State Kirchoff's law and derive the expression for the law. Also explain what is a gray surface?
Kirchhoff's Law states that the ratio of the emissive power of a body to its absorptive power is constant for the same wavelength, temperature, and angle of incidence. This means that a body that absorbs all radiation that falls upon it at a particular wavelength also emits radiation of that same wavelength equally efficiently.
Mathematically,
$\frac{E_{\lambda}}{A_{\lambda}} = f(\lambda,T,\theta)$
Where $E_{\lambda}$ is the emissive power at wavelength $\lambda$, $A_{\lambda}$ is the absorptive power at wavelength $\lambda$, $T$ is the temperature of the body, $\theta$ is the angle of incidence, and $f(\lambda,T,\theta)$ is a function that is constant for a given set of conditions.
The law can be derived from the principles of energy conservation, assuming that a body in thermal equilibrium radiates energy as much as it absorbs. When the radiation is in equilibrium with the temperature of the body, the emissive power and the absorptive power are equal (i.e., $E_{\lambda}=A_{\lambda}$), and the ratio is equal to 1.
A gray surface is a surface that has the same emissive power for all wavelengths and angles of incidence. In other words, the surface shows the same brightness or color from all directions, regardless of the wavelength of the incident light. This is because the emissive power of a gray surface is independent of the wavelength and incidence angle of the radiation.