The Inverse Square Law for Light
Slide a lamp along a rail and measure how the intensity of the light on a wall depends on the distance. Record your readings, plot the graphs and find the law.
You can also drag the lamp along the rail in the picture.
| # | r (m) | d (m) | I (W/m²) | I r² |
|---|
Record at least 3 readings at different distances.
Show the physics and the notes
Light spreads out. A small lamp sends out light in all directions. Here the lamp is in a reflector that sends all its light into a cone with a fixed angle, so that the beam makes a circle on the wall. As the lamp moves back, the cone gets wider and the same light is spread over a bigger circle. The diameter of the spot is proportional to the distance, d = 0.8 r, so the area of the spot is proportional to the square of the distance:
A = π (d/2)² = π (0.4 r)² = 0.503 r²
Intensity is the power of the light that falls on each square metre of the surface: I = P/A, in watts per square metre (W/m²). The power of the lamp is the same however far away the wall is, so
I = P / (0.503 r²) ∝ 1/r²
This is the inverse square law: double the distance and the intensity falls to a quarter; treble it and it falls to a ninth. It is true for any source that spreads its energy evenly in all directions, such as light, sound, radio waves and radiation from a radioactive source, as long as nothing absorbs the energy on the way.
How to verify it. (1) Record the intensity at 8 or more distances from 0.5 m to 5 m. (2) Plot I against r: it is a curve that falls steeply and then flattens, which does not tell you the power. (3) Linearise the data by plotting I against 1/r²: if the law is right the points lie on a straight line through the origin, and the gradient is P/0.503, which is 19.9 W for a 10 W lamp. (4) Check that I × r² is the same for every reading (the last column). A fit of ln I against ln r gives a gradient of −2 (the exponent).
Measuring the spot. Drag the ruler over the picture of the wall, through the centre of the spot, and read where the edges of the light cross it. The diameter should be 0.8 times the distance. The sensor at the centre of the spot reads the intensity.
The brightness on the screen cannot show the real range of intensities (from 4400% at 0.15 m to 4% at 5 m of the value at 1 m is a factor of 1100), so the picture is compressed. Use the numbers, not the brightness of the picture. With measurement noise on, each reading you record is off by a random amount of up to about 3%, as in a real experiment.
Learn more in the Inverse square law lesson. Original idea by Physics High.