The light incident on polarizing sheet \(P_{1}\) is linearly polarized at an angle of \(30.0^{\circ}\) with respect to the transmission axis of \(\mathrm{P}_{1} .\) Sheet \(\mathrm{P}_{2}\) is placed so that its axis is parallel to the polarization axis of the incident light, that is, also at \(30.0^{\circ}\) with respect to \(\mathrm{P}_{1}\). (a) What fraction of the incident light passes through \(P_{1} ?\) (b) What fraction of the incident light is passed by the combination? (c) By rotating \(\mathrm{P}_{2},\) a maximum in transmitted intensity is obtained. What is the ratio of this maximum intensity to the intensity of transmitted light when \(P_{2}\) is at \(30.0^{\circ}\) with respect to \(\mathrm{P}_{1} ?\)

Polarization is a property of light that describes the orientation of its wave vibrations. Natural light sources, such as the sun, emit light waves that vibrate in all possible directions perpendicular to the direction of propagation. When light is polarized, its waves vibrate in a single plane.

There are several methods to produce polarized light, one of the most common being the use of a polarizing filter or sheet. These materials allow only the light waves vibrating in a specific direction to pass through and absorb or block the other vibrations. The line along which a polarizing filter allows light to pass is known as its transmission axis.

Understanding polarization is essential for a range of applications, from reducing glare in photography to enhancing visibility for pilots and drivers with polarized glasses. It is also a fundamental concept in studying the behavior of electromagnetic waves in physics and engineering.

The intensity of transmitted light refers to the power of light passing through a medium or substance per unit area, generally measured in watts per square meter. When light encounters a polarizing material, its intensity is affected.

Using Malus' Law, we can predict how much of the incident light's intensity will be transmitted through a polarizer. It states that if you have linearly polarized light incident on a polarizer at an angle θ, the transmitted intensity (\(I_{t}\)) will be less than or equal to the incident intensity (\(I_{i}\)), depending on the cosine square of the angle (\(cos^{2}(θ)\)).

In practical terms, if the polarization axis of the light is aligned with the transmission axis of the polarizer, maximum light passes through. If misaligned by 90 degrees, then no light passes through, and at angles in between, the transmitted intensity will be proportionally reduced. These principles are used in many optical applications, including liquid crystal displays (LCDs) and analytical instruments like polarimeters.

Linearly polarized light is characterized by light waves that oscillate in parallel planes along a single direction, as opposed to unpolarized light where the vibrations occur in multiple directions. It is produced when unpolarized light passes through a polarizing filter, which aligns the vibrations of the light waves into one direction.

For example, if a beam of light passes through a polarizer, such as sheet \(P_{1}\), only the component of the light that vibrates in the direction of the polarizer's transmission axis will be transmitted. If this linearly polarized light then encounters another polarizer, sheet \(P_{2}\), with its transmission axis oriented at an angle θ to that of the light's polarization direction, the intensity of light that passes through \(P_{2}\) will again be determined by Malus' Law, further taking into account the alignment between the light and the second polarizer.

Understanding how linearly polarized light interacts with materials and other polarizing filters is crucial for technology such as photoelastic stress analysis and in the study of light-matter interactions.