Stefan’s law

Introduction

In this topic, we are going to learn about Stefan’s law. We will deal with several questions like what Stefan’s law is. We’ll also derive its formula using the simplest method to make your understanding much better about this topic. Further on, we will also deal with the most commonly asked questions on Stefan’s law. Going through the numerical process will make you build your concept stronger; and we’ll also deal with topics related to Stefan’s law.

Basic fundamental definitions.

In the following section, we will study some basic fundamental terms we use in Stefan’s law.

• Absorptive power or absorptive coefficient (a): The ratio of the amount of radiation absorbed by a surface (Qa) to the amount of radiation incident (Q) upon it is defined as the coefficient of absorption a = Qa/Q. It is unitless and dimensionless.
• Spectral absorptive power (a): a= Qa/Q; also called a monochromatic absorptivity coefficient.
• Emissive power (e): The amount of heat radiation emitted by a unit surface area in a unit second at a particular temperature. S.I. unit: J/m2- s or watt/m2.
• Spectral emissive power (e): The amount of heat radiation emitted by the unit area of the body in one second in the unit spectral region at a given wavelength.

Emissivity (e)

• Absolute emissivity or emissivity: Radiation energy given out by a unit surface area of a body in unit time corresponding to unit temperature difference w.r.t. the surroundings is called emissivity.

S.I. unit: W/m2 K

• Relative emissivity (er) :  er =  QGB/QIBB  =  eGB/eIBB = emitted radiation by grey body/ emitted radiation by ideal black body

GB= grey or general body, IBB= ideal black body

1. It has no unit
2. For ideal black body er = 1
3. Range 0 < er < 1

So, before moving to Stefan’s law, we first need to know about what an Ideal Black Body is.

Ideal Black Body

• A body surface that absorbs all incident thermal radiations at low temperature, irrespective of their wavelength and emits out all these absorbed radiations at high temperature, is assumed to be an ideal black body surface.
• The identical parameters of an ideal black body is given by a = a= 1 and r = 0 = t, er = 1.

• The nature of emitted radiations from the surface of an ideal black body only depends on its temperature.
• The radiations emitted from the surface of an ideal black body are called either full or white radiations.
• At any temperature, the spectral energy distribution for the surface of an ideal black body is always continuous, and according to this concept, if the spectrum of a heat source obtained is continuous, then it must be an ideal black body like a kerosene lamp, heating filament etc.
• There are two experimentally ideal black bodies.

1. Ferry’s ideal black body      (b) Wien’s ideal black body

• At low temperatures, the surface of the ideal black body is a perfect absorber, and at a high temperature, it proves to be a perfect emitter.
• An ideal black body need not be of black colour (e.g. Sun).

Define Stefan’s law.

The amount of radiation emitted per second per unit area by the ideal black body is directly proportional to the fourth power of its absolute temperature.

Amount of radiations emitted

E T4 

E = T4

Where T = temperature of ideal black body (in K)

(this law is true for only the ideal black body)

S.I. unit : E = watt/m2 

           = Stefan’s constant = 5.65 × 10-8 watt/m2K4 (universal constant)

Dimensions of : M1L0T–4

Total radiation energy emitted out by the surface of area A in time t:

Ideal black body   QIBB = AT4t 

and for any other black body,   QGB = e AT4t

Rate of emission of radiation

When the temperature of surrounding T0 (Let T0 < T)

Rate of emission of radiation  per unit area of ideal black body surface E1= T4

Rate of emission or absorption of radiation (per unit area) from surrounding E2= T04

The net rate of loss of radiation per unit area from ideal black body surface is

E = E1 – E2 = T4 – T04 = ( T4 – T04)

Net loss of radiation energy from the entire surface area in time t is QIBB =  A (T4 – T04)t

For any other body QGB =  er A (T4 – T04) t 

If, in time, the net heat energy loss for the ideal black body is dQ, and because of this, its temperature falls by dT. Rate of loss of heat (IBB)

RH = dQdt=σA(T4 – T04) J/s

It is also equal to emitted power or radiation emitted per second.

Rate of fall in temperature (rate of cooling)

RF = dTdt=σAms(T4 – T04)

m = mass of body,  s = specific heat capacity 

[because, dQdt=msdTdt ]

When a body cools by radiation, its cooling depends upon:

1. Nature of radiating surface: greater the emissivity (er), the faster will be the cooling.
2. Area of radiating surface: greater the area of the radiating surface, the faster the cooling.
3. Mass of radiating body: greater the mass of radiating body slower will be the cooling.
4. Specific heat of radiating body: greater the specific heat of radiating body slower will be the cooling.
5. The temperature of the radiating body: greater the temperature of the radiating body faster will be the cooling.

Conclusion

In this topic, we have learned about Stefan’s law. We now know what Stefan’s law is. Also, we have derived the equation for Stefan’s law numerically. We have seen what an ideal black body is and some general fundamental terms used in the above paragraph. Further here, we will also discuss some of the most recently asked problems on Stefan’s law. To understand more deeply, Kirchoff’s law is suggested to the user.