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Stefan Boltzmann Constant is the constant σ of the Stefan-Boltzmann law. This law was initially experimented by Stefan and later theoretically proved by Boltzmann. Its SI unit value is 5.67 × 10–8 W m–2 K–4.
Keyterms: Electromagnetic energy, Radiation, Temperature, Emissivity, Thermal equilibrium, Parameters, Energy
Stefan Boltzmann Law
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Energy can be transferred by radiation over enormous distances, without a medium (i.e., in vacuum). At absolute temperature T, the total electromagnetic energy radiated by a body is related to its size, ability to radiate (called emissivity), and temperature.
The energy emitted per unit time (H) for a perfect radiator is given by H = AT4, where A is the area and T is the absolute temperature of the body.
Stefan Boltzmann Law
According to Stefan Boltzmann's law:
The total amount of radiation energy produced by a black body per unit time from a specific area at absolute temperature is directly proportional to the black body's fourth power.
Black Body and Stefan Boltzmann Constant
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A black body is a hypothetical physical object having enough mass all the energy directed at it, containing it fully and not reflecting any of it back, regardless of the wavelength or angle of incidence of the incident radiations.
Since it absorbs all light that touches its surface, it is described as a "black body." A black body emits electromagnetic black-body radiation when it is at a constant temperature and consequently in thermal equilibrium.
Black Body
The most significant feature of the blackbody radiation curves is that they are universal. They depend on the temperature and are unaffected by the blackbody's size, shape, or composition. Planck's law states that the spectrum of emitted radiation is defined solely by the temperature of the black body and no other parameters.
At constant temperature, a black body has two desired properties:
- It is an ideal emitter: a black body emits as much or more thermal radiative energy than any other body at the same temperature.
- It is a diffuse emitter: the energy emitted iso-tropically, regardless of direction, when measured per unit area perpendicular to the direction of the back body.
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Planck's Law and Stefan Boltzmann Constant
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Planck's law asserts that when the temperature of a radiation-emitting body rises, the total radiated energy of the body increases, and the emitted radiation has shorter wavelengths, as established by German physicist Max Planck in 1900.
- B is the shadowy radiance of the body
- v is the frequency
- T is the absolute temperature
- k B (= 1.380649×10−23 JK−1) is the Boltzmann constant
- h (= 6.62607015×10−34 JHz−1) is the Planck constant
Mathematical Representation of Stefan Boltzmann Law
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From the statement of Stefan Boltzmann law, j* = σT4
where
- j* is known as the black-body radiant emittance
- σ (= 5.67 × 10-8 W/m2 k4) is called the Stefan Boltzmann constant and is a constant of proportionality
- T is the temperature of the black body
For a body that is not black, the radiant emittance is given by:
u = e σ AT4
Taking into consideration the surrounding temperature T 0, the net energy released per unit area is given by:
Δu = u – u0 = eσA [T4 – T04]
Where e is called emissivity, which is defined as the ability of a surface to radiate energy as thermal radiation. It can also be defined as the ratio of the thermal radiation from a surface to the radiation from an ideal black body at the same temperature, given by Stefan Boltzmann law.
Emissivity lies between 0 to 1, and for ideal black bodies, an emissivity of 1 is desired.
Also Check: Thermal Expansion
Value of Stefan Boltzmann Constant
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σ = 5.670367(13) × 10-8 W ⋅ m-2 K-4
From the above, the SI unit of Stefan's constant can be written as W ⋅ m-2 K-4.
Here,
- w stands for Watt.
- m for Metre.
- K for Kelvin.
Value of Stefan Boltzmann Constant
Dimensional formula for Stefan Boltzmann law constant will be [M] 1 [T]-3 [?]-4
Elementary Definitions
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- Total Emissive Power (e)
At a certain temperature, a body's total emissive power is defined as the total amount of radiant energy radiated every second per unit area of the body's surface. It is represented by the symbol e.
Its SI unit is J m-2 s-1.
Total emissive power of a black body is represented by the symbol E.
- Emissivity ε
Emissivity of a body is the ratio of the total emissive power of the body to the total emissive power of a black body. It is represented by the symbol ε. It means that.
ε =e/E or e=εE. For a block body, ε=1.
- Total Absorptive Power (a)
The total absorptive power of a body is defined as the ratio of the total radiant energy absorbed by the body in a certain interval of time to the total energy falling upon it in the same interval of time. It is represented by the symbol a.
Total absorptive power of a black body is represented by the symbol A. By definition A = 1.
Elementary Definitions
Applications of Stefan Boltzmann Constant
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The Stefan Boltzmann constant has many uses in physics. Some of them are:
- It is used to calculate the quantity of heat emitted by the dark body.
- It can convert temperature (K) to intensity (Wm-2) units, which is essentially power per unit area.
So, the Stefan Boltzmann constant, Dimensional formula, Stefan Boltzmann constant value, formula, terms and units, and applications are an important part of physics.
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Things to Remember
- Stefan’s law states that the total energy radiated per unit surface area per unit time by a black body at all the wavelengths is proportional to the fourth power of its absolute temperature.
- A blackbody is the perfect absorber and emitter of light. It absorbs any light that falls on it. A perfect blackbody is also a perfect radiator.
- The better an object is at absorbing light, the better it is at emitting it. So a perfect absorber should be the most efficient radiator possible; but at the same time, if an object is a perfect absorber it will not reflect any radiation, and so it will look black.
- The absolute temperature of an object is the temperature on a scale, where 0 is taken as absolute zero. This is also known as thermodynamic temperature; absolute temperature scales are Kelvin, degree units Celsius and Rankine degree unit Fahrenheit.
- Absolute zero is the temperature at which a system is in the state of lowest possible energy i.e., minimum energy. When molecules approach this temperature, their movements drop towards zero. It is the lowest temperature that a gas thermometer can measure. You know electronic devices will not work at this temperature. Finally, the Kinetic Energy of the molecules becomes negligible or ze.
Previous Year Questions
- Determine ratio of specific heats at constant volume. [AMUEEE 1998]
- Temperature of the sink in a carnot engine. [UPSEE 2019]
- Temperature of surrounding when Cu ball is heated. [AMUEEE 1998]
- Determine the entropy change in the body. [BITSAT 2012]
- Determine the temperature of the star. [BITSAT 2013]
- Detemining blackbody wavelength. [NEET 1989]
- Relation between the wavelength and temperature of black body. [KEAM 2004]
- Determine the thermal conductivity of the metal. [AP EAPCET]
- Determine the rate of heat radiated. [NEET]
- Determine the rate of heat transfer across the rod cross-section. [NEET 2009]
- Two Identical Bodies Are Made of A Material [NEET 2016]
- Steam at 100∘C is passed into 20g of Water [KCET 2014]
- Hammer Of Mass 200 Kg Strikes A Steel Block Of M [DUET 2019]
- Hydrogen, Helium and Other Ideal Diatomic Gas [NEET 2019]
- Describe the temperature vs time graph. [JEE MAIN 2019]
- Cp – Cv relation for diatomic gases. [AMUEEE1999]
- Find the value of ϒ for diatomic gases. [NEET 2019]
- What is calorie? [BHU UET 2010]
- What fraction of heat energy is converted to work? [KCET 2020]
- Heat evolved in neutralisation of HF. [UPSEE 2018]
Sample Questions
Ques. What are the Stefan constant units? (1 Mark)
Ans. The Unit of the Stefan-Boltzmann constant is 5.67 x 10-8 watt per meter squared per kelvin to the fourth (Wm2K-4).
Ques. Why is Stefan's law significant? (2 Marks)
Ans. The Stefan-Boltzmann Law describes how much energy the Sun emits in relation to its temperature (or allows scientists to figure out how hot the sun is based on how much power strikes the Earth in a square metre). The law also forecasts how much heat is radiated into space by the Earth.
Ques. What are the limitations of Stefan's law? (2 Marks)
Ans. Because of Stefan's Law, Newton's Law of Cooling is derived (or deduced). Newton's Law of Cooling's Limitations: When a body's excess warmth above its surroundings is small, this law applies (about 40°C) When the body cools, the ambient temperature is considered remaining constant.
Ques. A hot black body emits the energy at the rate of 16 J m-2 s-1 and its most intense radiation corresponds to 20,000 Å. When the temperature of this body is further increased and its most intense radiation corresponds to 10,000 Å, then find the value of energy radiated in Jm-2 s-1. (2 Marks)
Ans. Wein’s displacement law is, λm.T = b
i.e. T∝ [1/ λm]
Here, λm becomes half, the Temperature doubles.
Now from Stefan Boltzmann Law, e = sT4
e,/e2 = (T1/T2)4
⇒ e2 = (T2/T1)4 . e1 = (2)4.16
= 16.16 = 256 J m-2 s-1
Ques. What is the Stefan Boltzmann Constant Value? (3 Marks)
Ans. Stefan Boltzmann Constant Value
| Types of Units | Stefan Boltzmann’s Constant Value | Units |
|---|---|---|
| CGS units | σ ≈ 5.6704×105 | erg. cm2.s1.K4 |
| Thermochemistry | σ ≈ 11.7×108 | Cal. cm2. day1. K4 |
| US Customary units | σ ≈ 1.714×109 | BTU.hr1.ft2.°R4. |
Ques. What is Stefan-Boltzmann constant? Write its dimensional formula. (5 Marks)
Ans. The Stefan-Boltzmann law states that the intensity of the wavelength of the energy radiated from a body is increased with the increase in the difference between the temperature of the body and the surroundings.
The rate of heat energy loss or the power radiated by any body is proportional to the emissivity of the material of the body, the surface area of the body which can emit radiation and the fourth power of the temperature difference between the body and the surroundings.
Mathematically, this is shown as:
\(P \propto eAT^4\)
The proportionality is made into an equation by a constant known by the Stefan-Boltzmann constant. It has a value given as:
\(\sigma = 5.67 \times 10^{-8} W m^{-2}K^{4}\)
From the Stefan-Boltzmann law, we can deduce the constant as:
\(P = \sigma e AT^4 \)
Therefore,
\(\sigma =\frac{P}{EAT^4}\)
The ratio of the R.H.S. will always be a constant for any material. The actual value of the Stefan-Boltzmann constant is given by the equation:
\(\sigma = \frac{2\pi^5kB^4}{15h^2c^2}\)
\(\therefore \sigma = 5.67 \times 10^-8 Wm ^{-2}K^{-4}\)
Where, kB is the Boltzmann constant,
h is the Planck's constant,
c is the speed of light.
The dimensional formula for each physical quantity can be given as:
[kB] = JK-1
⇒ [kB] = [ML2T-2K1]
∴ [kb]4 = [M4L8T-8K4]
[h] = Js
⇒ [h] = [ML2T-1]
∴ [h]3 = [M3L6T-3]
[c] = ms-2
⇒ [c] = [LT-2]
∴ [c]2 = [L2T-4]
The dimensional formula for the constant is:
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