1.9: Light Decay with Distance

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Example 1.9.1 Some students measured the intensity of light from a 12 volt light bulb with 0.5 amp current at varying distances from the light bulb. The light decreases as the distance from the bulb increases, as shown in Figure 1.9.1A.

If the data are transformed as shown in Figure 1.9.1B then they become very interesting. For each point, $(d, I)$ of Figure 1.9.1A, the point $(1/d^{2} , I)$ is plotted in Figure 1.9.1B. Light intensity vs the reciprocal of the square of the distance from the light source is shown. It looks pretty linear.

Explore 1.9.1

One data point is omitted from Figure 1.9.1B. Compute the entry and plot the corresponding point on the graph. Note the factor $\times 10^{−3}$ on the horizontal scale. The point 2.5 on the horizontal scale is actually $2.5 \times 10^{−3} = 0.0025$.
Explain why it would be reasonable for some points on the graph to be close to (0,0) in Figure 1.9.1B.
Is (0,0) a possible data point for Figure 1.9.1B?

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Distance (cm)	20	25	30	40	50	70	100	150	200
Light Intensity (mW/cm²)	0.133	0.086	0.061	0.034	0.024	0.011	0.005	0.002	0.001

Figure $\PageIndex{1}$: A. Light intensity vs distance from light source. B. Light intensity vs the reciprocal of the square of the distance from the light source. The light intensity is measured with a Texas Instruments Calculator Based Laboratory (CBL) which is calibrated in milliwatts/cm² = mW/cm².

It appears that we have found an interesting relation between light intensity and distance from a light source. In the rest of this section we will examine the geometry of light emission and see a fundamental reason suggesting that this relation would be expected.

In Section 1.3 we saw that sunlight decayed exponentially with depth in the ocean. In Example 1.9.1 we saw that light decreased proportional to the reciprocal of the square of the distance from the light source. These two cases are distinguished by the model of light decrease, and the crux of the problem is the definition of light intensity, and how the geometry of the two cases affects light transmission, and the medium through which the light travels.

Light Intensity. Light intensity from a certain direction is defined to be the number of photons per second crossing a square meter region that is perpendicular to the chosen direction.

The numerical value of light intensity as just defined is generally large – outside our range of experience – and may be divided by a similarly large number to yield a measurement more practical to use. For example, some scientists divide by Avagadro’s number, $6.023 \cdot 10^{23}$. A more standard way is to convert photons to energy and photons per second to watts and express light intensity in watts per meter squared.

Light rays emanating from a point source expand radially from the source. Because of the distance between the sun and Earth, sun light strikes the Earth in essentially parallel rays and light intensity remains constant along the light path, except for the interference from substance along the path.

When light is measured ‘close to’ a point source, however, the light rays expand and light intensity decreases as the observer moves away from the source. Consider a cone with vertex V at a point source of light. The number of photons per second traveling outward within the solid is cone constant. As can be seen in Figure 1.9.2, however, because the areas of surfaces stretching across the cone expand as the distance from the source increases, the density of photons striking the surfaces decreases.

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Figure $\PageIndex{2}$: Light expansion in a cone.

Mathematical Model 1.9.1 Light Expansion: For light emanating uniformly from a point source and traveling in a solid cone, the number of photons per second crossing the surface of a sphere with center at the vertex of the cone is a constant, N, that is, independent of the radius of the sphere.

Because light intensity is ‘number of photons per second per square meter of surface’ we have

\[N=I_{d} A_{d}\]

where

$N$ is photons per second in a solid cone emanating from a light source at the vertex of the cone.

$d$ is distance from the light source.

$I_d$ is light intensity at distance d.

$A_d$ is the area of the intersection with the cone of the surface of a sphere of radius d and center at the vertex of the cone.

Illustrated in Figure 1.9.3 is a cone with vertex V, a sphere with center V and radius d and the portion of the surface of the sphere that lies within the cone. If the vertex angle is $\alpha$ then

\[A_{d}=\quad \text { Area }=2 \pi(1-\cos \alpha) d^{2}\]

Therefore in $N = I_{d} A_{d}$

\[I_{d}=K \frac{1}{d^{2}}\]

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Figure $\PageIndex{3}$: Two views of a cone and section of a sphere with center at the vertex of the cone.

Thus it is not a surprise that in Example 1.9.1, the intensity of light emanating from the 12 volt, 0.5 amp bulb was proportional to the reciprocal of the square of the light intensity.

For light emanating from a point source, the light intensity on the surface of a sphere with center at the point source and radius d is a constant, that is, the light intensity is the same at any two points of the sphere.

Exercise 1.9.1 Shown in Table 1.9.1 is data showing how light intensity from a linear light source decreases as distance from the light increases. Card board was taped on the only window in a room so that a one centimeter wide vertical slit of length 118 cm was left open. For this experiment, we considered 118 cm slit to be an infinitely long line of light. Other lights in the room were cut off. A light meter was pointed horizontally towards the center of the slit, 56 cm above the bottom of the slit, and light intensity was measured as the light meter was moved away from the window.

This experiment is easy to replicate. Try and use your own data for the next two parts. Best to have either a clear day or an overcast day; constant sunlight is needed.
Light emitted from a point surface is constant on surfaces of spheres with center at the light source. On which surfaces would you expect the intensity of light emitted from a linear source be constant?
(You may wish to do the next part before this one.) Explore the data and see whether you can find a relation between light intensity and some aspect of distance.
Formulate a model of how light decays with distance from a slit light source and relate your model to your observed relation.

Table for Exercise 1.9.1 Light intensity from a 1 cm strip of light measured at distances perpendicular to the strip.
Distance (cm)	3	4	5	6	7	8	9
Light Intensity (mW/cm²)	0.474	0.369	0.295	0.241	0.208	0.180	0.159
Distance (cm)	10	11	12	13	14	15	16
Light Intensity (mW/cm²)	0.146	0.132	0.120	0.111	0.100	0.092	0.086