Please see the attached image for the problem: The rate (per centimeter) at whic

Question

Please see the attached image for the problem:

The rate (per centimeter) at which light is absorbed as it passes through water is proportional to the intensity, or brightness, at that point. Write down this statement as a differential equation. Solve it in general form (that is derive the formula using symbols that you need) for the intensity as a function of the distance the light has traveled through the water; If 50% of the light is absorbed in 10 centimeters how much is absorbed in 20 centimeters?

Explanation / Answer

a)

If I is the intensity of light at distance x from surface and (I+dI) is the intensity of light at distance x+dx , and alpha is the absorbtion coefficient then the corrsponding differential eq is

dI/dx = -alpha*I

which means the variation of intensity through the thickness dx is proportional to the initial intensity of light. The minus sign comes from the fact that the intensity is decreasing.

b)

by rewiting the eq. as

dI/I =-alpha*dx   and integrating from 0 to x (from I0 to I(x)) one gets

ln[ I(x)/I0 ]= -alpha*x

or I(x) =I0*exp(-alpha*x)

where I0 is the intensity at point x=0 (surface ussualy) and I(x) is the intensity after travelling a distance x

c) ln (0.5*I0/I0) =-alpha*0.1

alpha = 6.931 (1/m)

I(0.2)/I0 = exp(-6.931*0.2) =0.25 =25%

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