Malus's Law

Malus's Law says the intensity of polarized light after a polarizing filter is I = I0 cos^2 θ. In College Physics I, it describes how the output depends on the angle between the light's polarization direction and the filter axis.

What is Malus's Law?

Malus's Law is the rule you use in College Physics I when polarized light passes through a polarizing filter and you want the transmitted intensity. The equation is I = I0 cos^2 θ, where I0 is the incoming intensity, θ is the angle between the light's polarization direction and the filter axis, and I is what gets through.

The idea behind it is simple: a polarizing filter only passes the electric field component that lines up with its transmission axis. If the incoming light is already polarized, the filter acts like a directional gate. When the polarization direction matches the filter axis, the light goes through most strongly. When the directions are perpendicular, the filter blocks the field component entirely and the transmitted intensity drops to zero.

The cosine-squared pattern comes from how light intensity depends on the square of the electric field amplitude. The filter does not just subtract some fixed amount of light. It selects one component of the wave, and that component scales with cos θ. Since intensity is proportional to field amplitude squared, the intensity becomes proportional to cos^2 θ.

A quick way to read the law is to think in steps. First, resolve the incoming polarized light into the part along the filter axis. Then square that amplitude relationship to get intensity. That is why the law gives 100 percent transmission at 0°, 50 percent transmission at 45° only if you start from a polarized beam and compare relative intensity, and 0 percent at 90°.

This is not the same thing as ordinary dimming. The angle matters because the light is polarized, which means its electric field has a specific direction. If the light were unpolarized, you would usually need an extra step before Malus's Law applies, such as passing through a first polarizer to create a known polarization direction.

Why Malus's Law matters in College Physics I – Introduction

Malus's Law is the cleanest way to connect polarization geometry to intensity calculations in optics. In College Physics I, it shows up whenever you trace what happens to light after it interacts with a polarizing filter, especially in problems about sunglasses, LCD screens, or a lab setup with stacked polarizers.

It also gives you a practical reading strategy for diagrams. If you see an angle between a polarization direction and a filter axis, you know to look for cos^2 θ rather than a linear cosine relationship. That helps you avoid one of the most common mistakes in optics problems, which is treating the angle factor like a simple projection without remembering that intensity depends on field amplitude squared.

The law also connects to the bigger idea that light can carry direction information in its electric field. That makes Malus's Law a bridge between wave behavior and measurable brightness. In lab work, you can rotate a polarizer and watch intensity change smoothly instead of staying constant, which is a strong visual check that the light is polarized.

If your class covers multiple polarizers, Malus's Law becomes a step in a chain. You may first use one polarizer to set the polarization direction, then apply the law again for a second polarizer at some angle. That turns the concept from a definition into a problem-solving tool.

How Malus's Law connects across the course

Polarization

Malus's Law only makes sense when light is polarized, because the equation depends on a definite direction for the electric field. If the light is unpolarized, you usually need another step before the law applies. So this term is the bridge between the abstract idea of polarization and the measurable change in brightness.

Polarizing Filter

A polarizing filter is the object that makes Malus's Law visible in a problem or lab. The filter has a transmission axis, and only the component of the electric field along that axis gets through. When you rotate the filter, you change θ, which changes the transmitted intensity.

axis of a polarizing filter

The axis of a polarizing filter is the direction you compare against the incoming polarization direction. That angle is the whole reason the cosine-squared factor appears. If the axis lines up with the polarization direction, transmission is strongest, and if it is perpendicular, transmission drops to zero.

Intensity

Malus's Law is an intensity law, not just a direction rule. It tells you how the brightness changes after the field passes through a polarizer. Because intensity depends on the square of the field amplitude, the angle dependence becomes cos^2 θ instead of a simple cos θ.

Is Malus's Law on the College Physics I – Introduction exam?

A quiz question on Malus's Law usually asks you to find the transmitted intensity after a polarizer is rotated to a given angle. You may be given I0 and θ, then asked to calculate I = I0 cos^2 θ or compare two orientations. The move is to identify the polarization direction, measure the angle to the filter axis, and use the square of the cosine, not just the cosine.

In a lab report or problem set, you might also explain why the intensity changes as the polarizer turns and describe any points where the output reaches a maximum or goes to zero. If the setup uses two polarizers, you may apply the law twice, once for each stage. That is a common way to connect the math to the physical behavior of the light.

Malus's Law vs Polarization

Polarization is the state of the light itself, meaning the direction of its electric field oscillation. Malus's Law is the rule for what happens to polarized light after it passes through a polarizer at some angle. One describes the property of the wave, the other describes the intensity change caused by a filter.

Key things to remember about Malus's Law

• Malus's Law gives the transmitted intensity of polarized light through a polarizing filter as I = I0 cos^2 θ.

• The angle θ is measured between the light's polarization direction and the axis of the polarizing filter.

• When the two directions line up, transmission is greatest; when they are perpendicular, transmission drops to zero.

• The cos^2 form comes from intensity being proportional to the square of the electric field amplitude.

• In College Physics I, you use this law to solve optics problems and interpret polarizer lab results.

Frequently asked questions about Malus's Law

What is Malus's Law in College Physics I?

Malus's Law is the equation I = I0 cos^2 θ for polarized light passing through a polarizing filter. It tells you how much intensity remains after the filter based on the angle between the light's polarization direction and the filter axis. In optics problems, it is the main calculation for rotated polarizers.

Why is it cos squared in Malus's Law?

The polarizer passes only the component of the electric field along its axis, and that component scales with cos θ. Intensity depends on the square of the field amplitude, so the result becomes cos^2 θ. That is why the law uses a squared cosine instead of a simple cosine.

What happens when the polarizer is at 90 degrees?

At 90°, the filter axis is perpendicular to the polarization direction, so the transmitted intensity is zero for ideal polarized light. The filter blocks the electric field component along that axis completely. In real lab settings, small imperfections can leave a tiny residual signal.

How is Malus's Law different from polarization?

Polarization is the direction of the light's electric field oscillation. Malus's Law is what you use after that, when polarized light passes through a polarizer at an angle. So polarization describes the wave, while Malus's Law predicts how a filter changes its brightness.