During radiant heat treatment of a thin-film material, its shape, which may be hemispherical (a) or spherical (b), is maintained by a relatively low air pressure (as in the case of a rubber balloon). Irradiation on the film is due to emission from a radiant heater of area A_h = 0.0052 m² , which emits diffusely with an intensity of I_e.h = 169,000 W/m² . sr.

  (a) Obtain an expression for the irradiation on the film as a function of the zenith angle θ.

  (b) Based on the expressions derived in part (a), which shape provides the more uniform irradiation G and hence provides better quality control for the treatment process?