Physics · Wave Optics · NEET
Theta is the angle between the direction of vibration of the incoming polarised light (its electric field direction) and the pass-axis of the polaroid. It is NOT the angle of incidence and NOT the angle with the surface. If the incoming light's polarisation is along the pass-axis, theta = 0 and cos^2(0) = 1, so all of it passes. If the incoming polarisation is at right angles to the pass-axis, theta = 90 degrees and cos^2(90) = 0, so none passes.
Malus's Law only applies to light that is already polarised. Unpolarised light has its electric vector pointing in all directions equally. When it hits a polaroid, you average cos^2(theta) over all angles, and the average of cos^2 over a full turn is 1/2. So the very first polaroid always cuts unpolarised light to exactly half, whatever way you turn it. Only after this first polaroid is the light polarised, and only then do you use I0 cos^2(theta) for the next polaroids.
I0 is the intensity of the polarised light that actually enters the polaroid you are analysing. In many NEET problems unpolarised light of intensity I hits polaroid P1 first, giving I/2 of polarised light. Then for P2, that I/2 is your I0. Read the question carefully: some questions define I0 as the original unpolarised beam, others (like the 2025 NEET question) define I0 as the intensity already after the first polaroid. The physics is the same; only the label changes.
The polaroid only lets through the component of the electric field along its pass-axis. That component is E0 cos(theta). But intensity is proportional to the square of the amplitude, so I is proportional to (E0 cos theta)^2 = E0^2 cos^2(theta) = I0 cos^2(theta). The single cos comes from resolving the field; the square comes from intensity being proportional to amplitude squared.
You get I0/2 and it stays I0/2 no matter how you rotate that single polaroid. That is because unpolarised light has no preferred direction, so turning the pass-axis makes no difference to the output. You only see the cos^2(theta) variation when a SECOND polaroid is added and you rotate one relative to the other.
Two Polaroids P1 and P2 are placed with their axes perpendicular to each other. Unpolarised light I0 is incident on P1. A third polaroid P3 is kept in between P1 and P2 such that its axis makes an angle 45 degrees with that of P1. The intensity of transmitted light through P2 is
A polaroid sheet is placed between two crossed polaroids, with its polarization axis at 22.5 degrees from the polarization axis of one of the polaroids. The intensity of the transmitted light is (I0 is the intensity of polarised light after passing through the first polaroid)
Try the real previous-year questions from this chapter — each with the answer and a full solution.
Malus's Law states that when plane-polarised light of intensity I0 falls on a polaroid, the transmitted intensity is I = I0 cos^2(theta), where theta is the angle between the polarisation direction of the light and the pass-axis of the polaroid.
It holds only for light that is already plane-polarised. For unpolarised light hitting the first polaroid, the output is always I0/2 whatever the orientation.
When two polaroids are crossed, theta = 90 degrees, so I = I0 cos^2(90) = 0. No light passes through crossed polaroids.
When I = I0/2, cos^2(theta) = 1/2, so cos(theta) = 1/sqrt2, giving theta = 45 degrees.
Unpolarised light contains all vibration directions equally. Averaging cos^2(theta) over all angles gives 1/2, so exactly half the intensity is transmitted no matter how the first polaroid is turned.
