Malus's law

Malus's law says the intensity of polarized light after a polarizer is proportional to cos² of the angle between the light’s polarization and the polarizer axis. In Honors Physics, it shows how polarization changes measured intensity.

What is Malus's law?

Malus's law is the rule Honors Physics uses to predict how much polarized light gets through a polarizer. If the incoming light is already polarized and the polarizer’s transmission axis makes an angle θ with that polarization direction, the transmitted intensity is

I = I0 cos²θ

where I0 is the intensity before the polarizer. The law tells you that the brightness does not drop in a straight-line way as the angle changes. Instead, it depends on the square of the cosine, so the intensity falls off fast as the polarizer turns away from alignment.

The physics behind it comes from the electric field of light. A polarizer only passes the component of the electric field that lines up with its transmission axis. If the incoming wave is perfectly aligned, almost all of it can pass. If the incoming polarization is 90° from the axis, the passed component is zero, so the transmitted intensity is zero.

A useful way to picture it is to think of the electric field as a vector. The polarizer does not change the light by magic, it removes the part of the field that points the wrong way. Since intensity depends on the square of field amplitude, the projected field component becomes a cos² relationship instead of just cos.

This is why angle matters so much in optics problems. Two polarizers lined up let much more light through than two polarizers turned perpendicular to each other. If unpolarized light hits the first polarizer, that first step creates polarization; after that, Malus's law describes what happens when the light passes a second polarizer at some angle.

You also see this law when you compare predicted brightness to measured brightness in a lab. If the angle is 0°, intensity stays at the maximum. At 60°, only one quarter of the original polarized intensity gets through because cos²60° = 1/4. At 90°, the transmitted intensity drops to zero for ideal polarizers.

Why Malus's law matters in Honors Physics

Malus's law is the calculation tool that turns polarization from a visual idea into a measurable one. In Honors Physics, that means you can predict how bright a beam will be after a polarizer instead of guessing from the picture.

It also connects wave behavior to lab data. When you rotate a polarizer and record intensity, the numbers should follow a cos² pattern. That makes the law useful for checking whether light is polarized, whether a filter is aligned correctly, or how well a polarizer works in an experiment.

This term also shows up in the bigger topic of electromagnetic radiation because light is not just a ray, it is an electromagnetic wave with a direction for its electric field. Once you understand that the polarizer filters the electric field component, the intensity formula makes sense instead of feeling random.

In class problems, Malus's law often appears with two polarizers, angular measurements, and intensity ratios. If you can track the angle, identify the incoming polarization, and apply cos²θ correctly, you can solve those questions cleanly.

How Malus's law connects across the course

Polarization

Polarization is the property that tells you which direction the light’s electric field oscillates in. Malus's law only works once light is already polarized, or after a first polarizer has created a polarized beam. If you do not know the polarization direction, you cannot choose the right angle θ for the formula.

Polarizer

A polarizer is the filter that selects one electric field direction and blocks the rest. Malus's law describes what happens after the light hits that filter, especially when the incoming polarization is at an angle to the polarizer’s transmission axis. In lab setups, rotating the polarizer is what changes the observed intensity.

Intensity

Intensity is the amount of power carried by the light per unit area, so it is what you measure when you say the beam gets dimmer. Malus's law gives the intensity after polarization, not just the electric field direction. The cos² relationship matters because intensity depends on the square of field amplitude.

Is Malus's law on the Honors Physics exam?

A quiz or problem set question usually gives you the initial intensity, the angle between the polarization direction and the polarizer axis, and asks for the transmitted intensity. Your job is to pick out the angle correctly, then use I = I0 cos²θ without mixing up degrees and radians on a calculator.

You may also see a lab question where you interpret a graph of intensity versus angle. A good response explains why the curve peaks at 0°, falls to half intensity near 45°, and reaches zero at 90° for an ideal polarizer. If there are two polarizers, you may need to treat the first one as creating polarized light before applying Malus's law to the second.

If the setup looks like a light filter or LCD-style panel, the key move is to connect angle to brightness. The formula is not just memorized, it tells you how the electric field component changes as the polarizer rotates.

Malus's law vs polarization

Polarization is the property of the light itself, while Malus's law is the rule that predicts how much of that polarized light passes through a polarizer at a given angle. Think of polarization as the state of the wave and Malus's law as the measurement rule.

Key things to remember about Malus's law

• Malus's law gives the transmitted intensity of polarized light after a polarizer: I = I0 cos²θ.

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

• When the axes line up, transmitted intensity is maximum, and when they are perpendicular, an ideal polarizer blocks the light completely.

• The law comes from the electric field component that points along the polarizer axis, and intensity depends on the square of that field.

• In Honors Physics, you use Malus's law to solve angle-and-intensity problems, interpret polarization labs, and analyze beams that pass through multiple polarizers.

Frequently asked questions about Malus's law

What is Malus's law in Honors Physics?

Malus's law says that the intensity of polarized light after a polarizer is I = I0 cos²θ. The angle θ is measured between the light’s polarization direction and the polarizer’s axis. It is the go-to formula for polarization and light-intensity problems.

Why is the formula cos squared and not just cosine?

The polarizer only passes the component of the electric field along its axis, which gives a cosine relationship for field amplitude. Intensity depends on the square of that amplitude, so the result becomes cos²θ. That is why brightness drops faster than the angle might suggest.

What happens when polarized light hits a polarizer at 90 degrees?

For an ideal polarizer, no light gets through when the polarization direction is perpendicular to the polarizer axis. In the formula, cos 90° = 0, so I = 0. Real polarizers are not perfect, but that is the ideal result used in class problems.

How do I use Malus's law in a lab or test problem?

First identify the incoming intensity and the angle between the polarization direction and the polarizer axis. Then plug into I = I0 cos²θ and calculate the transmitted intensity. In a lab, you may compare your measured data to the cos² pattern to see whether the setup matches the theory.