This all-in-one online Inverse Square Law Calculator performs calculations using the inverse square law formula that relates the intensity of radial emission from a point source in three-dimensional space to the distance from that source.
Note that both the intensities and distances specified in the calculator are assumed to be expressed in arbitrary but identical units of measurement. You can enter the values of any three known parameters in an input fields of this calculator and find the missing parameter.
What Is the Inverse Square Law?
The inverse square law is a fundamental principle in physics that describes how a specified physical quantity diminishes with increasing distance from its source.
The fundamental reason for this law is geometric in origin and is due to the fact that energy (or other conserved physical quantity) radiates isotropically outward from a point source in three-dimensional space.
This law applies widely across various fields, including optics, acoustics, gravity, and electromagnetism. The name of the law derives from the mathematical relationship it expresses: the intensity of a physical quantity is inversely proportional to the square of the distance from the source of that quantity.
Inverse Square Law Formula
Consider, for example, a point source of energy in three-dimensional space. As the energy spreads uniformly in all directions, it forms an expanding sphere centered on the source. The surface area of a sphere increases with the square of the radius (\(S=4πr^2 \)), meaning that the same total amount of energy is distributed over a larger area as distance increases.
To conserve energy, the intensity (energy per unit area) must decrease proportionally to \(1/r^2\). This inverse square relationship ensures that no energy is lost or gained as it radiates outward. It is simply spread thinner over a growing spherical surface, aligning with the principle of conservation of energy in a closed system.
Mathematically it is expressed by the formula:
$$ I=\frac{P}{4\pi r}$$
Where:
• \(I\) is the intensity or the amount of a physical quantity crossing a unit area per unit time,
• \(P\) is the power or the amount of a physical quantity produced at the point source per unit time,
• \(r\) is the distance from the source to the point under consideration.
The above formula can obviously be rewritten as:
$$\frac{I_1}{I_2} = \frac{r_2^2}{r_1^2}$$
where indices 1 and 2 denote the values of intensities and distances from the source at two points in space. Our Inverse Square Law Calculator uses exactly this formula.
Where Do We Find the Inverse Square Law?
The inverse square law appears across many areas of physics. In optics, light intensity from a point source diminishes with distance, which is crucial in photography, lighting design, and astronomy.
In acoustics, sound intensity drops according to the same principle, guiding audio engineering and speaker placement.
Gravitational and electrostatic forces also follow the law, as described by Newton’s law of gravitation and Coulomb’s law, respectively.
In radiation physics, it helps determine exposure and safety distances from sources of ionizing radiation.
In astronomy, it allows calculation of stellar distances and luminosities, making it fundamental to understanding the scale of the universe.
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