Radiation

aRadiationSOME TERMINOLOGY(1) Thermal Radiation: The process. of transferring of heat without disturbing the medium is called Radiation and transferred heat energy by the radiation is called radiant heat or thermal radiation. The thermal radiation is also called infrared radiation.(2) Total energy density: The total cnergy density of radiation at any point is the total amount of radiant energy per unit volume around that point for all the wavelength taken together. It is denoted by u and its unit is Joule/m 3 or ergs/cm3.(3) Spectral energy density: Spectral energy density for a particular wavelength is the amount of radiant energy per unit volume for a particular wavelength. It is denoted by u𝜆.(4) Total emissive power: The total emissive power of a body is the radiant energy emitted per unit time per unit surface area of a body for all wavelengths taken together. It is denoted by E.(5) Spectral emissive power: Spectral emissive power of a body at a particular wavelength is the radiant energy emitted per unit time per unit surface area of the body within a unit wavelength range. It is denoted by E𝜆.(6) Absorptive power: The absorption power of a body at a particular temperature and for a particular wavelength is defined as the ratio of the radiant energy absorbed per unit surface area per unit time to the total energy incident on the same area of the body in unit time within a unit wavelength range. It is denoted by a𝜆.KIRCHHOFF'S LAWIt states that the ratio of emissive power (e𝜆) to the absorptive power (a𝜆) for a given wavelength at a given temperature is the same for all bodies and equal to the emussive power of a perfectly black body (E𝜆) at that remporature, i.e, e𝜆

a𝜆=E𝜆ProofLet us consider a body placed in an isothermal enclosure. Let dQ be the amount of radiant energy of wavelength lying between 𝜆 and 𝜆+d𝜆 incident on unit surface area per second. If a𝜆 is the absorptive power of the body for the wavelength 𝜆 and at the temperature of enclosure, then the amount of radiant energy absorbed by unit surface area of the body per second will be a𝜆dQ. The remainder of the incident energy (1-a𝜆)dQ will be reflected or transmitted. If e𝜆 is the emissive power of the body for wavelength 𝜆 at the temperature of enclosure then the total radiation lying between wavelength 𝜆 and 𝜆+d𝜆 emitted by unit surface area of the body per second is e𝜆d𝜆. As the body is in the temperature equilibrium, so the energy radiated by the body = energy recerved by it.a

∴(1-a𝜆)dQ+e𝜆d𝜆	=dQ
or, e𝜆d𝜆=dQa𝜆
or, e𝜆 a𝜆	=dQ d𝜆

But dQ

d𝜆 depends only on temperature thus e𝜆

a𝜆 is same for all the substance for a given temperature i.e., the ratio of emissive power to the absorptive power for radiation of given wavelength is same for all bodies at a given temperature. If the body under consideration is perfectly black body, the absorptive power a𝜆=1 for all wavelengths and e𝜆 has maximum value which we denote by E𝜆. or. E𝜆=dQ

d𝜆Comparing above equations e𝜆

a𝜆=E𝜆i.e., the ratio of emission power to the absorptive power is equal to the emissive power of a perfectly black body at a given temperature which is Kirchhofis law. Here we see if e𝜆 increases then its corresponding value of a also increases. Hence from Kirchhoffs law, we can say that good emitter is good absorber of radiation. PRESSURE OF RADIATIONThe radiation posses the properties of light, so like light, it exerts small but definite pressure on the surfaces on which it is incident, called pressure of radiation. Maxwell proved on the basis of electromagnetic theory that pressure is equal to the energy density (i.e. amount of energy radiant per unit volume) for normal incidence on a surface.

Let a photon of energy h𝜈 move with the velocity of light c. According to the theory of relativity,h𝜈=mc2 or, m=h𝜈/c2 .......................(i)Then momentum of photon = mass × velocity=h𝜈

c2c=hv

c=e

cwhere e being energy of photon.Now the momentum incident on the surface PQ per unit per second is given byP=𝛴e

c ............................(ii)where 𝛴e= total energy incident on the surface per unit area per second =E( say )∴P=E

C (iii)If u is the energy density i.e., energy per unit volume, then the total energy passing through any area s of the surface normal to the radiation per second =u×s c.∴ Energy flux i.e. energy radiation per unit area per secondE=usc

s= uc ................. (iv)Substituting this value in (iii) we get,P=uc

c=uThus for normal incidence on a surface, the pressure of radiation is equal to the energy density.P=u STEFAN'S LAWStefan's law states that the rate of emission of radiant energy by unit area of a perfectly black body is directly proportional to the fourth power of its absolute temperature. i.e. E=𝜎T4 .................... (i)where 𝜎 is a constant called Stefan's constant.When a black body is at absolute temperature T and is surrounded by another black body at absolute temperature T0, then the amount of heat lost by the former black body per unit area per unit time is given by E=𝜎(T4-T40) .....................(ii)Proof:

Let us consider a cylinder enclosure ABCD of uniform cross-section with perfectly reflecting wall provided with a piston. Let it be filled with diffuse radiations of density u at uniform temperature T. If V be the volume of the enclosure U=uV ................................(i)Let a small amount of heat dQ flow in the enclosure from outside and piston moves out so that volume changes by dV. So the temperature and hence the energy density u changes, If dV is the change in internal energy of radiation and dW is the external work done, then by the lst law of thermodynamiesdQ=dU+dWUsing equation (1) adQ=d(uV)+PdV =Vdu+udV+PdVa

or, dQ	=Vdu+udV+1 3 udV since, P=1 3u
	=Vdu+4 3udV ..............(ii)

Now, from the 2nd law of thermodynamicsa

dS	=dQ T ................(iii)
∴dS	=V Tdu+4 3⋅u TdV .............(iv)

So, S is the function of two independent variables u and V.a

S	=f(u,V)
dS	=𝜕S 𝜕udu+𝜕S 𝜕VdV ...............(v)

Comparing (iv) and (v) we get𝜕S

𝜕u=V

T and 𝜕S

𝜕V=4u

3T As dS is a perfect differential, we have or, a

𝜕2S 𝜕u𝜕V=𝜕2S 𝜕V𝜕u
𝜕 𝜕u(𝜕S 𝜕V)=𝜕 𝜕V(𝜕S 𝜕u)
or, 𝜕 𝜕u(4u 3T)=𝜕 𝜕V(V T)	................. (vi)

As T is independent of V and is a function of u only, differencing equation (vi) we havea

4 31 T-4 3u T2𝜕T 𝜕u=1 T
1 31 T=4 3u T2𝜕T 𝜕ua
𝜕u u=4𝜕T T	....................... (vii)

Integrating we have∫𝜕u

u=∫4𝜕T

T logu=4logT+logA where logA be constant of integration logu=4logT+logA u=T4 ATotal rate of emission of radiant energy per unit area is related to the energy density by the relationa

E	=1 4uc
∴E	=1 4AcT4
	E = 𝜎 T4

where 𝜎 = 1

4 Ac is constant known as Stefan's Constant whose value is 𝜎 = 5.672 × 10-5 erg/cm2sec.K4 in CGS unit 𝜎= 5.672 × 10-8 Jm-2s-1k-4 in SI unit. Spectrum of black body radiation

Plank's Radiation LawAccording to Planck the amount of energy radiated from a blach body is not continuous but emits in discrete form i.e. radiation emitted in packet of energy. The packet of energy is called quanta. Each packet of energy is equal to h𝜈 where h is the Planck's constant and v is the frequency of radiation. The value of h=6.62×10-34JS. Planck derived a formula to explain the experimental observation of the distribution of energy in black body radiation. The formula was derived using the following postulates:1. A black body radiation chamber is filled up with simple harmonic oscillator which can have energy given byE=nhvwhere v is the frequency of oscillator. h is the plank's constant, n is integer, n=0,1,2,3……..2. The oscillators cannot radiate or absorb energy continuously but an oscillator of frequency v can only radiate or absorb energy in units or quanta of magnitude h𝜈. The average energy of a Planck's oscillator is given by.⏨E=hv

ehv

KT-1 ................(i) The number of resonators per unit volume in the frequency range v and v+dv is given byN=8𝜋v2

c3dv...............(ii) where c is the velocity of light.Combining cquation (i) & (ii) we get energy density per unit volume in the frequency range v & v+dv i.e.Evdv=8𝜋v2

c3hv

ehv

KT-1dvEvdv=8𝜋hv3

c31

ehv

KT-1dvThe above relation is known as Planck's relation.The energy density E𝜆d𝜆 belonging to d𝜆 can be obtained by using v =c

𝜆a

	v =c 𝜆
	dv=-c 𝜆2d𝜆
∴E𝜆d𝜆	=8𝜋h c3(c 𝜆)31 ehc KT𝜆-1(-c 𝜆2)d𝜆

Since the magnitude of dv=|-c

𝜆2d𝜆| = c

𝜆2d𝜆∴E𝜆d𝜆=8𝜋hc

𝜆51

ehc

KT𝜆-1d𝜆which gives the energy density for wave length range 𝜆 and 𝜆+d𝜆 in the spectrum of a black body.WIEN'S DISPLACEMENT LAWIt states that the product of the wavelength corresponding to maximum energy 𝜆m and absolute temperature T is constant i.e., 𝜆mT= constant This constant is called Wien's displacement constant. It's value is 0.2896cmK or 0.2896×10-2mK. The law shows that with increase in temperature, 𝜆m decreases. Wien"s has also shown that the energy Emax is directly proportional to the fifth power of the absolute temperature.Em∝ T5 or, Em=constant × T5Wien deduced the radiation law for the energy emitted at a given wavelength 𝜆 and at a given temperature T DEDUCTION OF RAYLEIGH- JEAN LAW OF SPECTRAL DISTRIBUTION OF ENERGY The number of modes of vibration per unit volume in the frequency range v and v+dv isNvdv=8𝜋v2

c3dvRayleigh and Jeans assumed that the law of equipartition of energy applicable to radiation also, i.e. they considered the average energy of an oscillator (i.e. per mode of vibration) as⏨E=KTThe energy density (i.e., energy per unit volume) within the frequency v and v+dv is given byUvdv= number of modes of vibration per unit volume in the freguency range v and v+dv× Average energy per mode of vibration.Uvdv=8𝜋v2

c3dvKTThis Reyleigh - Jean's law in terms of frequency. In terms of WavelengthU2d𝜆=8𝜋

c3(c

𝜆)2(-c

𝜆2d𝜆)KTSince the magnitude of dy i.e. dv=c

𝜆2d𝜆a

∴U𝜆d𝜆	=8𝜋 c3c2 𝜆2c 𝜆2d𝜆KT
U𝜆d𝜆	=8𝜋KT 𝜆4d𝜆

This is Rayleigh-Jeans law in terms of wavelength. WIEN'S LAW AND RAYLEIGH-JEANS LAW FROM PLANCK'SLAW OF RADIATIONAccording to Planck's law we have,E𝜆d𝜆=8𝜋hc

𝜆51

ehc

KT𝜆-1d𝜆For shorter wave length ehc

𝜆KT becomes large compared to unity and hence the Planck's law reduces toa

E𝜆d𝜆	=8𝜋hc 𝜆51 ehc KT𝜆d𝜆
∴E𝜆d𝜆	=8𝜋hc 𝜆51 ehc 𝜆KTd𝜆

which is Wien's law for shorter wavelength.a

∴E𝜆d𝜆	=8𝜋hc 𝜆51 ehc 𝜆KT-1d𝜆
	=8𝜋hc 𝜆51 1-hc 𝜆KT-1d𝜆

e-x=1-x

1!+ x2

2!.........( For higher wavelength , highert power can be neglected) =8𝜋hc

𝜆51

hc

𝜆KTd𝜆=8𝜋hc

𝜆5𝜆KT

hcd𝜆∴E𝜆d𝜆 = 8𝜋KT

𝜆4d𝜆which is Ragleigh Jean's law. Thus we see that Wien's law holds for shorter wavelength while the Rayleigh Jean's law for longer wavelengths. DEDUCTION OF STEFAN'S CONSTANT FROM PLANCK'S RADIATION LAWTotal radiant energy in unit volume of an isothermal enclosure isE=∞∫0E𝜆d𝜆=∫8𝜋hc

𝜆5(ehc

𝜆KT -1)d𝜆 ...........(i)But the total radiant energy in unit volume of an isothermal enclosure is also given byE=AT4 .....................(ii)where A is constant and T is the absolute temperature of enclosure.∴AT4=8𝜋hc∞∫0d𝜆

𝜆5(ek

𝜆KT-1) ..................... (iii) a Putting, x=hc

𝜆KT𝜆=hc

xKT∴d𝜆=-hcdx

x2KTSubstituting these value in equation (iii) we get,a

AT-1	=8𝜋hc∞∫0hcdx x2KT ((hc 𝜆KT)5 (ex -1)
	=8𝜋hc 15h4∫0-1 ex-1
∴AT4	=8𝜋K4T4𝜋4 15h3c3
∴A	=8𝜋K4T4𝜋4 15h3c3

Since Stefan's constant is given by, 𝜎=Ac

4a

∴𝜎=8𝜋5K4 15h3c3cT4 4=2𝜋4K4 15h3c2

∴𝜎=8𝜋5K4 15h3c3c 4