part b On page 50 the authors derived Beer\'s Law for attenuation in a medium wi

Question

part b

On page 50 the authors derived Beer's Law for attenuation in a medium with what we call linear absorption, i.e. the rate at which a beam of light is absorbed is proportional to the light intensity. They found that the light intensity decays exponentially. However, there are situations where the absorption is nonlinear, proportional to the square of the light intensity. In that case, the light is modeled by this equation: dI/dz = -aI^2 where a is a constant whose value depends on how strong the nonlinear absorption is in the medium. Assume that the units of I are Watts per square meter. (a) (2 points) What are the units of a? (b) (4 points) Assume that I = 1 at z = 0 and integrate the equation above (using methods similar to those on page 50) to find I(z) for z > 0. We won't worry much about units here; we just want to see if I decays more rapidly than exponentially or less rapidly. (c) (2 points) Does the intensity decay more rapidly or more slowly than the exponential decay that you get with linear absorption? Why? (Hint: Nonlinear absorption is strong only when the intensity is very large.)

Explanation / Answer

a) by the principle of homogenity ,the units on the left must be equal to the units on the right

unit on the left hand side is W/m3

so dimension of a = (W/m)^2 / W/m3 = W-m

b)dI/dz=- aI2

integrating both sides we have

dI/I2 = -a dz

-1/I = -az + C

at z=0 I=1

-1= 0+C

C=-1

so

-1/I =-az -1

I = 1/(az+1)

c)it decays less rapidly than exponential decay because intensity here is a linear coefficient of z

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