Question

Define the spectral transmissivity of a medium of thickness \(L\) in terms of \((a)\) spectral intensities and \((b)\) the spectral absorption coefficient.

Short answer

Answer: The spectral transmissivity of a medium of thickness L in terms of the spectral absorption coefficient is given by the equation \(\tau(\lambda) = e^{-\alpha(\lambda)L}\).

Step-by-step solution

Define spectral transmissivity in terms of spectral intensities

To define the spectral transmissivity in terms of spectral intensities, consider the intensity of incident radiation \(I_0(\lambda)\) and the transmitted intensity \(I_t(\lambda)\) after passing through a medium of thickness L. The spectral transmissivity, represented by \(\tau(\lambda)\), is the fraction of incident radiation that is transmitted through the medium. This can be expressed as follows: \[ \tau(\lambda) = \frac{I_t(\lambda)}{I_0(\lambda)} \]

Define spectral transmissivity in terms of the spectral absorption coefficient

To define the spectral transmissivity in terms of the spectral absorption coefficient, we need to incorporate the concept of absorption into the equation. Suppose the medium has a spectral absorption coefficient represented by \(\alpha(\lambda)\). Using Beer's Law, which relates the absorption of light to the properties of the medium through which it's traveling, we can derive the following expression for the transmitted intensity: \[ I_t(\lambda) = I_0(\lambda) e^{-\alpha(\lambda)L} \] Now, we can substitute the expression for \(I_t(\lambda)\) into the equation for the spectral transmissivity from Step 1: \[ \tau(\lambda) = \frac{I_t(\lambda)}{I_0(\lambda)} = \frac{I_0(\lambda) e^{-\alpha(\lambda)L}}{I_0(\lambda)} \] Finally, we can simplify the equation by canceling out \(I_0(\lambda)\) in the numerator and denominator: \[ \tau(\lambda) = e^{-\alpha(\lambda)L} \] Thus, the spectral transmissivity of a medium of thickness L is defined in terms of the spectral absorption coefficient as \(\tau(\lambda) = e^{-\alpha(\lambda)L}\).