Formula and Method for Malus's Law
Malus's Law, discovered in 1808 by French engineer Étienne-Louis Malus, describes how the intensity of polarized light changes as it passes through a polarizing filter (an analyzer). When light that is already linearly polarized reaches an analyzer whose transmission axis is rotated by an angle θ relative to the light's polarization direction, the transmitted intensity is I = I₁cos²θ. This calculator also handles the common two-polarizer setup, where unpolarized light first passes through a polarizer — cutting its intensity to I₁ = I₀/2 — and then through a second polarizer, the analyzer, held at angle θ.
How the calculation works
Enter the incident intensity I₀, choose whether that light is already polarized or unpolarized, and set the angle θ between the analyzer's transmission axis and the light's polarization direction (in degrees or radians). If the source is unpolarized, the calculator first halves the intensity to get I₁ = I₀/2, the intensity leaving the first polarizer; if the source is already polarized, I₁ simply equals I₀. It then applies Malus's Law, I = I₁ × cos²θ, to find the intensity leaving the analyzer, and also reports the overall fraction of the original I₀ that survives (I/I₀) and the attenuation in decibels, −10log₁₀(I/I₀).
Common mistakes
- Skipping the halving step: Malus's Law itself (I = I₁cos²θ) only applies to light that is already polarized when it reaches the analyzer — for unpolarized light passing through a first polarizer, apply the I₀/2 factor before applying cos²θ.
- Mixing angle units: cos²θ is very sensitive to whether θ is in degrees or radians; entering 30 as radians (≈1719°) instead of degrees gives a drastically wrong answer.
- Confusing axis angle with angle of incidence: θ is the angle between the light's polarization direction and the analyzer's transmission axis, not the angle at which the light beam strikes the filter.
Real-world applications
- Polarizing sunglasses and camera filters use angled polarizers to cut glare from light reflected off water, glass, and roads.
- LCD screens sandwich liquid crystals between two polarizers; rotating the light's polarization pixel by pixel controls how much light Malus's Law lets through.
- Photography and optical instruments use polarizing filters to dim light precisely without shifting its color, unlike absorptive neutral-density filters.
- Photoelasticity and stress analysis use crossed polarizers to reveal stress patterns in transparent plastics and glass.
Frequently Asked Questions
What is Malus's Law?
Malus's Law states that when polarized light of intensity I₁ passes through an analyzer (a second polarizer), the transmitted intensity is I = I₁cos²θ, where θ is the angle between the light's polarization direction and the analyzer's transmission axis.
Why does unpolarized light lose half its intensity through the first polarizer?
Unpolarized light contains equal amounts of every polarization angle. An ideal polarizer transmits only the component aligned with its axis, which averages out to exactly half the original intensity (I = I₀/2) no matter how the polarizer is oriented.
At what angle is transmission maximum or zero?
Transmission is maximum when the analyzer's axis is parallel to the light's polarization (θ = 0°, cos²0° = 1), and zero when the axes are perpendicular, or crossed (θ = 90°, cos²90° = 0).
Does Malus's Law apply to real, non-ideal polarizers?
Real polarizers absorb and scatter some light even at θ = 0°, so actual transmission is always somewhat lower than the ideal cos²θ prediction. Manufacturers specify this gap as the polarizer's extinction ratio or maximum transmittance, which this calculator does not account for.
Common Tip Percentages
| Service | Typical Tip Range | Notes |
|---|
| Restaurant (sit-down) | 15–20% | 20%+ for exceptional service |
| Food delivery | 10–20% | Consider extra for long distances |
| Taxi / rideshare | 10–20% | More for assistance with luggage |
| Hotel housekeeping | $2–5/night | Leave daily, not just at checkout |
| Barber / hair salon | 15–20% | On pre-discount total |
| Pizza delivery | 10–15% or $3–5 minimum | Higher in bad weather |
General Household Budget Ratios (50/30/20 Rule)
| Category | % of Net Income | Examples |
|---|
| Needs | 50% | Rent, groceries, utilities, insurance |
| Wants | 30% | Dining out, entertainment, hobbies |
| Savings / Debt | 20% | Emergency fund, retirement, extra payments |
Common Measurement Conversions for Everyday Use
| Quantity | US / Imperial | Metric Equivalent |
|---|
| 1 cup | 8 fl oz | 237 mL |
| 1 pint | 16 fl oz | 473 mL |
| 1 quart | 32 fl oz | 946 mL |
| 1 gallon | 128 fl oz | 3.785 L |
| 1 lb | 16 oz | 453.6 g |
| 1 ft | 12 in | 30.48 cm |