Inverse square

Distant lights appear dimmer

Introduction

A photometer made by making a grease spot on white paper can be used to study how the brightness of a lamp decreases with distance. A simple photometer is used to compare the brightness of lamps.

Material

• a meterstick
• five 60 Watt lightbulbs and holders. (It must be a spherical bulb not a flood light.)
• photometer
• two pieces of translucent white plastic at least 2.5 cm (1”) square and 5 mm thick
• a piece of aluminum foil the same size as the plastic
• transparent tape or a rubber band

Assembly

To make a plastic photometer, place the aluminum foil between the two pieces of plastic and hold them all together with the rubber band or tape.

To Do and Notice

Mount two 60 watt bulbs in holders and place them so that their centers are 1 meter apart. Hold the photometer half way between them. Notice how the two sides of the

Move the photometer toward and away from one lamp and notice that when the photometer is nearer to one lamp, that side of the photometer is brighter.

Replace one of the lights with 4 lights.

Hold the photometer in the middle, half way between the one light and the four, notice how the side of the photometer nearest the four lights is brighter.

Move the photometer back and forth until both sides have equal brightness.

Measure accurately the distance from the single lamp t the photometer and the quadruple set of lamps.

What’s Going On?

The light intensity from a point source or a spherical source decreases inversely proportional to the square of the distance. This inverse square proportionality can be used to quantitatively compare the brightness of the bulbs.

For two equally bright bulbs the light intensities are the same when the distances to the bulbs are the same.

For one bulb versus four bulbs however the light intensity balances when the photometer is closer to the single bulb.

Find the ratio of the distance from the photometer to the center of the single bulb d1 to the distance from the photometer to the center of the 4 bulbs d4.

For me: d1/d4 = 1/2

That is, the distance from the bulb to the photometer was 33 cm while the distance from the four bulbs was 67 cm.

Math Root

The intensity of a light, I, is the power per meter squared carried by the light.

I1 = P1/4pd12 where the 60 watts of power is spread over a sphere with surface area 4pd12.

Notice that for a source with a given power, the intensity decreases as the inverse of the square of the distance. This is the origin of the inverse square law.

The light power emitted by one bulb is proportional to the total power 60 watts. The light power emitted by 4 bulbs is 4 times this. P4/P1 = 4

The photometer is placed so that the intensities are equal:

I1 = I4

this means that

P1/4pd12 = P4/4pd42

or that doing the math:

d12 /d42 = P1/P4 = 1/4

Check out the theory and experiment.

theory

d12 /d42 = 1/4

experiment

d1/d4 = 1/2

They agree!

So What?

The Math root showed that the intensity of a source was inversely proportional to the square of the distance to the source.

The intensity of a light source obeys an inverse square law.

This applies to light bulbs as well as to stars and supernova.

If you know the power emitted by a supernova, and for one type of supernova, type Ia, we believe we do then you can measure the intensity at the earth and compute the distance to the supernova.