Show that if you have three polarizing filters, with the second at an angle of \(45.0^{\circ}\) to the first and the third at an angle of \(90.0^{\circ}\) to the first, the intensity of light passed by the first will be reduced to \(25.0 \%\) of its value. (This is in contrast to having only the first and third, which reduces the intensity to zero, so that placing the second between them increases the intensity of the transmitted light.)

Understanding Malus's Law is essential to grasp how polarizing filters manipulate the intensity of light. This scientific principle can be stated as the light intensity of polarized light passing through a polarizing filter is directly proportional to the square of the cosine of the angle between the light's initial polarization direction and the axis of the filter.

In a formula, we express this as: \[I = I_0 \times \text{cos}^2(\theta)\] where \(I\) is the transmitted light intensity, \(I_0\) is the initial light intensity, and \(\theta\) is the angle between the initial polarization direction and the polarizer's axis. It's important because it directly shows us how changing the angle of a polarizer will affect the intensity of the light passing through it. It's also essential when dealing with multiple polarizers, as this principle can be applied sequentially for each piece of polarizing material the light encounters.

When we discuss the intensity of light in physics, we are referring to the power carried by light waves per unit area, which is usually perceived as brightness to the human eye. It can be measured in watts per square meter (W/m^2), and it's a critical concept when working with polarizing filters.

As light encounters a polarizing filter, only the component of the light wave oscillating parallel to the axis of the filter is transmitted, thus intensity is affected. This is fundamentally why the intensity of light is reduced when passing through a series of polarizing filters set at an angle to each other, as demonstrated by the exercise solution.

Polarizing filters can also aid in reducing glare from reflective surfaces, as they selectively block the horizontally polarized, often-intense light reflections.

In the context of polarizing filters, the angle between polarizers is a measure of the rotation from the transmission axis of one polarizer to the transmission axis of the next. This angle has a profound impact on the intensity of the transmitted light, as predicted by Malus's Law.

The exercise illustrates this perfectly: at an angle of 45 degrees between the first and second polarizer, and again between the second and third polarizer, the transmitted intensity is reduced to 25%. This contrasts with placing polarizers at 90 degrees to each other, which typically results in no light being transmitted because the orientation of the polarizers is completely at odds.

Polarization of light refers to the direction in which the electric field of the light wave oscillates. Natural light is usually unpolarized, meaning that it vibrates in all directions perpendicular to the direction of propagation. However, when light encounters a polarizing filter, it emerges with vibrations in only one direction - it becomes polarized.

Polarizing filters are widely used in photography to limit light reflections and enhance image contrast. In our everyday lives, polarization is at work in sunglasses, which can reduce glare from horizontal surfaces, like water or roads, improving visual comfort and clarity.