7. Using the radiative transfer equation of a simple slab-like absorbing medium with no internal emission source, I = I0 exp(−n σ x), where I0 is the original photon flux, σ is the absorption cross section, n is the particle number density of medium, and x is the path length through the medium.
7a) What is the formula to calculate τ in terms of quantities that are provided above?
7b) What fraction of the original photon flux would be absorbed by the medium of τ = 5?
To calculate τ (optical depth) in terms of the given quantities, we can use the formula:
τ = n σ x.
This formula indicates that optical depth, τ, is equal to the product of the particle number density, n, the absorption cross-section, σ, and the path length through the medium, x.
Now, to find out what fraction of the original photon flux would be absorbed by the medium when τ = 5, we can use the formula provided in the question:
I = I0 exp(−n σ x),
where I represents the photon flux after passing through the medium.
Rearranging the equation, we get:
exp(−n σ x) = I / I0.
By substituting τ = n σ x, we can rewrite the equation as:
exp(−τ) = I / I0.
When τ = 5, we can plug this value into the equation:
exp(−5) = I / I0.
Using the exponential function's definition, we find:
I / I0 = exp(−5) ≈ 0.00674.
Therefore, approximately 0.674% (0.00674 as a decimal) of the original photon flux would be absorbed by the medium when τ = 5.