Question

A small helium-neon laser emits red visible light with a power of 5.80 mW in a beam of diameter 2.50 mm. (a) What are the amplitudes of the electric and magnetic fields of this light? (b) What are the average energy densities associated with the electric field and with the magnetic field? (c) What is the total energy contained in a 1.00-m length of the beam?

Short answer

The electric field amplitude is calculated from intensity; magnetic amplitude follows from electric. Energy densities follow, leading to total energy by integration over volume.

Step-by-step solution

Calculate Intensity of the Beam

The intensity \( I \) of the beam is the power per unit area. The area \( A \) of the circular cross-section of the beam is given by \( A = \pi \left( \frac{d}{2} \right)^2 \), where \( d \) is the diameter. Thus, \( A = \pi \left( \frac{2.50 \times 10^{-3}}{2} \right)^2 \). The intensity is then calculated as \( I = \frac{P}{A} = \frac{5.80 \times 10^{-3}}{A} \), where \( P \) is the power.

Calculate Electric Field Amplitude

The amplitude of the electric field \( E_{0} \) can be found using the relation \( I = \frac{1}{2} \varepsilon_{0} c E_{0}^2 \), where \( \varepsilon_{0} \) is the permittivity of free space and \( c \) is the speed of light in a vacuum. Solve for \( E_{0} \) from \( E_{0} = \sqrt{\frac{2I}{\varepsilon_{0} c}} \). Substitute the known values to calculate \( E_{0} \).

Calculate Magnetic Field Amplitude

The amplitude of the magnetic field \( B_{0} \) is related to the electric field by \( B_{0} = \frac{E_{0}}{c} \). Use the calculated \( E_{0} \) from Step 2 to find \( B_{0} \).

Calculate Average Energy Density for Electric Field

The average energy density \( u_{E} \) associated with the electric field is \( u_{E} = \frac{1}{2} \varepsilon_{0} E_{0}^2 \). Use the calculated \( E_{0} \) to compute \( u_{E} \).

Calculate Average Energy Density for Magnetic Field

The average energy density \( u_{B} \) associated with the magnetic field is \( u_{B} = \frac{1}{2} \frac{B_{0}^2}{\mu_{0}} \), where \( \mu_{0} \) is the permeability of free space. Use the previously found \( B_{0} \) to determine \( u_{B} \).

Calculate Total Energy in a 1.00-m Length of the Beam

The total energy \( U \) in a 1.00-m length is given by \( U = (u_{E} + u_{B}) \times (\text{area}) \times (\text{length}) \). Use the sum of \( u_{E} \) and \( u_{B} \), and multiply it by the cross-sectional area (from Step 1) and the length (1.00 m) to find \( U \).

Key concepts

Electric Field Amplitude

The electric field amplitude is a measure of the strength of the electric field in an electromagnetic wave. It is denoted by \( E_{0} \) and is linked to the intensity of the wave. For a given intensity \( I \), the electric field amplitude can be calculated using the formula:\[E_{0} = \sqrt{\frac{2I}{\varepsilon_{0} c}} \] where \( \varepsilon_{0} \) is the permittivity of free space, and \( c \) is the speed of light in a vacuum. Understanding the electric field amplitude gives insight into how much electric energy is being carried by the wave. A higher amplitude means a stronger electric field and more energy transfer.

Magnetic Field Amplitude

In electromagnetic waves, the electric component is accompanied by a magnetic component. The amplitude of the magnetic field \( B_{0} \) tells us the strength of this magnetic part. It is directly related to the electric field amplitude via the speed of light:\[B_{0} = \frac{E_{0}}{c} \] This relationship indicates that the magnetic field amplitude is determined by the electric field and the speed at which light travels. While the magnetic field is essential, it's typically much weaker than the electric field due to the nature of electromagnetic waves. However, it still plays a critical role in the wave's propagation and energy dynamics.

Energy Density

Energy density in electromagnetic waves refers to the amount of energy stored in a given volume by the electric and magnetic fields. There are separate formulas for the energy density of the electric field \( u_{E} \) and the magnetic field \( u_{B} \):- **Electric Field Energy Density**: \( u_{E} = \frac{1}{2} \varepsilon_{0} E_{0}^2 \) - **Magnetic Field Energy Density**: \( u_{B} = \frac{1}{2} \frac{B_{0}^2}{\mu_{0}} \)Where \( \mu_{0} \) is the permeability of free space. The total energy density is the sum of \( u_{E} \) and \( u_{B} \). Energy density provides a measure of how much energy is stored per unit volume in the fields of the wave, crucial for understanding energy distribution and transfer in electromagnetic processes.

Helium-Neon Laser

Helium-Neon lasers are a type of gas laser known for emitting red visible light. They are commonly used in pointers, barcode scanners, and holography. Such lasers operate by exciting a mixture of helium and neon gases through an electrical discharge. Characteristics of Helium-Neon lasers include:

• Typical wavelength: 632.8 nm (red light)
• Low output power, generally in milliwatts like 5.80 mW
• Stable and coherent light output

Understanding these properties is key to appreciating why Helium-Neon lasers are widely used in applications requiring precise and stable light.

Intensity of Light

The intensity of light is a measure of the energy that a light beam delivers per unit area per unit time. It is an indicator of how bright the light appears and is denoted by \( I \). The intensity is found by dividing the power of the light by the area it covers:\[I = \frac{P}{A} \] Where \( P \) is the power and \( A \) is the cross-sectional area of the beam. Intensity is a crucial concept in optics and electromagnetic waves because it determines how much energy the wave carries to a surface, closely tying in with both the electric and magnetic field amplitudes of the wave.