QUESTION 4 [TOTAL MARKS: 25] An ideal gas of photons enclosed in a volume V = 1 m³ is maintained at a constant temperature of T = 373 K. Q 4(a) [11 Marks] (i) Establish an expression giving the number of photons having frequencies in the interval v and v+dv. [3 Marks] (ii) Calculate the number of photons per unit volume in the wavelength interval dλ=1 nm near the wavelength = 540 nm. [4 marks] (iii) Calculate the total number of photons per unit volume. [4 marks] Q 4(b) [14 Marks] (i) Establish an expression giving the amount of radiant energy in the interval of frequencies between v and v+dv. [3 Marks] (ii) Calculate the radiant energy per unit volume in the wavelength interval dλ = 1 nm near the wavelength = 540 nm. [3 Marks] (iii) Calculate the total radiant energy density E/V emitted by the photon gas. [5 Marks] (iv) If the emission intensity ✓ (radiant power per unit surface area in all directions CE 4V over a hemisphere) is given by J = (c is the speed of light), show that J = σT where is the Stefan-Boltzmann constant and calculate its values. [3 Marks] . The number of photons N, distributed among the g, various quantum states of N₁ energies liv is given by Planck's distribution: 1 &v hv exp KT ⚫ The number dr, of quantum states of photons with frequencies between v and v+dy is given by dn,=" 8zv²V -dv. Π dx=0.244 and -dx 15 Stefan-Boltzmann constant: σ = =5.67×10 Wm²K+. 60h'c²