R}}$$