The Inverse-Square Law

The brightness of a star's light falls off with distance according to a simple mathematical law. We will test that law in the lab, and illustrate its key applications in astronomy.


In everyday life we describe light subjectively; for example, light is `good' if it enables us to do what we want to do, and `bad' if it doesn't. But light can be measured and described numerically. In particular, we can measure the intensity of light; if a given source produces one unit of light, two such sources will produce twice as much light, ten sources will produce ten times as much, and so on. Thus it makes sense to talk about the intensity of light in mathematical terms.

In this class we will need to measure the intensity of light in two different ways. First, we must consider the total amount of light a source — say, a star, or a light-bulb — gives off. Second, we must consider the amount of light from a source which reaches our location. The difference between these two kinds of intensity is part of everyday experience. For example, a 100 watt light-bulb is a fairly powerful source of light; placed a few feet from your desk, it provides plenty of reading light. But even a 1000 watt light-bulb won't provide enough light to read by if it's located a few hundred feet away.

It helps to give different names to these two ways of measuring intensity. The total amount of light a source emits is called its luminosity. A light-bulb's luminosity is roughly proportional to the number of watts it uses. (This is not an exact relationship because incandescent light-bulbs are not very efficient: in addition to light, they also give off lots of heat.) The amount of light we receive from a source is called its brightness. Brightness is the amount of light per unit area. The brightness of a source depends on how far away it happens to be, while the luminosity of a source does not.

GEOMETRY OF THE INVERSE-SQUARE LAW

A simple experiment illuminates (pun intended) the relationship between luminosity, brightness, and distance. As shown in the diagram below, we will set up a light-bulb, and on one side of the bulb we will set up a wall with a small hole. The light from the bulb spreads out in all directions. A certain amount of light passes through the hole and falls on a movable screen which is parallel to the wall. The total amount of light passing through the hole and falling on the screen does not depend on where we put the screen. But as we move the screen further away, this fixed amount of light must cover a larger area, and the brightness on the screen decreases.

To be specific, suppose we are using a 200 watt light-bulb. According to the manufacturer, this bulb has a light output of about 4000 lumens. (A lumen is a unit of luminosity — the proper definition would take a while to explain, but you can get a rough idea from the fact that this rather bright bulb is putting out 4000 of them.) Let's put the wall 1 foot away from the center of the light-bulb, and make the hole a square 1 inch on a side. Imagine a sphere with a radius of 1 foot = 12 inches centered on the light-bulb. This sphere has a surface area of 1,810 square inches; in other words, it would take 1,810 squares, each 1 inch on a side, to cover the entire sphere. The 4000 lumens put out by the light-bulb spreads evenly over the entire surface of the sphere, so each square inch gets just 4000 / 1810 = 2.2 lumens, which is also the amount of light passing through the 1 inch hole we've cut in the wall.

The inverse-square law in action. A certain amount of light passes through the hole at a distance of 1 foot from the light-bulb. At distances of 2 feet, 3 feet, and 4 feet from the bulb, the same amount of light spreads out to cover 4, 9, and 16 times the hole's area, respectively.

Now consider the light passing through the hole and falling on the screen. If we put the screen up right next to the hole, this light falls on a square 1 inch on a side. This square receives a total of 2.2 lumens, spread over 1 square inch, so the brightness of the light on the screen is 2.2 lumens / 1 square inch = 2.2 lumens per square inch. If we move the screen to a distance of 2 feet from the light-bulb, the light passing through the hole now falls on a square which is 2 inches on a side. The area of this square is 2 inches × 2 inches = 4 square inches, so the brightness on the screen is now 2.2 lumens / 4 square inch = 0.55 lumens per square inch. Moving the screen even further away spreads the light out more and reduces the brightness of the light even further. The numerical results for this simple experiment are summarized in the table below. In every case, the last column is just 2.2 lumens divided by the area of the illuminated square.

Distance from
bulb to screen
Size of square
on screen
Area of square
on screen
Brightness in square
1 foot (12 inches) 1 inch × 1 inch 1 square inch 2.20 lumens per square inch
2 feet (24 inches) 2 inches × 2 inches 4 square inches 0.55 lumens per square inch
3 feet (36 inches) 3 inches × 3 inches 9 square inches 0.244 lumens per square inch
4 feet (48 inches) 4 inches × 4 inches 16 square inches 0.138 lumens per square inch

We're now ready for the last step, which is to take away the wall between the light-bulb and the screen! When we do this, the brightness of the light falling on the screen does not change. The wall with its central hole helped us define the amount of light falling on the screen, and the bright outline of the hole helped us to see how that fixed amount of light spreads over a greater area as the screen is moved further from the bulb. But the light passing through the hole on its way to the screen `had no idea' that the wall was there, so it produces the same brightness on the screen no matter what. When we take away the wall, more of the screen is illuminated, but the brightness remains the same. The brightness depends on only two things: the luminosity of the light-bulb, and the distance from the bulb to the screen.

We can express the relationship between luminosity, brightness, and distance with a simple formula. Let L be the luminosity of a source which emits light in all directions, and D be the distance from the source to the point where we want to calculate the source's brightness. Then the brightness is
B =  L 

4 π D2
Here the denominator is just the area of a sphere of radius D. All that this formula says is that brightness is the luminosity divided by the area which is illuminated. Because the area of a sphere increases as the square of its radius, it's the square of D which appears in the denominator. That's why this is called the inverse-square law; brightness is inversely proportional to the square of the distance.

COMPARING BRIGHTNESSES WITH A NULL-PHOTOMETER

To test the inverse-square law, we need a way of measuring brightness. With modern technology, brightness can easily be measured electronically. Unfortunately, it's not easy to explain how this technology works; we would have to discuss the nature of electricity, some mysteries of quantum mechanics, and the physics of electromagnetic fields. So we will fall back on an earlier technology which can be understood at an intuitive level without a lot of explanation.

A null-photometer is a device for comparing the brightness of two light sources. It can't provide a direct measurement of brightness, but it can tell you when two sources have the same brightness. In practical terms, the null-photometer we will use is just a sheet of aluminum foil sandwiched between two slabs of wax; a band of foil is wrapped around the edge, with a window allowing you to view the sandwich edge-on.

The operation of a null-photometer is illustrated in the diagram below. To begin with, you orient the photometer so each side is pointing directly at one ot the two light sources you want to compare; the light must strike the wax slabs squarely, and not at an angle. Thus one side is illuminated by one source, and the other side is illuminated by the other source. You then look through the window. If both sources have the same brightness, both halves of the sandwich will be equally bright; this is called a `null' reading (hence the term null-photometer). If one source is brighter than the other, the corresponding side of the sandwich will be brighter than the other side. You eyes are pretty good at judging relative brightness; with a little care, you can determine a null reading quite accurately.

A null-photometer in operation. (a) With more light (arrows) coming from the left than from the right, the left half of the photometer's window is brighter. (b) With equal amounts of light coming from both sides, the two halves of the window have the same brightness.

THE INVERSE-SQUARE LAW IN THE LAB

To test the inverse-square law using a null-photometer, we need to express the law in a slightly different way. A null-photometer tells you if two light sources provide equal brightness; in mathematical terms, that is Ba = Bb, where Ba is the brightness produced by light source `a' and Bb is the brightness produced by light source `b'. Let's say that source `a' has luminosity La and is at distance Da, while source `b' has luminosity Lb and is at distance Db. Then if Ba = Bb, we must have
La

Da2
= Lb

Db2
or
La

Lb
= Da2

Db2
The equation on the left is derived by using our original formula for brightness, and canceling out the common factors. The equation on the right is derived from the one on the left by rearranging the terms; this form is convenient for an experimental test of the inverse-square law.

The basic procedure for our laboratory test of the inverse-square law is shown in the diagram below. We will set up two lights of known luminosities. The null-photometer is placed between the lights, and moved to the point where both halves of the window are equally bright. The distances from the photometer to the lights are then measured. Finally, the luminosities and distances are substituted into the equation just derived; if the law is correct, the two sides should be equal, or nearly equal if we allow for experimental error.

Experimental measurement. The null-photometer is placed between the two lights and moved until both halves of the window have the same brightness.

Testing the law

To test the law properly, we will set up several pairs of lights, with each pair separated from the others to avoid confusion. You will find this experiment easier if you work with a partner; one person can hold the photometer in position, while the other measures the distances to the lights. However, you and your partner should switch roles so that everyone gets a chance to do every measurement.

When you make measurement, hold the null-photometer between the lights, and move it back and forth along an imaginary line between them until both halves of the photometer's window appear the same. Your partner can check to make sure the photometer really is on a line between the two lights, and then measure the distances Da and Db. Ideally, these distances should be measured from the center of each light-bulb to the nearest side of the photometer, as shown in the diagram. Once both distances have been recorded, turn the photometer around so the left face is now the right face, and vice versa. Re-position the photometer between the lights, move it so both halves appear the same, and again measure the distances. Repeat two more times, turning the photometer each time. You should now have four separate measurements of the two distances. Each set of four measurements can be averaged to get a more precise value; you can also look at the range of values for each measurement to get some idea of the accuracy of your work.

Before moving on to the next pair of lights, be sure to record the luminosities La and Lb.

REVIEW QUESTIONS


Joshua E. Barnes      (barnes at ifa.hawaii.edu)
Updated: 1 November 2011
http://www.ifa.hawaii.edu/~barnes/ast110l_f11/inversesquare.html
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