Question

A plane polarized light is incident normally on the tourmaline plate. its \(\mathrm{E}^{\rightarrow}\) vectors make an angle of \(45^{\circ}\) with the optical axis of the plate. find the percentage difference between initial and final maximum values of \(\mathrm{E}^{\rightarrow}\) vectors. (A) \(19 \%\) (B) \(92 \%\) (C) \(50 \%\) (D) \(29 \%\)

Short answer

The percentage difference between the initial and final maximum values of the electric field vector is closest to (C) $50\%$.

Step-by-step solution

Recall Malus's law

Malus's law states that the intensity of the transmitted light through a polarizer depends on the angle between the electric field vector and the optical axis of the polarizer. Mathematically, it can be expressed as: \[I = I_0 \cdot \cos^2(\theta)\] where: - \(I\) = intensity of the transmitted light, - \(I_0\) = intensity of the incident light, and - \(\theta\) = angle between the electric field vector and the optical axis of the polarizer

Determine the initial and final maximum values of electric field vectors

Since we are given that the initial angle between the electric field vector and the optical axis of the plate is 45 degrees, we can use Malus's law to calculate the initial value of the electric field vector. The electric field vector is proportional to the square root of the intensity: \(E = \sqrt{I}\) The initial and final maximum values of the electric field vectors can be found by substituting the given angle into Malus's law: \[E_i = \sqrt{I_0 \cdot \cos^2(45^{\circ})}\] \[E_f = \sqrt{I_0}\]

Calculate the percentage difference between the initial and final electric field vectors

To calculate the percentage difference between the initial and final values of the electric field vectors, we can use the following formula: \[\frac{E_f - E_i}{E_i} \times 100\%\] Substitute the values of \(E_i\) and \(E_f\) from the previous step: \[\frac{\sqrt{I_0} - \sqrt{I_0 \cdot \cos^2(45^{\circ})}}{\sqrt{I_0 \cdot \cos^2(45^{\circ})}} \times 100\%\] Now, simplify the expression: \[\frac{\sqrt{I_0} - \sqrt{I_0 \cdot \frac{1}{2}}}{\sqrt{I_0 \cdot \frac{1}{2}}} \times 100\%\] \[\frac{\sqrt{I_0} - \sqrt{\frac{I_0}{2}}}{\sqrt{\frac{I_0}{2}}} \times 100\%\] Finally, calculate the numerical value: \[\frac{\sqrt{2} - 1}{1} \times 100\%\] \[\approx 41.42\%\] So, the percentage difference between the initial and final maximum values of the electric field vector is around \(41.42\%\). However, the options given are: (A) \(19 \%\) (B) \(92 \%\) (C) \(50 \%\) (D) \(29 \%\) The closest option to the calculated value is (C) \(50\%\). Although the exact value is not among the given answers, we can select the closest option as the final answer.

Key concepts

Malus's Law

Malus's Law is fundamental in understanding how light intensity changes when passing through a polarizer. When plane polarized light encounters a polarizer, the intensity of the transmitted light can be calculated using this important law. According to Malus's Law, the transmitted light intensity \( I \) is linked to the incident light intensity \( I_0 \) and the angle \( \theta \) between the light's electric field vector \( \vec{E} \) and the optical axis of the polarizer. The formula is: \[ I = I_0 \cdot \cos^2(\theta) \] This formula reveals that the intensity is highest when \( \theta = 0 \degree \) and decreases as the angle increases. It drops to zero at \( \theta = 90 \degree \). In practical terms, this law helps predict how much light will pass through a polarizing filter, a common tool in optical instruments and sunglasses. Understanding Malus’s Law is crucial for interpreting experiments involving polarized light and its interaction with materials.

Plane Polarized Light

Plane polarized light refers to light waves in which the electric field vibrations are confined to a single plane. This is unlike ordinary light, which vibrates in multiple planes perpendicular to its direction of travel. Plane polarization is achieved through methods such as using a polarizing filter, reflection, or scattering.

• Filters: Polarizers (like Polaroid sheets) absorb one orientation of electric field vectors, allowing only the other orientation to pass through.
• Reflection: At certain angles (like Brewster's angle), reflected light can become plane polarized.
• Scattering: Light scattering mechanisms can favor a specific polarization orientation under certain conditions.

By manipulating light in this way, plane polarized light is useful in various applications, such as reducing glare in photography, enhancing image contrast, and analyzing material properties. Understanding plane polarized light is also important for comprehending how polarizing filters control light transmission.

Optical Axis

The optical axis of a material like a polarizer or crystal is a line along which there is no double refraction of light. It represents a direction within the polarizer where the optical properties are uniform. When we talk about polarized light and polarizing devices, understanding the optical axis is crucial. The optical axis is the direction where electric field vectors can pass without any distortion in their orientation. Typically, when polarized light encounters a polarizer, aligning its electric field vector closer to the optical axis allows more light to pass through, maximizing intensity. Conversely, if the light's electric field vector is perpendicular to the optical axis, little to no light will get through. Appreciating the role of the optical axis is important for designing optical systems and analyzing how materials affect the polarization state of light. It helps in predicting the behavior of polarized light and its applications in areas such as microscopy, photography, and spectroscopy.

Electric Field Vectors

Electric field vectors \( \vec{E} \) describe the magnitude and direction of the electric field associated with a light wave. In the context of light polarization, these vectors are key in determining how light interacts with polarizers. Light can be thought of as a wave of electric and magnetic fields oscillating perpendicular to each other, and the electric field vector is often more relevant for discussing polarization. The orientation of \( \vec{E} \) with respect to the optical axis in a polarizing medium decides how much of the light is transmitted. When \( \vec{E} \) is parallel to the optical axis, maximum transmission occurs, owing to the alignment discussed in Malus's Law. When \( \vec{E} \) is at an angle, its component along the optical axis reduces, thereby reducing the transmitted intensity. Understanding the behavior of electric field vectors is essential not only for polarization but also for other phenomena like reflection, refraction, and diffraction, where the orientation of these vectors determines the outcome of interactions between light and materials.