The angle between the axes of two polarizing filters is \(45.0^{\circ} .\) By how much does the second filter reduce the intensity of the light coming through the first?

Polarizing filters are incredible tools used in photography and optics to select which light rays are absorbed and which are transmitted. Imagine a picket fence: only those things aligned with the slats can slip through. Similarly, a polarizing filter has a preferred direction, known as the axis of polarization, which permits only light waves vibrating parallel to it to pass.

These filters are often employed to eliminate glare from reflective surfaces, enhance contrast in photography, and protect delicate instruments from intense light. They can be rotated to adjust the level of polarization, thus changing the amount of light that passes through. Understanding this concept is key when we delve deeper into Malus' Law and light intensity variation due to these filters.

Light intensity, also known as illuminance or light level, is a measure of how much luminous flux is spread over a given area. Think of it as the brightness of light that reaches our eyes or a surface. In physics, intensity is a precise concept that plays a significant role in various phenomena, including the passage of light through polarizing filters.

It is also vital to note that when light passes through a polarizing filter, the filter's orientation affects the intensity of the transmitted light. If you squint your eyes on a bright day, you're reducing the effective intensity of light hitting your retina – a polarizing filter does something similar by blocking certain light waves.

The angle of polarization isn't just about the physical rotation of a filter; it refers to the angle between the direction of the electric field of the incoming light and the axis of the filter. Malus' Law, a fundamental rule in wave optics, precisely connects the intensity of the transmitted light through a polarizer to this angle. It is represented mathematically by the simple but powerful relationship: \[I_t = I_0 \times \text{cos}^2(\theta)\]Where \(I_t\) is the transmitted intensity, \(I_0\) the original intensity, and \(\theta\) is the angle of polarization.

The closer this angle is to 90 degrees, the less light makes it through, and at 0 degrees, the light is transmitted with maximum intensity. This phenomenon is beautifully observable in everyday life through the use of polarized sunglasses, which reduce glare by filtering out light polarized in certain directions.