Inverse Square Law

Inverse Square Law Refers to a physical quantity or intensity that is inversely proportional to the square of the distance from the source. Or in English: As the distance from the source increases, the strength of the field goes down really fast!

Inverse Square Law In math speak that means: you take the distance from the source. Square it! Then find the inverse (or reciprocal). And you’re done!

Examples A lot of things follow an inverse square law. In physics the most common examples are gravitational and electrical forces. The simplest example to talk about though, is light intensity.

Light Intensity Light Intensity refers to how bright a light it. (Also known as amplitude in the wave world). The inverse square law just states that as you get further away from the light source, the lower the intensity (or dimmer the light). You understand this intuitively. Think of a flashlight. When you hold the flashlight close to a surface you see a small and bright circle. If you pull it farther away the circle becomes larger but the light gets dimmer.

Light Intensity But why is it an inverse square instead of just an inverse relationship?

Light Intensity Light radiates outward in a sphere.

Light Intensity Each wave front travels outward forming an ever larger circle. As the wave moves outward, remember it has the same total amount of energy because energy is conserved so the total has to stay the same. But now, the energy is spread out over a larger area. This is why the light dims.

Square Relationship The square relationship is a little easier to see in 3D. If the distance from the source is doubled, the sides of the square also double. So you get FOUR times the area. If the distance triples, the sides of the square also triple. So you get NINE times the area. This is why it is called a SQUARE relationship. This number is the distance from the source squared.

Inverse Square Relationship But we don’t care as much about how the area has changed, but rather how much energy is concentrated in that area. The key is to remember that the same amount of energy passes through each layer of the sphere. It just gets spread out. The unit area closest to the sun has the same energy as the 4 unit area which is the same as the 9 unit area. So each square in the second layer has ¼ the energy of the first layer. And each square in the third layer has 1/9 the energy of the first layer. Both of these numbers are the INVERSE of the SQUARE of the distance from the source. That’s why it’s called an Inverse Square Law. 1E ¼E ¼E ¼E ¼E

Inverse Square Relationship Graphically that looks like THIS:

Inverse Square Law To summarize:

Inverse Square Law Remember THIS when solving problems: In math speak that means you take the distance from the source. Square it! Then find the inverse (or reciprocal). And you’re done!

Gravity This also applies to gravity and is referred to as the Universal Law of Gravity If you double (x2) your distance from earth, the gravitational force on you drops down to 1/22 or ¼. If you multiply your distance from the earth, by 100, the gravitational force drops down to 1/1002 Interestingly, the gravitational force of earth can never go down to zero. No matter how far away you get there is still a teeny tiny amount of gravity acting on you.

Electricity This also applies to electricity and is referred to as Coulomb’s Law. If you double (x2) the distance between two charges, the electrical force between them drops down to 1/22 or ¼. If you halve the distance between two charges,, the electrical force drops down to 1/(1/2)2 = 1/(1/4) = 4 Interestingly, the electrical force of charge can never go down to zero either. No matter how far away a charge get there is still a teeny tiny amount of electrical attraction or repulsion acting on the first charge.