Problem 24 - Physics

Question

A neodymium-glass laser emits short pulses of high-intensity electromagnetic waves. The electric field of such a wave has an rms value of \(E_{\mathrm{rms}}=2.0 \times 10^{9} \mathrm{~N} / \mathrm{C} .\) Find the average power of each pulse that passes through a \(1.6 \times 10^{-5}-\mathrm{m}^{2}\) surface that is perpendicular to the laser beam.

Step-by-Step Solution

Answer

The average power is approximately \( 8.5 \times 10^{10} \ \text{W} \).

Step 1: Understanding the Concept

The average power of an electromagnetic wave passing through a surface is related to the intensity of the wave and the area it passes through. The intensity is given by the formula \( I = \frac{1}{2} c \varepsilon_0 E_{\text{rms}}^2 \), where \( c \) is the speed of light and \( \varepsilon_0 \) is the permittivity of free space.

Step 2: Calculate the Intensity

Substitute \( E_{\text{rms}} = 2.0 \times 10^9 \ \text{N/C} \), \( c = 3.0 \times 10^8 \ \text{m/s} \), and \( \varepsilon_0 = 8.85 \times 10^{-12} \ \text{F/m} \) into the formula:\[I = \frac{1}{2} \times 3.0 \times 10^8 \ \text{m/s} \times 8.85 \times 10^{-12} \ \text{F/m} \times (2.0 \times 10^9 \ \text{N/C})^2\].Calculate \( I \).

Step 3: Compute the Intensity Result

Perform the calculations: \[I = \frac{1}{2} \times 3.0 \times 10^8 \times 8.85 \times 10^{-12} \times 4.0 \times 10^{18}\].Calculate \( I = 5.31 \times 10^{15} \ \text{W/m}^2 \).

Step 4: Compute the Average Power

Use the formula \( P = I \times A \) to calculate the average power, where \( A = 1.6 \times 10^{-5} \ \text{m}^2 \) is the area. Substitute \( I \) from Step 3:\[P = 5.31 \times 10^{15} \ \text{W/m}^2 \times 1.6 \times 10^{-5} \ \text{m}^2 \].

Step 5: Solve for Average Power

Calculate the result:\[P = 5.31 \times 1.6 \times 10^{10}\].Thus, the average power \( P = 8.5 \times 10^{10} \ \text{W} \).

Key Concepts

Electromagnetic Waves

Electromagnetic waves are waves that carry energy through space. They are produced by the movement of charged particles and do not require a medium to travel through. Electromagnetic waves encompass a broad spectrum: from radio waves to gamma rays. This spectrum is categorized based on the frequency and wavelength of the waves.

An important characteristic of electromagnetic waves is that they travel at the speed of light, which is approximately \(3.0 \times 10^8 \ \text{m/s}\). They consist of perpendicular electric and magnetic fields that oscillate and propagate through space. These fields are perpendicular to each other and to the direction of wave travel, meaning they are transverse waves.

• Electromagnetic waves can transfer energy across vast distances, like from the sun to the earth.
• They are utilized in numerous technologies, such as radio transmission, medical imaging, and laser applications.

Understanding electromagnetic waves is crucial for analyzing phenomena like light and lasers, and for calculating properties such as intensity, which directly affects the power output in applications like lasers.

Intensity of Light

The intensity of light is a measure of the energy carried by light waves as they travel through a given area. It is often expressed in watts per square meter \(W/m^2\). In practical terms, intensity tells us how bright or strong the light is at a particular point.

For electromagnetic waves like those emitted by a laser, intensity \(I\) can be calculated using the formula:
\[I = \frac{1}{2} c \varepsilon_0 E_{\text{rms}}^2\]
where

• \(c\) is the speed of light (\(3.0 \times 10^8 \ \text{m/s}\)),
• \(\varepsilon_0\) is the permittivity of free space (\(8.85 \times 10^{-12} \ \text{F/m}\)),
• and \(E_{\text{rms}}\) is the root mean square of the electric field strength.

The intensity indicates how much power the light transmits through an area, and this makes it a critical factor when assessing the power of laser pulses, as it is closely related to the concept of average power. For instance, in the given exercise, the intensity was used to determine the average power of the laser pulse.

Electric Field

The electric field is a vector field that represents the electric force per unit charge exerted at each point in space. It is produced by electric charges or by changing magnetic fields. In the context of electromagnetic waves, the electric field is a fundamental component, and its strength can significantly impact the intensity of the wave.

For lasers, the electric field is often described by its root mean square (RMS) value, which provides a measure of its average magnitude across the waveform:
\(E_{\mathrm{rms}} = \frac{E_{0}}{\sqrt{2}}\)
where \(E_{0}\) is the peak electric field.

• The RMS value gives us a practical measure to calculate the intensity of the light wave.
• In laser applications, the more intense the electric field, the higher the potential energy carried by the laser beam, useful in applications like cutting or medical treatments.

The electric field, particularly its RMS value, is crucial for calculating wave properties like intensity and ultimately the power of emitted laser pulses, as seen in the context of the neodymium-glass laser exercise.