Magnetic flux linked with a stationary loop 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
(A) $ \dfrac{{{a^2}{T^4}}}{{3R}} $
(B) $ \dfrac{{{a^2}{T^2}}}{{3R}} $
(C) $ \dfrac{{{a^2}{T^3}}}{{6R}} $
(D) $ \dfrac{{{a^2}{T^3}}}{{3R}} $

Hint: The magnetic field of the conducting material is changing along with time as specified. We want to determine the flux through the conducting material to find the heat generated. The heat produced is the integral of the emf of the circuit. Emf is the derivative of the flux through the circuit. Hence, we can find heat generated this way.

Complete step by step solution:
When the loop starts entering the region of the magnetic field, emf will be induced in the loop due to Faraday’s law of electromagnetic induction. Since the magnetic flux of the loop is given as:
 $ \phi = at\left( {T - t} \right) $
The EMF induced in the loop will be
 $ \Rightarrow \varepsilon = - \dfrac{{d\phi }}{{dt}} $
Put the value of the magnetic flux in the above equation and then differentiate it:
 $ \Rightarrow \varepsilon = \dfrac{d}{{dt}}\left( {at\left( {T - t} \right)} \right) $
 $ \Rightarrow \varepsilon = - at + 2at = at $
And so, the heat generated in the conducting material will be,
 $ H = \int\limits_0^T {\dfrac{{{\varepsilon ^2}}}{R}dt} $ ………. $ \left( {\because H = \int {P.dt} } \right) $
Put the value of emf we derived in the above equation we get,
 $ \Rightarrow H = \int\limits_0^T {\dfrac{{{a^2}{t^2}}}{R}} dt $
 $ \Rightarrow H = \left[ {\dfrac{{{a^2}{t^3}}}{{3R}}} \right]_0^T $
On further solving the equation we get,
 $ \Rightarrow H = \left[ {\dfrac{{{a^2}{T^3}}}{{3R}} - 0} \right] $
 $ \Rightarrow H = \dfrac{{{a^2}{T^3}}}{{3R}} $
As a result, the amount of heat generated in the loop during that time is $ \dfrac{{{a^2}{T^3}}}{{3R}} $ .
Hence, the correct answer is option (D).

Additional information:
Magnetic flux is the rate of flow of the magnetic field through a specified area. Magnetic flux is proportional to the number of magnetic field lines going through the virtual surface. If the magnetic field is uniform throughout the material, the magnetic flux passing through a surface of the vector area is the dot product between the magnetic field and the surface.

Note:
The magnetic flux only gets affected by charges that are present inside the closed surface. If we consider the magnetic field, the magnetic field gets affected by the charges outside the closed surface too. The formula of magnetic flux is different for surfaces with uniform and non-uniform magnetic fields.