Which equation correctly expresses the attenuation of intensity through a material, given I = I0 e^(-μ x)?

Which equation correctly expresses the attenuation of intensity through a material, given I = I0 e^(-μ x)?

• A. I = I0 e^(+μ x)
• B. I = I0 e^(-μ x)
• C. I = I0 / (1 + μ x)
• D. I = I0 - μ x

Answer: (B)

Attenuation of intensity through a material follows exponential decay because the amount of beam remaining after traveling a small distance is proportional to what you started with, with a constant probability per unit length of interaction. If a thin slice dx reduces the intensity by a fraction μ dx, the change in intensity satisfies dI = -μ I dx, or dI/dx = -μ I. Solving this differential equation with the initial intensity I0 at x = 0 gives I = I0 e^{-μ x}. The negative sign ensures that increasing thickness x lowers the intensity, and μ is the linear attenuation coefficient with units of 1/length. This form is the Beer-Lambert law for attenuation. For very small x, you can approximate I ≈ I0 (1 - μ x). The other forms don’t match the underlying physics: a plus sign would imply growth, a simple inverse or linear subtraction doesn’t reflect the multiplicative, probabilistic survival of the beam.

Attenuation of intensity through a material follows exponential decay because the amount of beam remaining after traveling a small distance is proportional to what you started with, with a constant probability per unit length of interaction. If a thin slice dx reduces the intensity by a fraction μ dx, the change in intensity satisfies dI = -μ I dx, or dI/dx = -μ I. Solving this differential equation with the initial intensity I0 at x = 0 gives I = I0 e^{-μ x}. The negative sign ensures that increasing thickness x lowers the intensity, and μ is the linear attenuation coefficient with units of 1/length. This form is the Beer-Lambert law for attenuation. For very small x, you can approximate I ≈ I0 (1 - μ x). The other forms don’t match the underlying physics: a plus sign would imply growth, a simple inverse or linear subtraction doesn’t reflect the multiplicative, probabilistic survival of the beam.