What is the correct formula for total emissive power of the gray body (${{E}_{g}}$)
(A). $({{E}_{g}})={{T}^{4}}$
(B). $({{E}_{g}})={{T}^{2}}$
(C). $({{E}_{g}})=\dfrac{1}{{{T}^{4}}}$
(D). None of the above

Hint: Emissive power is the measure of energy emitted at a particular temperature. A body absorbs or emits heat at different wavelengths. A black body absorbs radiation from all wavelengths. It can also emit radiation in all wavelengths. Emissive power depends on the heat energy emitted, the area of the cross section of the body and the time taken.

Complete answer:
The total amount of heat energy emitted per unit area per unit time is called emissive power at that temperature. The energy is emitted as radiation in different wavelengths. It is given by-
${{E}_{g}}=\dfrac{Q}{At}$
Here,$Q$ is the heat energy emitted
$A$ is the area of cross section
$t$ is time taken
From the above equation, we can see that emissive power does not depend on temperature, i.e.
${{E}_{g}}\propto {{T}^{0}}$
A body is said to be perfectly black when it can emit radiation for all wavelengths and the coefficient is 1.
Since the emissive power is not related to the temperature but rather ${{E}_{g}}=\dfrac{Q}{At}$,

Hence, the correct option is (D). none of these.

Note:
The ratio of the emissive power of a body to the emissive power of a perfectly black body is called coefficient of emission. It is given by- $\dfrac{{{E}_{g}}}{{{E}_{b}}}$. Energy is emitted by matter in the form of heat due to different motion, vibrational as well as rotational motion of particles like atoms, molecules. The energy radiated is in the form of radiation.