Problem 50 - Physics

Question

A beam of polarized light has an average intensity of \(15 \mathrm{~W} / \mathrm{m}^{2}\) and is sent through a polarizer. The transmission axis makes an angle of \(25^{\circ}\) with respect to the direction of polarization. Determine the rms value of the electric field of the transmitted beam.

Step-by-Step Solution

Answer

The rms value of the electric field is approximately 0.963 V/m.

Step 1: Understanding Intensity of Polarized Light

The average intensity of polarized light incident on a polarizer can be expressed using the formula: \[ I = \frac{1}{2} c \epsilon_0 E^2_{\text{rms}} \]where \( c \) is the speed of light in a vacuum (\( 3 \times 10^8 \text{ m/s} \)), \( \epsilon_0 \) is the permittivity of free space (\( 8.85 \times 10^{-12} \text{ C}^2/\text{Nm}^2 \)), and \( E_{\text{rms}} \) is the root mean square of the electric field.

Step 2: Intensity After Passing Through the Polarizer

The intensity of the light after passing through the polarizer is given by Malus's Law:\[ I' = I_0 \cos^2(\theta) \]where \( I_0 = 15 \, \mathrm{W/m^2} \) is the initial intensity and \( \theta = 25^\circ \) is the angle between the light's initial polarization direction and the axis of the polarizer.

Step 3: Calculating Transmitted Intensity

Plug in the values to calculate the transmitted intensity:\[ I' = 15 \, \mathrm{W/m^2} \times \cos^2(25^\circ) \]First, calculate \( \cos(25^\circ) \approx 0.906 \). Then:\[ \cos^2(25^\circ) = (0.906)^2 \approx 0.821 \]\[ I' = 15 \, \mathrm{W/m^2} \times 0.821 = 12.315 \, \mathrm{W/m^2} \]

Step 4: Calculating the RMS Value of the Electric Field

Use the formula for intensity to find the rms value of the electric field of the transmitted beam:\[ I' = \frac{1}{2} c \epsilon_0 E^2_{\text{rms}} \]Rearrange to solve for \( E_{\text{rms}} \):\[ E_{\text{rms}} = \sqrt{\frac{2 I'}{c \epsilon_0}} \]Substitute \( I' = 12.315 \, \text{W/m}^2 \), \( c = 3 \times 10^8 \, \text{m/s} \), \( \epsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/\text{N} \cdot \text{m}^2 \):\[ E_{\text{rms}} = \sqrt{\frac{2 \times 12.315}{3 \times 10^8 \times 8.85 \times 10^{-12}}} \]\[ E_{\text{rms}} \approx \sqrt{\frac{24.63}{26.55 \times 10^{-4}}} \approx \sqrt{0.927} \approx 0.963 \, \text{V/m} \]

Step 5: Final Step: Conclusion

The root mean square (rms) value of the electric field of the transmitted beam is approximately \(0.963 \, \text{V/m} \).

Key Concepts

Malus's Law

When dealing with polarized light, Malus's Law is a crucial principle. It describes how light intensity changes as it passes through a polarizer. A polarizer filters light based on its orientation, and this law helps us predict the resultant intensity. Malus's Law states:

• When polarized light encounters a polarizer, the transmitted intensity is determined by the incident light's intensity and the angle between the light's initial direction of polarization and the polarizer's axis.
• The mathematical expression for Malus's Law is given by: \[ I' = I_0 \cos^2(\theta) \]where \( I_0 \) is the initial intensity and \( \theta \) is the angle.

Through this formula, we can determine how much light continues through the polarizer and how much is blocked. This is particularly useful in various applications, such as controlling light in cameras or sunglasses.

Root Mean Square (RMS)

Root Mean Square, or RMS, is a statistical measure often used in physics to describe the magnitude of a varying quantity. In the context of light and electric fields, RMS values are crucial for simplifying sinusoidal functions like those found in wave physics.

• RMS provides a way to calculate a consistent and realistic value of the field even when it oscillates between positive and negative values.
• The RMS value can be calculated for periodic functions, such as sine waves, and it's equivalent to the square root of the arithmetic mean of the squares of the values.

In electrical engineering and physics, RMS is commonly used to gauge the effective value of an alternating current (AC). For electric fields in light waves, the RMS value offers insights into the power or intensity that the field possesses.

Intensity of Light

Intensity refers to the power carried by light as it travels through a given area in a unit of time. For any electromagnetic wave, the intensity is a product of the square of the RMS electric field value and constant factors like the speed of light and permittivity of space.

• Intensity is expressed in watts per square meter (W/m²). It indicates how much energy the light beam conveys to a surface.
• The equation \[ I = \frac{1}{2} c \epsilon_0 E^2_{\text{rms}} \] relates intensity \( I \) to the RMS electric field \( E_{\text{rms}} \), where \( c \) stands for the speed of light, and \( \epsilon_0 \) denotes the permittivity of free space.

By understanding intensity, one can determine how powerful the light is after undergoing transformations like passing through polarizers as described by Malus's Law.

Electric Field

The electric field is a physical field produced by electrically charged objects. It is a significant factor in understanding electromagnetic waves, like light.

• In the context of light, the electric field oscillates perpendicular to the direction of travel of the wave.
• The strength of this field determines the amplitude and intensity of the light wave.
• Calculating the electric field, especially the RMS value, is crucial in determining the energy and intensity of light waves as they interact with materials and devices.

Electric fields are not confined to light alone. They play a role in various domains, such as electronics, electrical engineering, and plasma physics, indicating their broad application and importance.