Question

A new glass material is developed to minimize the transmission of light through a window with a glass panel of thickness 5 mm. The refractive index of the glass material is 1.5 and the absorption coefficient can be changed from 0.3 cm$^{-1}$ to 1 cm$^{-1}$. In the given range of absorption coefficients, the ratio of the maximum to the minimum fraction of the light coming out of the other side of the glass panel is _________. (Round off to two decimal places)

Hint:

For absorption-only problems, reflectance does not matter—the transmission ratio depends only on $e^{-\alpha x}$.

Answer 1

Correct Answer: 1.4

The intensity of light transmitted through an absorbing medium follows Beer–Lambert law: \[ I = I_0 e^{-\alpha x} \] where $\alpha =$ absorption coefficient, $x =$ thickness of glass. Convert thickness to cm: \[ x = 5\ \text{mm} = 0.5\ \text{cm} \] Two extreme absorption coefficients: \[ \alpha_{min} = 0.3\ \text{cm}^{-1},\qquad \alpha_{max} = 1.0\ \text{cm}^{-1} \] Transmitted fractions: \[ T_{max} = e^{-\alpha_{min} x} = e^{-0.3 \times 0.5} = e^{-0.15} \] \[ T_{min} = e^{-\alpha_{max} x} = e^{-1.0 \times 0.5} = e^{-0.5} \] Required ratio: \[ R = \frac{T_{max}}{T_{min}} = \frac{e^{-0.15}}{e^{-0.5}} = e^{0.35} \] \[ R = e^{0.35} = 1.419 \] Rounded to two decimals: \[ \boxed{1.42} \]