Kirchhoff's Law of Heat Radiation & Stefan's Boltzmann law

2. PAPER-1

3. SOME IMPOTANT FACTS :-
EMISSIVE POWER:- The Emissive Power eλ of a body at a particular
temperature(T) for a wavelength λ is defined as the Energy emitted per
second per unit surface area of the body within a unit wavelength range.
The Energy radiated per second per unit surface area lying between λ
and λ +dλ is given by eλ dλ . The Emissive power of a perfectly black body is
maximum and is denoted by Eλ .
ABSORPTIVE POWER:- The Absorptive Power aλ of a body at a particular
temperature(T) and for a particular wavelength λ is defined as the ratio of the
amount of Energy Absorbed in a given time by the surface to the amount of
energy incident on the surface in the same line.

6. Suppose aλ is the Absorptive power of a body. Let dQ be the
quantity of heat energy lying within the wavelengths λ and λ+dλ, incident on unit area of
the surface in one second, then the quantity of heat is Absorbed by the surface is given by
aλ dQ. The remaining energy will be either reflected or
transmitted or both.
If is the emissive power, then the amount of energy emitted per second per unit area
between wavelength λ and λ+dλ, by the virtue of temperature, is given by .
Thus, the total energy given per second per unit area of the surface =
In Equilibrium state,
Or
For a Perfectly black body,

7. Substituting Value Of dQ from
We Get,
This is Known As Kirchhoff’s Law.
APPLICATIONS:- Kirchhoff’s Law tell us that good absorbers are good emitters. If a
body Absorbs Radiation of a particular wavelength strongly, It also emits the same
radiation strongly. A Sodium Vapour which emits 2 yellow lines strongly is also a good
absorber of light of these two wavelength.

8. The rate of Emission of Gradient energy by unit area of perfectly
black body is directly proportional to the fouth power of Absolute
Temperature.(Stefan Law)
When a black body at absolute temperature T is surrounded by
another black body at Temperature To Absolute then not only it
looses the amount of energy but also gain an amount of energy
.
Thus, the total amount of heat lost by the system per unit time is
given by the expression,

9. A cylindrical enclosure ABCD of Uniform area
of cross section with perfectly reflecting balls and
provided with perfectly frictionless piston P.
When the cylinder is filled with diffused radiation of
Energy density u at uniform temperature T,
Where V is volume of the enclosure then the total energy is given by the expression-
When a Small amount of heat energy dQ flows in the enclosure from outside and at a same
time, the piston P moves outwards so that the volume changes by a small amount of dV.
In this process, the temperature and a result desity u changes by infinitely small amount du,
Which produces changes in internal energy. According to the first law of thermodynamic
if dW is the external work done by radiation in the expansion of volume by an amount to dV

10. Then the Equation become,
In this Case, Pressure is diffused radiation,
From 2nd law of themodynamic, dQ=TdS
For the Above System, the entropy is found to the function of two
independent variables u and V. i.e.,

11. Therefore, the entropy is perfect differential
Comparing the two equations and obtaining the coefficient of du and dV
Since, dS is perfect differential, Therefore,
T is independent of V and is function of u.

12. Thus,
On integrating, we get-

13. Total Rate of Emissive power of Radiant Energy per unit area is relative to
energy density by relation,
Where c is velocity of light.
is Stefan constant, its value is

14. For low temperature difference between T and To, newton’s law of Cooling is
expressed as an approximation of Stefan-Boltzmann law i.e.,
When
Then,
Hence,

15. Sol.- Given,
We know that , by Stefan-boltzmann formula,
Q.- Two Large closely spaced concentric spheres(both are black body radiator)
are maintained at temperature 200 K and 300 K respectively. The space in
between the two sphere is evacuated. Calculate the net rate of energy transfer
between the two spheres.

16. 1. White clothes are preferred in summer and dark coloured clothes in
winter.
2. Cooking utensils are blackened at bottom and polished on the upper
surface. Black surface will absorb the Whole of the Heat from the furnace
upper surface not allow the heat inside the utensil to flow out.
3. The thermocouple junction exposed to heat is blackened to absorb
maximum amount of Heat.
4. Polished Reflector are used in electric heaters to reflect maximum heat in
the room.
5. The hot water pipes used in room are painted black so that they can
radiate maximum amount of heat to the room. The same pipes outside the
rooms are painted white so that they does not lose heat to the surrounding.