Beer–Lambert law from Maxwell equations?

[Gavroy]

hi

is it possible to derive the Beer-Lambert law directly from Maxwell's equations? cause i have to derive it and i have only seen some geometrically motivated derivations but i need a proper one.

 

[chrisbaird]

Exponential decay laws (of which the Beer-Lambert law is one example) are not unique to electromagnetics, so Maxwell's equations will not help you. Exponential decay laws are simply the solution to the differential equation where the change in a variable is proportional to the variable (for instance, if the number of donuts I eat every hour is proportional to the number of donuts left, then the total number of donuts as a function of time will decay exponentially).

You could show that plane waves with complex-valued k (which includes absorption) are a solution to Maxwell's equations, and that plane waves with non-zero imaginary part decay exponentially, and therefore obey the Beer-Lambert law. But I would not consider that deriving the law from Maxwell's equation. It's like asking someone to prove 2+2=4 using Maxwell's equations. While number addition is surely obeyed and used in Maxwell's equations, it is a mathematical entity that holds true beyond electromagnetics.