# Black Body

#### Perfectly Black Body:

• A body which absorbs all the radiant heat incident upon it is called a perfectly black body.
• Thus the coefficient of absorption of a perfectly black body is equal to 1. In fact, the blackness of such a body is due to the fact that it does not reflect or transmit any part of heat incident upon it.
• No body exists in nature, which can be called a perfectly black body. For practical purposes, lamp black which absorbs nearly 98 % of the heat incident upon it is considered as a perfectly black body.

#### Characteristics of Perfectly Black Body:

• A perfectly black body which absorbs all the radiant heat incident upon it.
• For perfectly black body the coefficient of absorption is equal to 1.
• The blackness of such a body is due to the fact that it does not reflect or transmit any part of heat incident upon it. Thus coefficient of reflection and coefficient of transmission are zero.

#### Ferry’s Black Body:

• A body which absorbs all the radiant heat incident upon it is called a perfectly black body.

#### Construction:

• A perfectly black body can be artificially constructed by taking a double-walled, hollow metal sphere Having a small hole.
• The inner surface of the sphere is coated with lamp black and it has a conical projection on the opposite side of the hole.

#### Working:

• The radiation entering the sphere through this hole suffers multiple reflections.
• During every reflection, about 98% of the incident radiant heat is absorbed by the sphere. Therefore the radiation is completely absorbed by the sphere within a few reflections.
• In this way, the sphere acts as a perfectly black body whose effective area is equal to the area of the hole.

#### Spectrum of a Black Body:

• A black body emits radiations of all possible wavelengths from zero to infinity. These radiations are of electromagnetic nature.
• These radiations do not depend on nature of the surface of the black body but depend only on its absolute temperature.
• Black body radiations extend over the whole range of wavelength of electromagnetic waves. The distribution of energy over this entire range of wavelength or frequency is known as black body radiation spectrum.
• A sensitive instrument called bolometer is used to find energy density between the wavelengths λ and λ  + dλ, By rotating the prism of the instrument this energy density is found for all the range of wavelengths at a constant high temperature of the perfectly black body.

#### Characteristics of the Spectrum of a Black Body:

• The emissive power of perfectly black body increases with increase in its temperature for every wavelength.
• Each curve has characteristic form and each of them has a maxima i.e. maximum emissive power corresponding to a certain wavelength.
• The position of maxima shifts towards ultraviolet region (shorter wavelength) with the increase in the temperature.
• λm T = Constant (Wien’s displacement law)
• The area under each curve gives the total radiant power per unit area of the black body at that temperature and it is directly proportional to T4 (Verification of Stefan’s law)

#### Wien’s Displacement Law:

• For a black body, the product of its absolute temperature and the wavelength corresponding to maximum radiation of energy is constant.

Thus, λm T = Constant

The value of the constant of Wien’s displacement law is 2.898 x 10-3 mK.

#### Significance of Wien’s Displacement Law:

• This law can be used to surface temperature of stars. This is the only method to determine the temperature of celestial bodies.
• It explains colour change in solid on heating from dull red (longer wavelength) to yellow (shorter wavelength)to white (all wavelengths of visible spectra).

#### Stefan’s Law:

• The heat energy radiated per unit time per unit area of a perfectly black body is directly proportional to the fourth power of its absolute temperature.
• Explanation: Let Eb, the heat radiated per unit time per unit area of a perfectly black body whose absolute temperature is T.

So by Stefan’s Law,

Eb ∝ T4

Eb   = σ  T4

where σ  is a constant known as Stefan’s constant.

The value of σ in S.I. system is 5.67 × 10-8 Jm-2 K-4s-1. or 5.67 x 10-8 Wm-2 K-4

The value of σ  in c.g.s system is 5.67 × 10-5 erg cm-2-4s-1.

Dimensions of σ are [M1L0T-3K-4]

#### Expression for the Rate of Loss of Heat to the Surrounding:

• Let T be the absolute temperature of a perfectly black body. Let To be the absolute temperature of the surrounding.

So by Stefan’s Law,

Heat radiated per unit time per unit area of a perfectly black body   = σ  T4

Let A be the surface area of the perfectly black body. Then,

Heat lost by the body  per   unit   time = A σ T4

where σ is a constant known as Stefan’s constant.

Heat   received  from the surrounding per unit time = A σ To4

Net rate of loss of heat = A σ T4 – A σ To4

= A σ(  T4 – To4)

This is an expression for the rate of loss of heat to the surrounding.

#### Newton’s Law of Cooling:

• The rate of loss of heat by a body is directly proportional to its excess temperature over that of the surroundings provided that this excess is small.
• Explanation: Let θ and θo, be the temperature of a body and its surroundings respectively. Let dQ / dt be the rate of loss of heat, So from Newton’s Law of Cooling,

where k is a constant.

#### Alternate Statement:

• By Newton’s law of cooling, mathematically

Where, θ andθ, are the temperature of the body and its surroundings respectively and

dQ / dt is the rate of. loss of heat. K is constant.

Let ‘m’ be the mass of the body, c be its specific heat.

• Thus, the rate of fall of a temperature of a body is directly proportional to its excess temperature over that of the surroundings.

#### Derivation of Newton’s Law of Cooling from Stefan’s Law:

• Let us consider a body whose surface area is A having absolute temperature T and kept in the surrounding having absolute temperature To.  Let e be the emissivity (or coefficient of emission) of the surface of the body.

Let ( T  -To) =  x,  where  x  is Small.

∴ T   =  To  +    x.

Let dQ/ dt be the rate of loss of heat by the body. We know that

E / Eb = e

∴  E  = e Eb

Where E & Eb, are the emissive powers of the body and perfectly black body respectively.

Using Stefan’s Law we know that for a perfectly black body rate of loss of heat =  Aσ(  T4   –  To4 )

Therefore, for given body,

As  x /To is small so higher powers of  x /To will be very small and hence those terms can be neglected.

• This is Newton’s Law of cooling i.e. the rate of loss of heat of a body is directly proportional to its excess temperature over the surroundings provided the excess is small. Thus Newton’s Law of Cooling is derived (or deduced) from Stefan’s Law.

#### Limitations of Newton’s Law of Cooling:

• This law is applicable when the excess temperature of a body over the surroundings is very small (about 40OC)
• When body is cooling the temperature of the surrounding is assumed to be constant. which is not true.
• The law is applicable for higher temperature using forced convection.

• The specific heat of a solid or liquid is determined by the method of mixtures.
• The solid is heated to a high temperature. It is dropped in a calorimeter containing water (or liquid) at room temperature. Finally the maximum temperature of the mixture is noted.
• Now  as the temperature of the mixture begins to increase, the mixture begins to lose heat by conduction and radiation. Loss of heat by conduction can be minimised by surrounding the mixture by a bad conductor of heat such as cotton, wool etc. However, the loss of heat by radiation cannot be stopped.
• Therefore the maximum temperature of the mixture is always less than the temperature it would reach if radiation were absent. This correction to be made in the final temperature of the mixture is called radiation correction.

#### Method of Applying Radiation Correction:

• A stopwatch is started at the moment the solid is dropped into the liquid and time t taken by the mixture to reach the maximum temperature is θ noted.
• The mixture is then allowed to cool for time t / 2. Let θ ’ be the temperature of the mixture after time t / 2.
• Then, radiation correction = Δθ  =  ½ (θ – θ  )
• Thus corrected maximum temperature of the mixture =  θ + Δθ

#### Solar Constant:

• The solar constant is the rate at which solar radiant energy is intercepted by the earth per unit area at the outer limits of earth’s atmosphere at the earth-sun mean distance.
• The solar constant, S = 1353 W/m².

#### Calculation of Surface Temperature of the Sun:

• The central portion of the sun is very hot. It has a temperature of 107 K. It can be estimated using concepts of nuclear reactions.
• The outer surface of the sun is comparatively cooler this region is called the photosphere. Its temperature can be estimated using solar constant.
• Let T be the absolute temperature of the surface of the sun. Let Rs be its radius. By Stefan’s law the total power radiated per second is given by

Where σ = Stefan’s constant

Let r be the earth-sun mean distance. r = 1.496 × 10¹¹ m,

Now the energy radiated by sun is distributed over a sphere of surface area 4πr²

By definition of solar constant

#### Greenhouse Effect:

• Earth’s surface absorbs thermal energy from the sun and becomes a source of thermal radiation. The wavelength of the radiation lies in infrared region.
• A large part of the radiation is absorbed by green house gases like carbon dioxide, methane, nitrous oxide, Chlorofluorocarbons, troposphere ozone. Due to which the atmosphere of the earth heats up and the atmosphere gives more energy to the earth resulting in warmer surface.
• Above process repeats till no radiations are available for absorption. This heating of the surface and the atmosphere of the earth is called greenhouse effect.
• Significance of greenhouse effect is that it keeps the earth warmer which leads to the biodiversity. In absence of this effect the temperature of the earth would be -18° C.
• But due to human activities the quantities of greenhouse gases is increasing rapidly making the earth warmer. This increase may disturb life of plants and animals. It may result in melting of ice in polar regions, which may lead to rise in sea level submerging the costal regions.