Question

1.1 Planck's Thermal Radiation Formula (See SQ1) Planck's formula of thermal radiation can be expressed in terms of frequency as df (a) From Eq.(1) (right-hand side), work out the units of u(f,T). Hence, state clearly the meaning (b) From Eq(l), show that the Stefan-Boltzmann law follows, i.e, total energy goes like oT and find an expression for the prefactor σ. Hint: The following integral (that you will see in statistical mechanics course) may be useful e-idr = 15 Optional (No bonus point): Show Eq. (2).] (c) Consider the factor h f in Eq. (1) which gives the mean (thermal average) energy of an oscillator of characteristic frequency f. The same term also appears in Einstein's theory of heat capacity of solids. From Eq. (3), find the behavior for (6) hf kgT (low frequency and/or high temperature) and (ii) hf > kgT (high frequency and/or low temperature). (d) Einstein's theory of heat capacity assumes that there are 3N identical but independent oscillators Using Eq. (3), derive an expression for heat capacity Cmolar of a solid consisting of one mole of atoms. Hence, find the behavior of Cmolar(T) for (G) knT > hf and (ii) knT<hf.

Answer 1