Light intensity of two different waves λ 1 and λ

We measure the light intensity of two different waves λ1 and λ2 in the liquid at the surface and at depth d. We get I01=I02 on the surface and Id1=2Id2 at depth d. It is necessary to determine the depth at which the light intensity λ1 exceeds the light intensity λ2 by 10 times.

Answer:

From the Bouguer-Lambert law it follows that the light intensity is related to the absorption of the medium as follows: I = I0e^(-alphad), where I0 is the initial light intensity, alpha is the absorption coefficient of the medium, d is the depth of light penetration.

We use the relationship for the intensities on the surface and at depth d: I01/I02 = e^(-alphad), Id1/Id2 = e^(-alpha2d)

Let's express alphad from the first equation: alphad = ln(I01/I02)

I pad the alphad into the second equation: Id1/Id2 = e^(-2ln(I01/I02)) = (I01/I02)^(-2)

Let's express d: d = (1/alpha)*ln(I01/I02)

Let's find d for which I1/I2 = 10: I01/I02 * e^(-alphad) / e^(-2alphad) = 10 I01/I02 * e^alphad = 10 * e^2alphad ln(I01/I02) + alphad = ln(10) + 2alpha*d d = (ln(10) - ln(I01/I02)) / alpha

Answer: d = (ln(10) - ln(I01/I02)) / alpha.

The formula can be simplified by noting that ln(10) = 2.3026. Then d = 2.3026 / alpha - ln(I01/I02) / alpha.

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To solve this problem, we need to use the Bouguer-Lambert law, which states that the intensity of light passing through a certain thickness of a medium decreases exponentially according to the law:

I = I0 * exp(-k*d),

where I0 is the initial light intensity, k is the absorption coefficient of the medium, d is the thickness of the medium.

We are also given that I01=I02 at the surface, and at depth d Id1=2Id2, which means that the ratio of light intensities at different depths for waves λ1 and λ2 will be equal to:

I01/I02 = 1,

Id1/Id2 = 2.

In order to find the depth at which the intensity of light with wavelength λ1 exceeds the intensity of light with wavelength λ2 by 10 times, it is necessary to solve the following equation:

I01 * exp(-k1d) = 10 * I02 * exp(-k2d),

where k1 and k2 are the absorption coefficients for waves λ1 and λ2, respectively.

Let us express k1 from the equation:

k1 = (1/d) * ln(I01/(10*I02)),

and from the k2 equation:

k2 = (1/d) * ln(I02/I01).

Let us substitute the values ​​of the absorption coefficients into the equations and solve them relative to the depth d:

d = (1/(k1-k2)) * ln(I01/(10*I02)).

Thus, we obtain a calculation formula for determining the depth d at which the intensity of light with wavelength λ1 exceeds the intensity of light with wavelength λ2 by 10 times.

Answer: d = (1/(k1-k2)) * ln(I01/(10*I02)).

Substituting the known values, we get a numerical answer. A detailed solution to this problem is presented in the picture below:

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