As light passes through a material, its intensity decreases exponentially with distance:
Ix=I0e−αx I_x = I_0 e^{-\alpha x} Ix=I0e−αxwhere IxI_xIx is the light intensity at a distance xxx, I0I_0I0 is the initial light intensity, and α\alphaα is the absorption coefficient. Some students investigate this absorption by measuring the intensity of light passing through different thicknesses of tinted glass.
The table below shows their results for different thicknesses of glass and the corresponding natural logarithms of the light intensity.
| Thickness of glass x/mmx / \text{mm}x/mm | Light intensity Ix/W m−2I_x / \text{W m}^{-2}Ix/W m−2 | ln(Ix/W m−2)\ln(I_x / \text{W m}^{-2})ln(Ix/W m−2) |
|---|---|---|
| 1.0 | 105 | |
| 2.0 | 66.7 | |
| 3.0 | 42.5 | 3.75 |
| 4.0 | 27.1 | 3.30 |
| 5.0 | 17.3 | 2.85 |
Calculate and fill in the missing ln(Ix/W m−2)\ln(I_x / \text{W m}^{-2})ln(Ix/W m−2) values for x=1.0 mmx = 1.0\text{ mm}x=1.0 mm and 2.0 mm2.0\text{ mm}2.0 mm to 3 significant figures.
Show how the exponential equation holds a linear relationship when taking the natural log of both sides, and explain what the gradient and vertical intercept representing y=mx+cy = mx + cy=mx+c correspond to experimentally.
Based on their completed graph, the students obtain a straight-line fit with a vertical intercept of 5.105.105.10 and a gradient of −0.45 mm−1-0.45\text{ mm}^{-1}−0.45 mm−1. Use these values to find the initial light intensity I0I_0I0 and the absorption coefficient α\alphaα.
The students use an LDR in a potential divider circuit as a simple light meter. The circuit consists of a constant +6.0 V+6.0\text{ V}+6.0 V supply connected in series across a fixed resistor R1=4.0 kΩR_1 = 4.0\text{ k}\OmegaR1=4.0 kΩ and the LDR R2R_2R2. The output voltage VoutV_{\text{out}}Vout is measured across the LDR.
The resistance of the LDR was 120 Ω120\,\Omega120Ω when 1.0 mm1.0\text{ mm}1.0 mm of glass was used and 10.0 kΩ10.0\text{ k}\Omega10.0 kΩ when 5.0 mm5.0\text{ mm}5.0 mm of glass was used.
Calculate the minimum and maximum output voltages, and state the range of the output voltage obtained in this experiment.
Explain how this potential divider circuit responds to an increase in the thickness of glass in terms of light intensity, LDR resistance, and output voltage.
