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A beam of monichromatic light falls normally on the surfcae of a plane-parallel plate of thickness l. The absorption coefficient of the substance the plate is made of varies linearly along the normal to its surfcae from x_(1) to x_(2). The coefficient of reflection at each surface of the plate is equal to rho. Neglecting the secondary reflections, find the transmission coefficient of such a plate. |
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Answer» Solution :Apart from the factor `( 1- rho)` on each end face of the plate, we shell get a factor due to absorptions. This factor can be calculated by assuming the plate to consist of a LARGE number of very thin slab within each of which the ABSORPTION coefficient can be assumed to be CONSTANT. Thus we sheel get a produced like `............E^(-CHI(x)dx) e^(-chi(x+dx)dx) e^(-chi(x+2dx)dx)..........` This produced in nothing but `e^(-int_(0)^(l)chi(x)dx` Now `chi (0) = chi_(1), chi (l) = chi_(2)` and variation with `x` is linear so `chi(x) = chi_(1) + (x)/(l) (chi_(2) - chi_(1))` Thus the factor becomes `e^(-int_(0)^(l) [chi_(1) + (x)/(l) (chi_(2) - chi_(1))] dx) =e^(-(1)/(2) (chi_(1) + chi_(2))l` |
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