Proof of the Inverse Square Law

The inverse square law is a fundamental principle in physics that describes the relationship between the intensity of a physical quantity and the distance from its source. It states that the intensity of a physical quantity decreases with the square of the distance from the source.

To understand and prove the inverse square law, we can consider the example of the intensity of light from a point source. Let's break down the proof into the following sections:

1. Definition of Intensity:
- The intensity of a physical quantity is the amount of that quantity per unit area or per unit solid angle.
- In the case of light, it refers to the amount of light energy passing through a unit area per unit time.

2. Intensity at a Distance:
- When a point source emits light, the light spreads out in all directions, forming a sphere with the source at its center.
- As the sphere expands, the same amount of light energy is distributed over a larger surface area.
- Therefore, the intensity of light decreases as the distance from the source increases.

3. Calculation of Intensity:
- The intensity of light at a particular distance can be calculated using the formula:
Intensity = Power/Area
- Assuming the point source emits light uniformly in all directions, the power remains constant regardless of distance.
- The surface area of a sphere is given by the formula:
Area = 4πr^2, where r is the radius or distance from the source.
- Substituting these values into the intensity formula, we get:
Intensity ∝ Power/4πr^2

4. Inverse Square Relationship:
- To prove the inverse square law, we need to show that the intensity is inversely proportional to the square of the distance.
- Let's consider two distances, r1 and r2, where r2 is twice the distance of r1.
- Plugging these distances into the intensity formula, we get:
Intensity1 ∝ Power/4πr1^2
Intensity2 ∝ Power/4πr2^2
- Dividing the two equations, we find:
Intensity2/Intensity1 = (Power/4πr2^2)/(Power/4πr1^2)
Intensity2/Intensity1 = r1^2/r2^2
- Simplifying further, we obtain:
Intensity2/Intensity1 = (r1/r2)^2
- This equation shows that the ratio of intensities is equal to the square of the ratio of distances, proving the inverse square law.

In conclusion, the inverse square law is a fundamental principle that describes the relationship between the intensity of a physical quantity and the distance from its source. The proof demonstrates that the intensity decreases with the square of the distance, as the same amount of energy is spread over a larger surface area. This concept is applicable to various physical phenomena, including light, sound, gravity, and electromagnetic radiation.