Irradiance and Lambert’s Law

It is a direct experience, although some intuitive explanation, the sun heats less the further from the zenith, either during the day or through the seasons. Among the various causes, the most prevalent is the height of the Sun above the horizon or equivalently, the value of the angle between the distance to this star segment and the ray pointing to the zenith. This variation produces that the same amount of radiation of energy emitted covers on different amount of surface.

Two ways of looking at the phenomenon:


A more enlightened area, the radiation is scattered more, so that each unit area receives less energy.
The German physicist Johann Heinrich Lambert, studied these phenomena irradiance, finding the equations relating these variables and named after Lambert’s law.
To avoid overloading the paper with formulas, there is a brief development of this law in Wikipedia: Lambert’s Law

To verify this mathematical relationship between the angle of incidence and the amount of radiation received, we made this experience illuminating, with a LED, an LDR sensor from different angles, from zenith (0 °) to the horizon (90 °).

Using Lambert equation:


where we get the irradiance E and where both I It is the intensity of the light source, and r which is the distance from the source, they are constant throughout the experience. Therefore the result depends only on the angle of incidence of light \alpha on the sensor.

The assembly comprises a servomotor, holding a rod on whose end is the LED. This engine allows us to vary the inclination of the light beam, approximately, grade by grade. The illuminated sensoris at the height of the motor shaft.


The connection diagram is:


The program does is to turn by a degree of the rod and take the time to load capacitor connected to the LDR. With that we have a reference value of the resistance variation of the LDR. We have already seen how to take charge time of a capacitor at the input: Load a capacitor .

The results are these, where the blue line is the value of that time and the red line is the cosine of the incident angle. You can see the trend of LDR resistance change that fits the cosine predicted by theory.


Arduino code:


1 comment

  1. Thank you for sharing those setups. Do you post the list of the materials used ? where to find them? I teach HS kids and I want to try them. A link to a vendor would be great. Thank you.

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