Gaussian Beams (use green light (λ = 532 nm for wavelength) for calculations). Carry out the algebra to derive the effect of a lens of focal length, f, on an input Gaussian beam with flat wave-front (Ri = [infinity]) and beam waist ωi = ωo at the lens entrance. What are R(d) and w(d) for all distances, d, to the right of the lens? (Work out the details to obtain general results.)

Along 8-cm diameter steam pipe whose external surface temperature is 900c connects two buildings. the pipe is exposed to ambient air at 70c with a wind speed of 50 km/hr blowing across the pipe. determine the heat loss from the pipe per unit length. (b) air at 500c enters a section of a rectangular duct (15 cm x 20 cm) at an average velocity of 7 m/s. if the walls of the duct are maintained at 100c. a) the length of the tube for an exit temperature of the air to be 40 0c. b)the rate of heat transfer from the air. c) the fan power needed to overcome the pressure drop in this section of the duct.