Inverse Square Law Calculator: Calculate Intensity, Distance & Source Strength (I = k/r²)

The inverse square law is a fundamental principle in physics that states that a specified physical quantity (such as intensity, force, or field strength) is inversely proportional to the square of the distance from the source. This law applies to many phenomena including light, sound, radiation, gravitational force, and electric fields. Our Inverse Square Law Calculator simplifies these calculations, allowing you to determine intensity, distance, or source strength using the relationship: I = k / r², where I is the intensity, k is the source strength (constant), and r is the distance from the source.

Whether you're calculating light intensity for photography, sound levels for acoustics, radiation exposure for safety, or gravitational/electric forces, understanding the inverse square law is crucial. Our calculator helps you solve for any of the three variables, making complex physics calculations simple and accessible.

Our Inverse Square Law Calculator offers three calculation modes:

1. Calculate Intensity (I): Enter the source strength (k) and distance (r) from the source. The calculator determines the intensity using I = k / r².
2. Calculate Distance (r): Enter the intensity (I) and source strength (k). The calculator determines the distance using r = √(k / I).
3. Calculate Source Strength (k): Enter the intensity (I) and distance (r). The calculator determines the source strength using k = I × r².

Select your calculation mode, enter the known values with your preferred units, and click Calculate to get instant results with detailed step-by-step solutions. The calculator supports multiple units for distance (meters, kilometers, centimeters, millimeters, feet, inches, miles), intensity (watts per square meter, lux, foot-candles), and source strength (watts × square meters, lumens).

The mathematical form of the inverse square law is:

Components of the Inverse Square Law:

• Intensity (I): The physical quantity being measured at a point, such as light intensity, sound intensity, radiation intensity, or field strength. Measured in units like W/m² (watts per square meter), lux (lumens per square meter), or appropriate units for the specific phenomenon.
• Source Strength (k): A constant that represents the strength or power of the source. This depends on the total power, energy, or strength of the source. For point sources, k = Power / (4π) for spherical radiation. Measured in units like W⋅m² (watts × square meters) or lumens.
• Distance (r): The distance from the source to the point where intensity is measured. Measured in meters, centimeters, kilometers, feet, inches, or miles.
• Mathematical Relationship: Intensity decreases with the square of distance. Doubling the distance reduces intensity to 1/4 (one-fourth) of the original value. Tripling the distance reduces intensity to 1/9 (one-ninth).

Rearranging the Formula:

You can rearrange the inverse square law to solve for any variable:

• Intensity: I = k / r²
• Distance: r = √(k / I)
• Source Strength: k = I × r²

Physical Interpretation:

The inverse square law arises from geometric spreading. For a point source radiating uniformly in all directions, the energy spreads over the surface of a sphere with area 4πr². As the distance doubles, the same amount of energy spreads over four times the area, resulting in one-quarter the intensity. This fundamental relationship applies to any phenomenon where energy, force, or field spreads uniformly from a point source.

The inverse square law applies to numerous physical phenomena:

• Light Intensity: Light from a point source (like a light bulb) follows the inverse square law. If you move twice as far from a light source, the light intensity becomes 1/4 as bright. This is crucial in photography, lighting design, and optical measurements.
• Sound Intensity: Sound waves from a point source spread spherically, following the inverse square law. Moving twice as far from a sound source reduces sound intensity to 1/4. This principle is used in acoustics, audio engineering, and noise control.
• Radiation Intensity: Radioactive sources, X-ray machines, and other radiation sources follow the inverse square law. This is critical for radiation safety, medical imaging, and nuclear safety calculations.
• Gravitational Force: Newton's law of universal gravitation follows an inverse square relationship: F = G(m₁m₂) / r². The gravitational force between two objects decreases with the square of distance.
• Electric Force: Coulomb's law for electric forces between charges follows the inverse square law: F = k(q₁q₂) / r². Electric field strength also follows this relationship.
• Magnetic Field: Magnetic field strength from a magnetic dipole follows an inverse cube law at large distances, but many magnetic phenomena involve inverse square relationships.
• Electromagnetic Radiation: The intensity of electromagnetic waves (radio, microwave, infrared, visible, ultraviolet, X-rays, gamma rays) follows the inverse square law for point sources.
• Thermal Radiation: Heat transfer by radiation follows the inverse square law, important in thermal engineering and heat transfer calculations.

The inverse square law has practical applications in many fields:

• Photography and Lighting: Understanding how light intensity changes with distance helps photographers position lights correctly and calculate proper exposure. Moving a light twice as far requires four times the power for the same illumination.
• Sound Engineering: Audio engineers use the inverse square law to position microphones and speakers correctly, calculate sound levels at different distances, and design sound systems for venues.
• Radiation Safety: Medical professionals, nuclear workers, and safety engineers use the inverse square law to calculate safe distances from radiation sources and determine exposure limits.
• Astronomy: Astronomers use the inverse square law to calculate the luminosity of stars based on their apparent brightness and distance. The law explains why distant stars appear dimmer.
• Wireless Communication: Radio and wireless signal strength decreases with distance following the inverse square law (in free space), important for designing communication systems and antenna placement.
• Medical Imaging: X-ray and gamma-ray imaging systems must account for the inverse square law when calculating radiation doses and positioning sources.
• Security Systems: Motion sensors, security lights, and other systems rely on understanding how signal or light intensity changes with distance.
• Industrial Applications: Quality control, material testing, and industrial processes that involve radiation, light, or sound must account for distance effects.

The inverse square law has dramatic effects on intensity as distance changes:

This table demonstrates how quickly intensity decreases with distance. Doubling the distance always results in one-quarter the intensity, making the inverse square law a powerful tool for understanding and predicting how physical quantities change with distance.

The inverse square law is a fundamental principle in physics that explains how intensity, force, and field strength decrease with distance from a source. Our Inverse Square Law Calculator provides a powerful and accurate tool for determining intensity, distance, or source strength, making complex physics calculations simple and accessible for students, engineers, and professionals.

By simplifying inverse square law calculations with comprehensive unit support and detailed step-by-step solutions, this calculator empowers users to solve problems in lighting, acoustics, radiation safety, astronomy, and many other fields. For related calculations, explore our Gravitational Force Calculator for gravitational force calculations or our Wavelength to Energy Calculator for electromagnetic radiation energy conversions.