Q.246. Using Wien's formula, demonstrate that
(a) the most probable radiation frequency
(b) the maximum spectral density of thermal radiation
(c) the radiosity
Ans. a ) The most probable radiation frequency is the frequency for which
The maximum frequency is the root other than ω = 0 of this equation. It is
where x_{0} is the solution of the transcendental equation
(b) The maximum spectral density is the density corresponding to most probable frequency. It is
where x_{0} is defined above,
(c) The radiosity is
Q.247. The temperature of one of the two heated black bodies is T_{1} = 2500 K. Find the temperature of the other body if the wavelength corresponding to its maximum emissive capacity exceeds by Δλ = 0.50 gm the wavelength corresponding to the maximum emissive capacity of the first black body.
Ans. For the first black body
Then
Hence
Q.248. The radiosity of a black body is M_{e} =3.0 W/cm^{2}. Find the wavelength corresponding to the maximum emissive capacity of that body.
Ans. From the radiosity we get the temperature of the black body. It is
Hence the wavelength corresponding to the maximum emissive capacity of the body is
Q.249. The spectral composition of solar radiation is much the same as that of a black body whose maximum emission corresponds to the wavelength 0.48μm. Find the mass lost by the Sun every second due to radiation. Evaluate the time interval during which the mass of the Sun diminishes by 1 per cent.
Ans. The black body temperature of the sun maybe taken as
Thus the radiosity is
Energy lost by sun is
This corresponds to a mass loss of
The sun loses 1 % of its mass in
Q.250. Find the temperature of totally ionized hydrogen plasma of density p = 0.10 g/cm^{3} at which the thermal radiation pressure is equal to the gas kinetic pressure of the particles of plasma. Take into account that the thermal radiation pressure p = u/3, where u is the space density of radiation energy, and at high temperatures all substances obey the equation of state of an ideal gas.
Ans. For an ideal gas p = n k T where n = number density of the particles and i.s Boltzman constant In a fully ionized hydrogen plasma, both H ions (protons) and electrons contribute to pressure but since the mass of electrons is quite small only protons contribute to mass density. Thus
and
Where is the proton or hydrogen mass.
Equating this to thermal radiation pressure
Then
molecular weight of hydrogen = 2 x10^{3} kg .
Thus
Q.251. A copper ball of diameter d = 1.2 cm was placed in an evacuated vessel whose walls are kept at the absolute zero temperature. The initial temperature of the ball is T_{o } = 300 K. Assuming the surface of the ball to be absolutely black, find how soon its temperature decreases η = 2.0 times.
Ans. In time dt after the instant t when the temperature of the ball is T, it loses
Joules of energy. As & result its temperature falls by  dT and
Where ρ = density of copper, C = its sp.heat
Thus
or
Q.252. There are two cavities (Fig. 5.39) with small holes of equal diameters d = 1.0 cm and perfectly reflecting outer surfaces. The
distance between the holes is l = 10 cm. A constant temperature T_{1} = 1700 K is maintained in cavity 1. Calculate the steadystate temperature inside cavity 2. Instruction. Take into account that a black body radiation obeys the cosine emission law.
Ans. Taking account of cosine low of emission we write for the energy radiated per second by the hole in cavity # 1 as
where A is an constant, dΩ is an element of solid angle around some direciton defined by the symbol Ω . Integrating over the whole forward hemisphere we get
We find A by equating this to the quantity is stefan Boltzman constant and d is the diameter of th hole.
Then
Now energy reaching 2 from 1 is
where is the solid angle subtended by the hole of 2 at 1. {We are assuming
This must equal
which is the energy emitted by 2. Thus equating
or
Substituting we get T_{2} = 0.380 k K = 380 K .
Q.253. A cavity of volume V = 1.0 1 is filled with thermal radiation at a temperature T = 1000 K. Find:
(a) the heat capacity C_{v};
(b) the entropy S of that radiation.
Ans. (a ) The total internal eneigy of the cavity is
Hence
(b) From first law
so
Hence
Q.254. Assuming the spectral distribution of thermal radiation energy to obey Wien's formula where a = 7.64 ps•K , find for a temperature T = 2000 K the most probable
(a) radiation frequency;
(b) radiation wavelength.
Ans. We are given
(a) then
so
(b) We determine the spectral distribution in wavelength.
But
so
(we have put a minus sign before dλ to subsume just this fact dλ is ve where dω is +ve.)
This is maximum when
or
Q.255. Using Planck's formula, derive the approximate expressions for the space spectral density u_{ω} of radiation
(a) in the range where hω << kT (RayleighJeans formula);
(b) in the range where Nω >> kT (Wien's formula).
Ans. From Planek’s formula
(a) In a range (long wavelength or high temperature).
for small x.
(b) In the range (high frequency or low temperature) :
and
Q.256. Transform Planck's formula for space spectral density u_{ω}. of radiation from the variable ω to the variables v (linear frequency) and λ (wavelength).
Ans. We write
Then
Also
Q.257. Using Planck's formula, find the power radiated by a unit area of a black body within a narrow wavelength interval Δλ = = 1.0 nm close to the maximum of spectral radiation density at a temperature T = 3000 K of the body.
Ans. We write the required power in terms of the radiosity by considering only the energy radiated in the given range. Then from the previous problem
But
so
Using the data
and
Q.258. Fig. 5.40 shows the plot of the function y (x) representing a fraction of the total power of thermal radiation falling within
the spectral interval from 0 to x. Here is the wavelength corresponding to the maximum of spectral radiation density). Using this plot, find:
(a) the wavelength which divides the radiation spectrum into two equal (in terms of energy) parts at the temperature 3700 K;
(b) the fraction of the total radiation power falling within the visible range of the spectrum (0.400.76 Rm) at the temperature 5000 K;
(c) how many times the power radiated at wavelengths exceeding 0.76 Jim will increase if the temperature rises from 3000 to 5000 K.
Ans. (a) From the curve of the function y(x) we see that y = 05 when x = 1.41
Thus
(b) At 5000 K
So the visible range (0.40 to 0.70) ^im corresponds to a range (0.69 to 1.31) of x
From the curve
y (0.69 ) = 0.07
y( 131) = 0.44
so the fraction is 0.37
(c) The value of x corresponding to 0.76 are
The requisite fraction is then
Q.259. Making use of Planck's formula, derive the expressions determining the number of photons per 1 cm^{3 } of a cavity at a temperature T in the spectral intervals
Ans. We use the formula
Then the number of photons in the spectral interval (ω, ω + dω) is
using
we get
Q.260. An isotropic point source emits light with wavelength λ = 589 nm. The radiation power of the source is P = 10 W. Find:
(a) the mean density of the flow of photons at a distance r = = 2.0 m from the source;
(b) the distance between the source and the point at which the mean concentration of photons is equal to n = 100 cm^{ 3}.
Ans. (a) The mean density of the flow of photons at a distance r is
(b) If n(r) is the mean concentration (number per unit volume) of photons at a distance r form the source, then, since all photons are moving outwards with a velocity c, there is an outward flux of cn which is balanced by the flux from the source. In steady state, the two are equal and so
so
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