6.6 - Inverse square law

6.6 - Inverse square law

Use the inverse square law to connect lamp distance with light intensity and predict photosynthesis rates. This Higher lesson develops the calculations and the conditions under which the model works.

6.6 - Higher: the inverse square law

This whole part is Higher-only statement 6.6.

Light from a simple source spreads over a larger area as it travels away. The area over which it is spread increases with the square of distance, so intensity falls with the square of distance. If distance doubles, the same light is spread over four times the area and intensity becomes one quarter, not one half.

Inverse square law

For light from the model source,

I∝1d2I \propto \frac{1}{d^2}

For a calculated relative intensity on the scale used in Pearson questions,

Irel=1d2I_{\text{rel}} = \frac{1}{d^2}

To compare two positions without choosing a unit-dependent scale,

I2I1=(d1d2)2\frac{I_2}{I_1} = \left(\frac{d_1}{d_2}\right)^2

Use one distance unit consistently. A value calculated from 1/d21/d^{2} is a relative intensity, normally labelled in arbitrary units; it is not a lux reading. A light meter measures illuminance directly in lux.

Worked example 1: calculate relative intensity

A sample is 40 cm from the lamp. Calculate its relative light intensity using 1/d21/d^{2} and give the answer to three significant figures.

Requested quantity: relative light intensity; d=40 cmd = 40\ \text{cm}.

Equation:

Irel=1d2I_{\text{rel}} = \frac{1}{d^2}

Substitution and working:

Irel=1402=11600=0.000625I_{\text{rel}} = \frac{1}{40^2} = \frac{1}{1600} = 0.000625

Answer: 0.0006250.000625 arbitrary units (3 significant figures). This is smaller than the value at 20 cm, which is sensible because the sample is farther from the same lamp.

6.6 - From light intensity to photosynthesis rate

A limiting factor prevents a process from becoming faster. When light is limiting, supplying more light energy can increase photosynthesis; once another factor limits the rate, the simple proportional prediction no longer applies.

Worked example: connect distance to biological rate

At 15 cm, a photosynthesis-rate proxy is 24 arbitrary rate units. The lamp is moved to 30 cm. Predict the new rate, assuming light remains the factor limiting the rate and all other relevant conditions stay constant.

Distance has doubled, so calculate the intensity fraction:

I30I15=(1530)2=(12)2=14\frac{I_{30}}{I_{15}} = \left(\frac{15}{30}\right)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}

While light is limiting, photosynthesis rate is directly proportional to light intensity. The predicted rate is therefore:

24×14=6.0 arbitrary rate units24 \times \frac{1}{4} = 6.0\ \text{arbitrary rate units}

The prediction is bounded by its assumptions. At very high intensity another factor may prevent rate from continuing to rise in direct proportion, and a real bench lamp may depart from the ideal model because of its shape, reflectors or ambient light.

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