Inverse Square Law Calculator

Calculate how intensity, power, and force change with distance using fundamental physics principles.

The inverse square law describes how physical quantities like light intensity, sound intensity, and gravitational force decrease with the square of distance from the source.

Examples

Click on any example to load it into the calculator.

Light Bulb Intensity

Light

Calculate how light intensity decreases as you move away from a 100W light bulb.

Source Power: 100 W

Initial Distance: 1 m

Target Distance: 3 m

Medium Factor: 1.0

Speaker Sound Level

Sound

Determine sound intensity change from a speaker at different distances.

Source Power: 50 W

Initial Distance: 2 m

Target Distance: 10 m

Medium Factor: 0.95

Gravitational Force

Gravity

Calculate gravitational force change between two objects at different distances.

Source Power: 1 W

Initial Distance: 1 m

Target Distance: 5 m

Initial Intensity: 1 W/m²

Medium Factor: 1.0

Radio Wave Propagation

Radio

Analyze radio signal strength at various distances from a transmitter.

Source Power: 1000 W

Initial Distance: 10 m

Target Distance: 50 m

Medium Factor: 0.8

Other Titles
Understanding the Inverse Square Law: A Comprehensive Guide
Explore the fundamental physics principle that governs how light, sound, gravity, and other forces diminish with distance. This guide covers the mathematical foundations, real-world applications, and practical implications of this universal law.

What is the Inverse Square Law?

  • Mathematical Foundation
  • Physical Interpretation
  • Universal Applications
The inverse square law is one of the most fundamental principles in physics, describing how the intensity of a physical quantity decreases with the square of the distance from its source. This law applies to any phenomenon that radiates uniformly in all directions from a point source, including light, sound, gravity, and electromagnetic radiation. The mathematical expression is I = P/(4πr²), where I is intensity, P is power, and r is distance.
Why Does the Square Matter?
The 'square' in inverse square law comes from geometry. As you move away from a point source, the energy spreads out over an ever-increasing spherical surface area. Since the surface area of a sphere is 4πr², the intensity (power per unit area) must decrease proportionally. This means that if you double the distance, the intensity drops to one-fourth of its original value. Triple the distance, and it drops to one-ninth.
The Universal Nature of the Law
This law is universal because it's based on fundamental geometric principles rather than specific physical properties. Whether you're dealing with light from a star, sound from a speaker, gravitational force from a planet, or electromagnetic radiation from an antenna, the same mathematical relationship applies. This universality makes it one of the most powerful tools in physics and engineering.
Limitations and Exceptions
While the inverse square law is remarkably universal, it has important limitations. It assumes a point source radiating uniformly in all directions. Real sources often have directional characteristics, and the law may not apply near the source where the point source approximation breaks down. Additionally, in absorbing or scattering media, additional factors must be considered.

Key Applications of the Inverse Square Law:

  • Light Intensity: How brightness decreases with distance from a light source
  • Sound Intensity: How loudness decreases as you move away from a speaker
  • Gravitational Force: How the force between two masses changes with distance
  • Electromagnetic Fields: How radio, microwave, and other EM wave strength varies with distance
  • Radiation: How nuclear radiation intensity decreases with distance from a source

Step-by-Step Guide to Using the Calculator

  • Understanding Inputs
  • Interpreting Results
  • Practical Applications
Using the inverse square law calculator requires understanding what each input represents and how to interpret the results in practical terms. This step-by-step guide will help you use the calculator effectively for various applications.
1. Determine Your Source Power
The source power is the total energy per unit time being emitted by your source. For a light bulb, this is the electrical power rating. For a speaker, it's the acoustic power output. For gravitational calculations, you might use a normalized value of 1. Be sure to use consistent units throughout your calculation.
2. Set Your Reference Distance
The initial distance is your reference point. This could be 1 meter for standard calculations, or it could be a specific distance where you know the intensity. For example, if you know a light bulb produces 100 W/m² at 1 meter, use 1 meter as your initial distance.
3. Choose Your Target Distance
The target distance is where you want to know the intensity. This could be 5 meters away from a speaker, 10 meters from a light source, or any other distance of interest. The calculator will show you how the intensity changes at this new distance.
4. Consider Medium Effects
The medium factor accounts for absorption, scattering, or other effects that might reduce the intensity beyond what the pure inverse square law predicts. For air or vacuum, use 1.0. For water, fog, or other absorbing media, use values less than 1.0.
5. Interpret Your Results
The calculator provides several useful outputs: the intensity at the target distance, the power ratio (how much the power has changed), the intensity ratio, and the decibel change. The decibel change is particularly useful for sound applications, as human hearing is logarithmic.

Common Distance-Intensity Relationships:

  • Double distance = 1/4 intensity (6 dB decrease)
  • Triple distance = 1/9 intensity (9.5 dB decrease)
  • Ten times distance = 1/100 intensity (20 dB decrease)
  • Half distance = 4x intensity (6 dB increase)
  • One-third distance = 9x intensity (9.5 dB increase)

Real-World Applications and Practical Examples

  • Lighting Design
  • Audio Engineering
  • Astronomy
  • Engineering
The inverse square law has countless practical applications across many fields. Understanding this principle is essential for professionals in lighting, audio, astronomy, telecommunications, and many other disciplines.
Lighting and Photography
In photography and lighting design, the inverse square law explains why moving a light source closer to a subject dramatically increases illumination. A light moved from 4 meters to 2 meters away will provide 4 times more light. This principle is crucial for studio lighting, architectural lighting, and understanding natural lighting conditions.
Audio and Acoustics
Sound engineers use the inverse square law to predict how sound levels will change with distance. This is essential for concert hall design, speaker placement, and understanding why you need to be much closer to hear a whisper than a shout. The law also explains why outdoor concerts need much more powerful sound systems than indoor venues.
Astronomy and Space Science
Astronomers rely heavily on the inverse square law to understand stellar brightness, calculate distances to stars, and design telescopes. The law explains why stars appear much dimmer than they actually are and why we need increasingly sensitive instruments to observe distant objects in the universe.
Telecommunications and Radio
Engineers designing radio, television, and cellular networks must account for the inverse square law when planning coverage areas and signal strength. This law determines how far signals can travel and how much power is needed for reliable communication at different distances.

Professional Applications:

  • Concert Sound: Calculating speaker power needed for different venue sizes
  • Street Lighting: Determining lamp placement and wattage for uniform illumination
  • Satellite Communication: Planning signal strength for Earth-orbiting satellites
  • Medical Imaging: Understanding X-ray and radiation therapy dose distribution
  • Environmental Monitoring: Measuring pollutant dispersion from point sources

Common Misconceptions and Correct Methods

  • Distance vs. Intensity
  • Linear vs. Square Relationships
  • Real-World Complications
Despite its fundamental nature, the inverse square law is often misunderstood or misapplied. Understanding common misconceptions helps ensure accurate calculations and proper interpretation of results.
Misconception: Linear Distance Relationship
A common mistake is assuming that doubling the distance halves the intensity. This would be a linear relationship, but the inverse square law shows that doubling the distance reduces intensity to one-fourth, not one-half. This square relationship is much more dramatic than many people expect.
Misconception: All Sources Follow the Law
While the inverse square law applies to point sources radiating uniformly in all directions, many real sources don't meet these criteria. Directional antennas, focused light sources, and extended sources may follow different relationships. Always consider whether your source approximates a point source.
Misconception: Ignoring Medium Effects
The pure inverse square law assumes propagation through a perfect medium. In reality, air, water, fog, and other media can absorb, scatter, or otherwise affect the propagation of energy. These effects can be significant, especially over long distances or in challenging environments.
Correct Application Methods
To use the inverse square law correctly, always verify that your source approximates a point source, account for directional characteristics if present, consider medium effects, and use consistent units throughout your calculations. When in doubt, measure at multiple distances to verify the relationship.

Verification Methods:

  • Measure intensity at multiple distances and plot on log-log graph
  • Compare calculated vs. measured values to identify deviations
  • Use the law to predict unknown values and verify with experiments
  • Consider environmental factors that might affect propagation
  • Account for source size and directional characteristics

Mathematical Derivation and Advanced Concepts

  • Geometric Foundation
  • Energy Conservation
  • Vector Fields
  • Advanced Applications
The inverse square law can be derived from fundamental principles of geometry and energy conservation. Understanding this derivation provides insight into why the law is so universal and helps identify when it might not apply.
Geometric Derivation
The law follows directly from the geometry of spheres. If a point source emits energy uniformly in all directions, the energy must spread over the surface of an expanding sphere. Since the surface area of a sphere is 4πr², the energy per unit area (intensity) must be P/(4πr²), where P is the total power. This geometric relationship is independent of the type of energy being radiated.
Energy Conservation Principle
The inverse square law is also a consequence of energy conservation. In the absence of absorption or generation, the total energy passing through any spherical surface centered on the source must be constant. As the sphere grows larger, the same total energy is spread over a larger area, resulting in decreased intensity.
Vector Field Interpretation
For forces like gravity and electromagnetism, the inverse square law can be understood in terms of vector fields. The field lines spread out as they move away from the source, and their density (which determines field strength) decreases with the square of distance. This interpretation connects the law to fundamental field theories in physics.
Advanced Applications and Extensions
Beyond simple point sources, the inverse square law can be extended to handle extended sources, directional sources, and complex geometries. These extensions are essential for practical applications in engineering, astronomy, and other fields where real sources don't perfectly match the ideal point source model.

Mathematical Relationships:

  • I = P/(4πr²) - Basic inverse square law formula
  • I₂/I₁ = (r₁/r₂)² - Intensity ratio relationship
  • dB = 10 log₁₀(I₂/I₁) - Decibel change calculation
  • F = GMm/r² - Gravitational force (Newton's law)
  • E = kQ/r² - Electric field strength (Coulomb's law)