Malus's Law

Malus's Law says the intensity of polarized light after a polarizer is I = I0 cos^2(θ), where θ is the angle between the light's polarization and the polarizer axis. In Principles of Physics II, it shows how polarization controls brightness.

What is Malus's Law?

Malus's Law is the rule that tells you how much polarized light gets through a polarizer in Principles of Physics II. If the incoming light is already polarized and hits a polarizer at an angle θ to the polarizer’s transmission axis, the transmitted intensity is I = I0 cos^2(θ).

The main idea is that a polarizer does not just block light randomly. It only passes the component of the electric field that lines up with its axis. If the light is aligned with the axis, θ = 0 and all of it gets through. If the light is perpendicular to the axis, θ = 90°, the cosine is zero, so no light gets through.

This is why the law is written with cosine squared. Light intensity depends on the square of the electric field amplitude, so when the field component is reduced by a factor of cos(θ), the intensity drops by cos^2(θ). That is the step that makes the law feel different from a simple geometry problem.

One common setup is unpolarized light entering a polarizer first. In that case, the first polarizer does not follow Malus's Law directly in the same way, because unpolarized light has no single polarization direction. The first polarizer usually cuts the intensity in half, then any later polarizer uses Malus's Law relative to the now polarized beam.

You will also see the law in two-polarizer problems. If the axes are at 90° to each other, the second polarizer blocks the polarized light from the first one completely. If the axes are at 45°, the second polarizer transmits some light, but not half, because the intensity falls with cos^2(45°) = 1/2.

This makes Malus's Law a direct link between wave behavior and measurable brightness. It turns polarization from an abstract direction in a wave into a number you can calculate on a lab sheet or exam problem.

Why Malus's Law matters in Principles of Physics II

Malus's Law shows up any time the course turns polarization into a calculation instead of a picture. In Principles of Physics II, that usually means solving optics problems where you have to predict how bright a beam will be after one or more polarizers, or explain why a filter changes what you see.

It also gives you a clean way to connect the direction of the electric field to the measured intensity. That connection matters because polarization is not about the light disappearing, it is about selecting one component of the wave and rejecting the rest. Once you can track that component, many optics problems become straightforward.

The law is also a bridge to lab work. If you measure intensity while rotating a polarizer, the numbers should follow a cos^2 pattern. That pattern is how you check whether light is polarized, whether a polarizer is oriented correctly, or whether your setup is behaving the way the theory predicts.

It also shows up in real optical devices. Sunglasses that reduce glare, camera filters, LCD screens, and 3D movie glasses all depend on controlling polarization. Malus's Law is the math underneath those effects, so it helps explain why rotating a filter can make a scene dim, bright, or nearly dark.

How Malus's Law connects across the course

Polarization

Polarization tells you the direction of the light wave's electric field. Malus's Law only makes sense once the light has a definite polarization direction, because the angle θ is measured between that direction and the polarizer's axis. If you are asked why a beam changes brightness when you rotate a filter, polarization is the first idea to identify.

Polarizer

A polarizer is the device that selects one direction of the electric field and blocks the others. Malus's Law describes what happens after polarized light enters that device. In multi-polarizer problems, the orientation of each polarizer gives you the angle you plug into the cosine squared rule.

Intensity

Intensity is the measurable brightness of the transmitted light, and Malus's Law gives a direct formula for how it changes. The intensity falls with cos^2(θ), not just cos(θ), because intensity depends on the square of the field amplitude. That distinction is a common point where problems get tricky.

circular polarization

Circular polarization is different from the linear polarization used in the basic form of Malus's Law. A linearly polarized beam has one fixed oscillation direction, while circular polarization rotates in time. When a problem introduces circular polarization, you usually need a different description before you can decide how a polarizer will affect the beam.

Is Malus's Law on the Principles of Physics II exam?

A quiz or problem set question usually gives you the incoming intensity, the polarization direction, and the polarizer angle, then asks for the transmitted intensity. Your job is to identify whether the light is already polarized or unpolarized, choose the right starting intensity, and apply I = I0 cos^2(θ) with the correct angle.

You may also see a lab-style question asking you to predict or interpret a graph of intensity versus polarizer angle. In that case, look for the cos^2 shape, not a straight line. If two polarizers are involved, track the beam after each one in order, because the output of the first polarizer becomes the input for the next step. A common mistake is using the angle between the first and last axis instead of the angle between each pair of successive elements.

Malus's Law vs Polarization

Polarization is the property of the light wave, while Malus's Law is the rule that tells you how a polarizer changes the intensity of polarized light. If you mix them up, you may describe the wave direction correctly but miss the actual intensity calculation.

Key things to remember about Malus's Law

• Malus's Law is the cosine-squared rule for the intensity of polarized light after a polarizer: I = I0 cos^2(θ).

• The angle θ is measured between the light's polarization direction and the polarizer's transmission axis.

• If the light is already aligned with the axis, the transmitted intensity is maximum, and if it is perpendicular, the intensity drops to zero.

• For unpolarized light, the first polarizer usually cuts the intensity in half before Malus's Law applies to later polarizers.

• The law is how you turn polarization into an intensity calculation in optics problems, labs, and device explanations.

Frequently asked questions about Malus's Law

What is Malus's Law in Principles of Physics II?

Malus's Law says that the intensity of polarized light after passing through a polarizer is I = I0 cos^2(θ). The angle θ is between the light's polarization direction and the polarizer axis. In Physics II, it is the standard rule for calculating how polarization changes brightness.

Does Malus's Law apply to unpolarized light?

Not in the same direct way. Unpolarized light has no single polarization direction, so the usual Malus's Law setup does not fit until the light passes through a first polarizer and becomes polarized. After that, the law can be applied to the next polarizer.

Why is the intensity proportional to cos squared instead of just cos?

The polarizer selects the component of the electric field along its axis, which scales like cos(θ). Intensity depends on the square of the field amplitude, so the result is cos^2(θ). That is why the brightness drops faster than the field component alone.

What happens if two polarizers are 90 degrees apart?

If the first polarizer produces light polarized in one direction and the second is turned 90 degrees relative to it, the second blocks that polarization completely. The transmitted intensity is zero in the ideal case. This is a classic way to check whether you are tracking the polarizer axes correctly.