If the density of states function in the conduction band of a particular semiconductor is a constant equal to K, derive the expression for the thermal-equilibrium concentration of electrons in the conduction band, assuming Fermi-Dirac statistics and assuming the Boltzmann approximation is valid.

Avertical steel rod of length 2.0 m and diameter 25 mm is rigidly fixed at its upper end and has a flange at its lower, free, end. what will be the maximum stress produced in the rod when a load of 20 kn is applied, (a) gradually to the lower end, (b) suddenly to the lower end, (c) by a collar falling from a height of 50 mm on to the flange?