Malus Law Calculator

Calculate the intensity of polarized light with our easy-to-use tool based on Malus's Law.

Select the variable you want to calculate, enter the known values, and get an instant result.

Practical Examples

See how the Malus Law calculator works with these common scenarios.

Example 1: Calculate Final Intensity

finalIntensity

A polarized light beam with an intensity of 100 W/m² passes through an analyzer with its axis at a 30° angle to the beam's polarization.

Initial Intensity (I₀): 100 W/m²

Angle (θ): 30°

Example 2: Perpendicular Polarizers

finalIntensity

A polarized light beam (intensity 50 W/m²) passes through an analyzer oriented at 90° to the polarization axis.

Initial Intensity (I₀): 50 W/m²

Angle (θ): 90°

Example 3: Find Initial Intensity

initialIntensity

After passing through an analyzer at a 45° angle, the light intensity is measured to be 25 W/m². What was the initial intensity?

Final Intensity (I): 25 W/m²

Angle (θ): 45°

Example 4: Determine the Angle

angle

If an initial intensity of 200 W/m² is reduced to 50 W/m² after passing through an analyzer, what is the angle between the polarizers?

Initial Intensity (I₀): 200 W/m²

Final Intensity (I): 50 W/m²

Other Titles
Understanding Malus Law: A Comprehensive Guide
Dive deep into the principles of light polarization and the mathematical relationship that governs it.

What is Malus Law?

  • The Core Principle of Polarization
  • The Role of Polarizers and Analyzers
  • The Mathematical Formula
Malus's Law, named after Étienne-Louis Malus, is a fundamental principle in optics that describes how the intensity of a plane-polarized beam of light changes as it passes through a second polarizer, often called an analyzer. It provides a simple yet powerful relationship between the initial intensity of the light and its final intensity after passing through the analyzer, based on the angle between their polarization axes.
The Formula: I = I₀ cos²(θ)
The law is expressed by the formula I = I₀ cos²(θ), where I is the final intensity, I₀ is the initial intensity of the polarized light, and θ is the angle between the light's initial plane of polarization and the axis of the analyzer. When unpolarized light passes through the first polarizer, its intensity is halved (I = I₀/2), and it becomes polarized. Malus's law applies to this newly polarized light as it meets a second polarizer.

Step-by-Step Guide to Using the Malus Law Calculator

  • Selecting the Calculation Mode
  • Entering Input Values Correctly
  • Interpreting the Results
Our calculator is designed to be intuitive. Follow these steps to get accurate results:
1. Choose What to Calculate
Start by using the dropdown menu to select the variable you wish to find: 'Final Intensity (I)', 'Initial Intensity (I₀)', or 'Angle (θ)'. The required input fields will automatically enable or disable based on your choice.
2. Provide the Known Values
Fill in the active input fields. For instance, if you are calculating the final intensity, you will need to provide the initial intensity and the angle. Ensure your inputs are positive numbers and the angle is within the valid range (0-90 degrees).
3. Get Your Result
Click the 'Calculate' button. The result will be displayed clearly in the 'Result' section. You can use the 'Reset' button to clear all fields and start a new calculation.

Real-World Applications of Malus Law

  • Technology in Sunglasses and Camera Lenses
  • LCD Screens and Displays
  • Scientific and Medical Imaging
Malus's Law is not just a theoretical concept; it's the science behind many technologies we use daily.
Polarized Sunglasses
Polarized sunglasses use this principle to reduce glare. They contain a filter with a vertical polarization axis, which blocks horizontally polarized light reflected from surfaces like water or roads, thereby reducing eye strain and improving visibility.
Liquid Crystal Displays (LCDs)
LCD screens, found in TVs, monitors, and smartphones, rely on the ability to control light passing through polarizers. Liquid crystals can change the polarization angle of light passing through them when a voltage is applied. By placing these crystals between two polarizers, the amount of light passing through can be precisely controlled to create images.

Common Misconceptions and Correct Methods

  • Unpolarized vs. Polarized Light
  • Angle Measurement is Crucial
  • Intensity cannot Increase
A common point of confusion is the initial state of the light. Malus's Law (I = I₀ cos²(θ)) applies specifically to light that is already polarized before it reaches the second polarizer (the analyzer). If you start with unpolarized light, it first passes through a polarizer, and its intensity is cut in half (Ipolarized = Iunpolarized / 2). This new intensity then becomes the I₀ for Malus's Law.
The Angle θ
The angle θ is the relative angle between the two polarization axes, not the absolute angle of one polarizer relative to a fixed point. If the first polarizer is at 20° and the analyzer is at 70°, the angle θ used in the formula is 70° - 20° = 50°.

Mathematical Derivation and Examples

  • Deriving the Cosine-Squared Relationship
  • Worked Example: Calculating Intensity
  • Worked Example: Finding the Angle
The electric field of a plane-polarized light wave can be represented as a vector. When this light encounters an analyzer with a transmission axis at an angle θ to the light's polarization direction, only the component of the electric field vector parallel to the analyzer's axis is transmitted. This component has an amplitude of E₀ cos(θ), where E₀ is the original amplitude. Since light intensity is proportional to the square of the electric field's amplitude (I ∝ E²), the transmitted intensity is I = I₀ cos²(θ).

Calculation Examples

  • Given I₀ = 80 W/m² and θ = 60°, the final intensity is I = 80 * cos²(60°) = 80 * (0.5)² = 80 * 0.25 = 20 W/m².
  • Given I₀ = 100 W/m² and I = 25 W/m², the angle is θ = arccos(sqrt(25/100)) = arccos(sqrt(0.25)) = arccos(0.5) = 60°.