Intensity of Light Calculator (eq 35-21)
An SEO-friendly tool to calculate light intensity through polarizers based on Malus’s Law.
Intensity vs. Angle
Intensity at Common Angles
| Angle (θ) | Final Intensity (I) | Transmittance |
|---|
What is the Intensity of Light on a Screen Calculation?
The “calculate intensity of light on screen using eq 35-21 [iₒ]” refers to a fundamental principle in physics known as Malus’s Law. This law describes how the intensity of polarized light changes when it passes through a second polarizing filter (an “analyzer”). When initially unpolarized light passes through the first filter, its intensity is cut in half. The light is now polarized. When this polarized light hits the second filter, the final intensity depends on the square of the cosine of the angle (θ) between the two filters’ transmission axes. This calculator is a practical tool for physicists, engineers, and students working in optics.
Understanding this principle is crucial in many fields. Photographers use polarizing filters to reduce glare and enhance skies, engineers use it in the design of LCD screens, and scientists use it in various optical experiments. One common misunderstanding is assuming the intensity decreases linearly with the angle; in reality, the relationship is a cosine squared curve, meaning the change is non-linear.
The Formula for Light Intensity (Malus’s Law)
The calculation is a two-step process. First, when unpolarized light of an initial intensity (I₀) passes through a polarizer, its intensity is halved.
I₁ = 0.5 * I₀
Next, this polarized light (with intensity I₁) passes through a second polarizer (analyzer) set at an angle θ relative to the first. The final intensity (I) is given by Malus’s Law:
I = I₁ * cos²(θ)
Combining these gives the complete formula used in this calculator:
I = (0.5 * I₀) * cos²(θ)
| Variable | Meaning | Unit (Typical) | Typical Range |
|---|---|---|---|
| I | Final Transmitted Intensity | W/m² (Watts per square meter) | 0 to 0.5 * I₀ |
| I₀ | Initial Unpolarized Intensity | W/m² (Watts per square meter) | Any positive value |
| I₁ | Intensity after the first polarizer | W/m² (Watts per square meter) | Exactly 0.5 * I₀ |
| θ (theta) | Angle between the two polarizers’ axes | Degrees (°) | 0° to 360° |
Practical Examples
Example 1: 45-Degree Angle
Imagine a beam of unpolarized light with an intensity of 1000 W/m² hits a pair of polarizers. The angle between their axes is 45°.
- Input – Initial Intensity (I₀): 1000 W/m²
- Input – Angle (θ): 45°
- Step 1 (After 1st Polarizer): I₁ = 0.5 * 1000 = 500 W/m²
- Step 2 (After 2nd Polarizer): I = 500 * cos²(45°) = 500 * (0.707)² = 500 * 0.5 = 250 W/m²
- Result: The final intensity on the screen is 250 W/m². This is the default calculation shown in the calculator.
Example 2: 90-Degree Angle (Crossed Polarizers)
Now, let’s take the same light source but set the polarizers perpendicular to each other. This is known as having “crossed polarizers.”
- Input – Initial Intensity (I₀): 1000 W/m²
- Input – Angle (θ): 90°
- Step 1 (After 1st Polarizer): I₁ = 0.5 * 1000 = 500 W/m²
- Step 2 (After 2nd Polarizer): I = 500 * cos²(90°) = 500 * (0)² = 0 W/m²
- Result: The final intensity is 0 W/m². Ideally, no light passes through, resulting in darkness. You can explore this and other scenarios with our Malus’s Law Calculator.
How to Use This Light Intensity Calculator
- Enter Initial Intensity: In the first field, input the intensity of your unpolarized light source (I₀). The unit is typically W/m², but the calculation is unit-agnostic.
- Enter Angle: In the second field, provide the angle (θ) in degrees between the transmission axes of the first and second polarizers.
- Review Results: The calculator instantly updates. The primary result is the final intensity (I). You can also see intermediate values like the intensity after the first polarizer (I₁) and the overall transmittance percentage.
- Analyze Chart and Table: Use the dynamic chart and table to see how intensity changes over a range of angles for your given initial intensity.
- Copy Results: Use the “Copy Results” button to easily save or share the inputs and outputs of your calculation. For more on light properties, see our guide on what light polarization is.
Key Factors That Affect Light Intensity
- Initial Intensity (I₀): This is the most direct factor. A brighter initial source will result in a brighter final output, as the final intensity is directly proportional to the initial intensity.
- Angle Between Polarizers (θ): This is the most critical factor. As the angle approaches 90° (perpendicular), the intensity drops to zero. As it approaches 0° or 180° (parallel), the intensity is maximized. Our Brewster’s Angle Calculator explores another angle-dependent light phenomenon.
- Number of Polarizers: Adding more polarizers will further reduce intensity. The formula becomes I_n = I_{n-1} * cos²(θ_n).
- Quality of Polarizers: Real-world polarizers are not perfect. Some light might be absorbed or reflected, and they might not block 100% of light when crossed, leading to results slightly different from the ideal formula.
- Wavelength of Light: While Malus’s law itself doesn’t depend on wavelength, the efficiency of polarizing materials can be wavelength-dependent.
- Medium’s Refractive Index: If the light is traveling through a medium other than a vacuum, its properties can be affected. Check our Snell’s Law Calculator for more on this.
Frequently Asked Questions (FAQ)
1. What is “eq 35-21”?
This is likely a reference to an equation number in a specific physics textbook, such as “Fundamentals of Physics” by Halliday, Resnick, and Walker. In that context, it often refers to Malus’s Law for calculating light intensity through polarizers.
2. Why is the intensity halved after the first polarizer?
Unpolarized light has electric field vectors oscillating in all random directions. A polarizer only allows the component of light parallel to its axis to pass through. On average, this results in exactly half the intensity getting through.
3. What happens if the angle is 0 or 180 degrees?
At 0° or 180°, the polarizers are aligned. Cos(0°) = 1 and Cos(180°) = -1, but both squared equal 1. So, cos²(θ) = 1, and the final intensity is I = I₁, meaning there is no additional loss at the second polarizer.
4. What happens at 90 degrees?
At 90°, the polarizers are “crossed.” Cos(90°) = 0, so the final intensity is zero. In theory, no light passes through. This is a key principle in blocking light and is used in LCDs.
5. Are the units important?
While W/m² is standard, the formula works for any intensity unit (like lux or candela) as long as you are consistent. The output unit will be the same as the input unit.
6. Can the final intensity ever be greater than the initial intensity?
No. A passive polarizing filter always removes light energy, so the intensity will always be less than or equal to the initial intensity.
7. What if I have three polarizers?
If you place a third polarizer between two crossed (90°) polarizers at a 45° angle, light will reappear! The final intensity would be I = (0.5 * I₀) * cos²(45°) * cos²(45°). You can find out more about material properties in our index of refraction database.
8. Is this related to sunglasses?
Yes! Polarized sunglasses are a real-world application of this principle. They are essentially polarizing filters that block horizontally polarized light, which is the primary component of glare reflecting off surfaces like water or roads. Dive deeper in our beginner’s guide to wave optics.