Question

Two polarizers are out of alignment by \(30.0^{\circ} .\) If light of intensity \(1.00 \mathrm{~W} / \mathrm{m}^{2}\) and initially polarized halfway between the polarizing angles of the two filters passes through the two filters, what is the intensity of the transmitted light?

Short answer

Answer: The intensity of the transmitted light after passing through the two polarizers is approximately 0.586 W/m^2.

Step-by-step solution

Understand Malus's Law

According to Malus's Law, the intensity of transmitted light through a polarizer can be found using the following formula: \[I_t = I_0 \cos^2(\theta)\] where \(I_t\) is the transmitted intensity, \(I_0\) is the initial intensity, and \(\theta\) is the angle between the polarizing axes of the two polarizers.

Adjust for Initial Polarization

Since the light is initially polarized halfway between the polarizing angles of the two filters, we need to divide the angle difference by 2 before passing through the first filter. The angle to be considered for Malus's Law in this case is \(\theta / 2\): \[\theta^{'} = \frac{\theta}{2}\]

Calculate the Intensity After the First Filter

Now, we will apply Malus's Law to find the intensity of the light after passing through the first filter: \[I_1 = I_0 \cos^2(\theta^{'})\] where \(I_1\) is the intensity after the first filter and \(\theta^{'} = \frac{30.0}{2} = 15.0^{\circ}\). Plugging in the given values, we get: \[I_1 = (1.00 \: \mathrm{W/m^2}) \cos^2(15.0^{\circ})\]

Calculate the Intensity After the Second Filter

Since the polarizing axes of the two polarizers differ by \(30.0^{\circ}\), after passing through the first filter, the angle between the polarized light and the second filter is \(15.0^{\circ}\). Again, we will apply Malus's Law for the transmitted light through the second filter: \[I_t = I_1 \cos^2(15.0^{\circ})\] Plug in the value of \(I_1\) to get the final transmitted intensity: \[I_t = (1.00 \: \mathrm{W/m^2}) \cos^2(15.0^{\circ}) \cos^2(15.0^{\circ})\]

Solve for Transmitted Light Intensity

Finally, calculate the transmitted light intensity: \[I_t = (1.00 \: \mathrm{W/m^2}) \cos^2(15.0^{\circ}) \cos^2(15.0^{\circ}) \approx 0.586 \: \mathrm{W/m^2}\] So, the intensity of the transmitted light after passing through the two filters is approximately \(0.586 \: \mathrm{W/m^2}\).

Key concepts

Polarized Light Intensity

In the world of optics, understanding how light behaves when it encounters different materials is crucial. One such behavior is polarization, a process where light waves oscillate in particular directions. When light becomes polarized, its intensity can be measured in terms of the amount of light that passes through a polarizing filter.

This is where Malus's Law comes into play, which is a fundamental principle that allows us to calculate the polarized light intensity after it passes through a polarizing filter. The law states that the transmitted light intensity, denoted as \(I_t\), is equal to the initial intensity \(I_0\) multiplied by the square of the cosine of the angle between the light's initial polarization direction and the axis of the filter—expressed mathematically as \(I_t = I_0 \times \text{cos}^2(\theta)\).

In the case of the exercise given, the original intensity of light is 1.00 W/m². If we assume the light is initially polarized at an angle that is halfway between the orientations of two polarizing filters and one filter is out of alignment by \(30.0^{\text{\textdegree}}\), this angle must be accounted for to determine the final intensity. By applying Malus's Law twice, once for each filter, we can find the remaining intensity of the light after it has been doubly filtered.

Angle of Polarization

The angle of polarization, often denoted by theta \(\theta\), is a key factor in dictating the behavior of polarized light as it interacts with polarizing filters. It is the angle between the direction of the electric field of the incoming light wave and the direction of the polarizing axis of the filter.

In practice, when light encounters a polarizer, only the component of the light wave that aligns with the polarizing axis is transmitted, while the remainder is absorbed. As the angle of polarization between the light and the polarizer's axis changes, the intensity of the transmitted light changes accordingly, following a cosine squared relationship stated in Malus's Law.

Therefore, when light that is initially polarized passes through a second filter out of alignment with the first, it is necessary to consider the angle of polarization with respect to the second filter. In our textbook example, the light is initially polarized at an angle that requires division by 2, reflecting the fact that it is midway between the two filters' polarizing angles, resulting in an effective angle of polarization of \(15.0^{\text{\textdegree}}\). By doing so, we ensure that the calculation of the transmitted intensity accurately reflects the physical scenario.

Physics of Light Transmission

The physics behind light transmission through polarizers is rooted in the wave nature of light. Polarizers are optical filters that allow light waves vibrating in a certain orientation to pass through, while blocking light waves vibrating in other orientations.

When unpolarized light, which vibrates in all planes perpendicular to its direction of travel, encounters a polarizer, it emerges with vibrations in only one plane—thus becoming polarized. If another polarizer with a different orientation is placed in the path of this polarized light, the intensity of the light that can pass through both filters depends on the relative angle between their axes. This is explained by the cosine squared law given by Malus's Law, showing how the transmitted intensity diminishes as the angle of polarization moves away from the filter's polarizing axis.

This scientific principle is used in a variety of applications, from glare reduction in sunglasses to scientific instruments that measure the concentration of certain molecules. In the case study provided, understanding the physics of light transmission allows for an accurate computation of the remaining light intensity after it passes through two misaligned polarizers.