The inverse square law with light

Demonstration

Observe the behaviour of light from a point source. Develop the idea of an inverse square relationship and then consider Newton's gravitational law.

Apparatus and Materials

• Light meter (or photographic exposure meter)
• Lamp, 12 V 24 W
• Lamp holder on base
• Power supply for lamp

Health & Safety and Technical Notes

Use standard tungsten lamps. Avoid halogen lamps unless suitably filtered for UV light.

With some exposure meters it may be necessary to reflect back some of the light to the reverse side of the meter in order to make the readings: a piece of white card will do this effectively.

Some exposure meters have scales marked so that adjacent markings represent light values increasing by a factor of 2 for each stop. Care must be taken when interpreting these.

Read our standard health & safety guidance

Procedure

1. Switch the lamp on in a darkened room and direct the meter towards it from distances of 30, 60, 90 cm. Note the meter readings in each case. The readings will be seen to fall off in the proportion 1: 1/4: 1/9, etc.

Teaching Notes

• This is intended as a quick simple demonstration. The units of measurement are not important.
• Newton was worrying about how to explain Kepler’s Laws using gravity. If the Earth’s gravitational field strength did not vary with distance from the Earth, the time for the Moon’s orbit would be about 11-12 hours.
• By reducing the field strength at greater distances, the time predicted for the orbit can be extended. Using the simplest scheme of halving gravity when the distance doubles, does not fit the results for the time of orbit of the Moon around the Earth. (It would give 1/60 of our g at the Moon since the Moon is 60 Earth radii away from Earth.)
• I began to think of gravity extending to the orb of the Moon ... From Kepler's rule ... I deduced that the forces which keep the planets in their orbs must be reciprocally as the squares of their distances from the centres about which they revolve; and thereby compared the force requisite to keep the Moon in her orb with the force of gravity at the surface of the Earth, and found them to answer pretty nearly. (Newton, recalling his work of 1666.)
• But then he hesitated. It is sometimes said that this was because he did not have an accurate value for the radius of the Earth but that is probably untrue. It is more likely because he saw a difficulty about the gravitational pull of the whole Earth. He put the work aside for many years and turned his attention to developing mathematics and experimenting on light and colour.
• The inverse-square law is a sensible one to try because it is the way in which anything thins out if it sprouts straight lines from a source and continues out without getting lost. Light from a small lamp does that.
• At double the distance, the same light spreading out through clear air without being absorbed, falls on four times the area at double the distance but gives only a quarter of the illumination.

• Another crude example is the ‘butter gun’. Suppose the owner of a restaurant invents a gadget to butter many slices of toast efficiently. It is a small, motor-driven sprayer that squirts out a fine spray of melted butter from its muzzle, in straight lines in a wide cone.
• Suppose the spray just covers one piece of toast at a distance of 30 cm in a standard time. Then four pieces of toast could be placed at 60 cm from the sprayer but the toast would have to sprayed for four times as long, or be more thinly spread. This is the inverse-square law of buttering.