Q.Explain what is meant by polarization and derive Malus' law.

[!TLDR] Polarization restricts a transverse wave's field oscillation to one direction; passing already-polarized light through a second polarizer at angle $\theta$ to its axis transmits only the component $E_0\cos\theta$, so intensity scales as $I=I_1\cos^2\theta$ -- Malus' Law. [!ANSWER] Malus' Law: $I=I_1\cos^2\theta$, where $\theta$ is the angle between the incoming polarization direction and the polarizer's axis.

Since light is a transverse electromagnetic wave, its oscillating electric field, at any instant, can point in ANY direction within the plane perpendicular to the direction of propagation. Ordinary light from a source is UNPOLARIZED: many independent atomic emitters each contribute their own wave train with its own random field direction in that plane, so overall the field direction is randomly distributed over all angles. A POLARIZER is a material (such as a Polaroid sheet) that transmits only the component of the electric field lying along one particular fixed direction, its POLARIZING AXIS, absorbing the perpendicular component; light emerging from a polarizer therefore oscillates along that single direction only -- it is PLANE POLARIZED light. This is POLARIZATION: the restriction of a transverse wave's oscillation direction to a single plane.

To derive Malus' Law, consider light that has ALREADY been polarized (say by a first polarizer), with electric field $E_1=E_{10}\sin(kx-\omega t)$ oscillating along a fixed direction, of intensity $I_1\propto|E_{10}|^2$. This light now passes through a SECOND polarizer, whose polarizing axis is set at angle $\theta$ to the first polarizer's (i.e. to the current direction of $E_1$). Only the COMPONENT of $E_1$ that lies along this second axis can pass through; resolving $E_{10}$ into components parallel and perpendicular to the second axis, the parallel component has magnitude $E_{20}=E_{10}\cos\theta$, and it is this component alone that is transmitted (the perpendicular component, $E_{10}\sin\theta$, is fully absorbed by the second polarizer).

Since intensity is proportional to the square of amplitude, the emergent intensity is $I_2\propto|E_{20}|^2=|E_{10}\cos\theta|^2=|E_{10}|^2\cos^2\theta$. Writing this as a ratio to the incoming intensity $I_1\propto|E_{10}|^2$: $I_2=I_1\cos^2\theta$ -- this is MALUS' LAW.

Two limiting cases follow immediately and check the formula's sense: at $\theta=0°$ (the second polarizer's axis exactly PARALLEL to the incoming polarization), $\cos^2\theta=1$ and the full intensity passes through unchanged, exactly as expected since no component is being blocked at all. At $\theta=90°$ (the axes exactly PERPENDICULAR/crossed), $\cos^2\theta=0$ and no light emerges at all, exactly as expected since the entire field lies along the direction the second polarizer blocks. (Note that Malus' Law, as derived, applies to light that is ALREADY polarized passing through a further polarizer; UNPOLARIZED light passing through the very first polarizer instead always loses exactly half its intensity, by a separate averaging argument over all random incoming field directions, regardless of that first polarizer's own orientation.) [!ANSWER] Malus' Law: $I=I_1\cos^2\theta$, where $\theta$ is the angle between the incoming polarization direction and the polarizer's axis.

Define polarization as restricting a transverse wave's oscillation to one plane via a polarizer's selective transmission, then resolve an already-polarized field into components parallel/perpendicular to a second polarizer's axis and square the parallel (transmitted) component's amplitude to get Malus' Law.

Deriving Malus' Law for UNPOLARIZED light incident directly on a single polarizer (which instead follows the separate, orientation-independent halving rule), rather than correctly starting from ALREADY polarized light incident on a SECOND polarizer, which is what Malus' Law specifically describes.