Problem 32
Question
Two polaroids are kept crossed to each other. Now one of them is rotated through an angle of \(45^{\circ}\). The percentage of incident light now transmitted through the system is (a) 159 (b) 2596 (c) \(50 \%\) (d) 6096
Step-by-Step Solution
Verified Answer
The transmitted percentage of incident light is 25%, but the options provided don't match this result.
1Step 1: Understand the Concept of Polaroids
When two polaroids are crossed at a \(90^\circ\) angle, no light passes through. Rotating one affects the transmitted intensity based on the angle of rotation.
2Step 2: Apply Malus's Law
Malus's Law states that the intensity of light passing through two polaroids is given by I = I_0 \cos^2(\theta), where I_0 is the initial intensity and \(\theta\) is the angle between the polarizer axes. Here \(\theta = 45^\circ\).
3Step 3: Compute the Transmitted Intensity
The intensity of light after passing through the first polaroid becomes I_0/2. After rotation by \(45^\circ\), the transmitted intensity through the second polaroid is: \[I = \frac{I_0}{2} \cos^2(45^\circ) = \frac{I_0}{2} \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{I_0}{4}\]
4Step 4: Determine Percentage of Incident Light Transmitted
The transmitted intensity \(I\) is \(\frac{I_0}{4}\), so the percentage of incident light is \frac{1}{4} \times 100 = 25\%.
Key Concepts
PolaroidsIntensity of LightAngle of Rotation
Polaroids
Polaroids are optical devices that have the ability to filter light waves, allowing only those vibrating in a specific direction to pass through. They are particularly useful in various applications such as reducing glare in sunglasses and in experimental setups involving light and optics.
When two polaroids are placed with their polarization axes at right angles (also known as crossed polaroids), they block out all light. The first polaroid, called a polarizer, filters the light, so it only allows light vibrations in one direction. The second polaroid, known as the analyzer, can then either permit or block this light depending on its angle of alignment to the first polaroid.
Understanding how polaroids interact can also help in determining the extent of light intensity that passes through two such filters. This knowledge is critical in scientific fields that depend on precise light intensity control and measurement, like photography, television, and LCD displays.
When two polaroids are placed with their polarization axes at right angles (also known as crossed polaroids), they block out all light. The first polaroid, called a polarizer, filters the light, so it only allows light vibrations in one direction. The second polaroid, known as the analyzer, can then either permit or block this light depending on its angle of alignment to the first polaroid.
Understanding how polaroids interact can also help in determining the extent of light intensity that passes through two such filters. This knowledge is critical in scientific fields that depend on precise light intensity control and measurement, like photography, television, and LCD displays.
Intensity of Light
The intensity of light is a measure of how much energy a light wave carries per unit area, and it determines how bright the light appears to be. When light passes through a polaroid, its intensity is significantly affected.
In our specific scenario, the intensity of light after passing through the first polaroid becomes half of its original value. This is because only one component of the light wave is allowed to pass through while the other is absorbed. The formula to determine the transmitted light intensity through a polaroid system is given by Malus's Law:
\[ I = I_0 \cos^2(\theta) \]
Where \( I_0 \) is the initial intensity before entering the polaroids, and \( \theta \) is the angle between the axes of the two polaroids. This relationship emphasizes that as the angle \( \theta \) changes, the transmitted intensity adjusts accordingly, being zero at 90 degrees and at a maximum when \( \theta \) is zero.
In our specific scenario, the intensity of light after passing through the first polaroid becomes half of its original value. This is because only one component of the light wave is allowed to pass through while the other is absorbed. The formula to determine the transmitted light intensity through a polaroid system is given by Malus's Law:
\[ I = I_0 \cos^2(\theta) \]
Where \( I_0 \) is the initial intensity before entering the polaroids, and \( \theta \) is the angle between the axes of the two polaroids. This relationship emphasizes that as the angle \( \theta \) changes, the transmitted intensity adjusts accordingly, being zero at 90 degrees and at a maximum when \( \theta \) is zero.
Angle of Rotation
The angle of rotation is a key variable when examining light transmission through polaroids. It refers to the degree at which one polaroid is rotated relative to another. This rotation angle impacts the intensity of the light that is ultimately transmitted through both polaroids.
According to Malus's Law, the transmitted light's intensity is directly related to the square of the cosine of the rotation angle between the polaroids' axes. For example, if the polaroids are initially crossed (i.e., at a 90-degree angle), rotating one polaroid by 45 degrees results in an intensity of:
\[ I = \frac{I_0}{2} \cos^2(45^\circ) = \frac{I_0}{4} \]
This equation shows that at a 45-degree rotation, the transmitted light's intensity becomes one-fourth of the initial light intensity across the polaroid system. Thus, the correct percentage of the transmitted light regarding our exercise is 25%, illustrating how the angle of rotation can be pivotal in light control applications.
According to Malus's Law, the transmitted light's intensity is directly related to the square of the cosine of the rotation angle between the polaroids' axes. For example, if the polaroids are initially crossed (i.e., at a 90-degree angle), rotating one polaroid by 45 degrees results in an intensity of:
\[ I = \frac{I_0}{2} \cos^2(45^\circ) = \frac{I_0}{4} \]
This equation shows that at a 45-degree rotation, the transmitted light's intensity becomes one-fourth of the initial light intensity across the polaroid system. Thus, the correct percentage of the transmitted light regarding our exercise is 25%, illustrating how the angle of rotation can be pivotal in light control applications.
Other exercises in this chapter
Problem 31
Two waves of same frequency and same amplitude from two monochromatic source are allowed to superpose at a certain point. If in one case the phase difference is
View solution Problem 32
We shift Young's double slit experiment from air to water. Assuming that water is still and clear, it can be perdicted that the fringe pattern will (a) remain u
View solution Problem 33
In Young's double slit experiment, the separation between slit is halved and the distance between the slits and screen is doubled. The fringe width is (a) uncha
View solution Problem 33
In Young's double slit experiment, the two slits act as coherent sources of equal amplitude \(A\) and wavelength \(\lambda\). In another experiment with the sam
View solution