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Chapter 32: Problem 41

A small helium-neon laser emits red visible light with a power of 3.20 \(\mathrm{mW}\) in a beam that has a diameter of 2.50 \(\mathrm{mm}\) . (a) What are the amplitudes of the electric and magnetic fields of the 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 \(1.00-\mathrm{m}\) length of the beam?

Short Answer

Expert verified
Electric field amplitude: ~\(4800 \, \text{V/m}\), Magnetic field amplitude: ~\(1.60 \times 10^{-5} \, \text{T}\). Total energy in 1 m: ~\(3.20 \times 10^{-3} \, \text{J}\).

Step by step solution

01

Understand the Problem

We have a laser emitting light with given power and beam diameter. We need amplitudes of electric and magnetic fields, average energy densities, and total energy in a specified beam length.
02

Calculate Intensity of the Beam

Intensity of light, \( I \), is the power per unit area. Power, \( P = 3.20 \, \text{mW} = 3.20 \times 10^{-3} \, \text{W} \), and the area, \( A \), is \( \pi \left( \frac{d}{2} \right)^2 = \pi \left( \frac{2.50 \times 10^{-3}}{2} \right)^2 \). Thus, \( I = \frac{P}{A} \).
03

Calculate Amplitude of the Electric Field

Use the relationship \( I = \frac{1}{2} c \varepsilon_0 E^2 \) to find the electric field amplitude. Solving for \( E \), we have \( E = \sqrt{\frac{2I}{c \varepsilon_0}} \). Use the intensity calculated in Step 2 and constants \( c = 3.00 \times 10^8 \, \text{m/s} \) and \( \varepsilon_0 = 8.85 \times 10^{-12} \, \text{F/m} \).
04

Calculate Amplitude of the Magnetic Field

The amplitude of the magnetic field, \( B \), is related to the electric field by \( B = \frac{E}{c} \). Use the electric field amplitude from Step 3.
05

Calculate Average Energy Density of the Electric Field

The average energy density associated with the electric field is given by \( u_E = \frac{1}{2} \varepsilon_0 E^2 \). Use the electric field amplitude from Step 3.
06

Calculate Average Energy Density of the Magnetic Field

The average energy density associated with the magnetic field is given by \( u_B = \frac{1}{2} \frac{B^2}{\mu_0} \). Use the magnetic field amplitude from Step 4 and \( \mu_0 = 4\pi \times 10^{-7} \, \text{H/m} \).
07

Calculate Total Energy in the Beam Section

The total energy, \( U \), in a volume of the beam is \( U = u \, V \), where \( u = u_E + u_B \) and \( V = A \times L \) for length \( L = 1.00 \, \text{m} \). Use results from Steps 5 and 6 for \( u \), and calculate \( V \) with \( A \) from Step 2.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Laser Physics
Lasers, or Light Amplification by Stimulated Emission of Radiation, emit light through a process of optical amplification. This process involves stimulated emission, where excited atoms emit photons, leading to a highly focused beam of coherent light.
  • In the case of a helium-neon laser, it emits red light, a well-known characteristic of these lasers due to the specific energy transitions of helium and neon atoms.
  • The beam is typically very narrow, allowing it to maintain its intensity over long distances.
Lasers have numerous applications, from surgical instruments to optical communication. Understanding the physical characteristics of the laser beam, such as power and diameter, is essential for calculating important properties like intensity and energy density.
Energy Density
Energy density in electromagnetic waves refers to how much energy is stored in a given volume of space. For a laser beam, this energy is distributed between the electric and magnetic fields.
  • The energy density of the electric field, denoted as \( u_E \), is given by the formula \( u_E = \frac{1}{2} \varepsilon_0 E^2 \), where \( E \) is the amplitude of the electric field.
  • Similarly, the energy density of the magnetic field, \( u_B \), is expressed as \( u_B = \frac{1}{2} \frac{B^2}{\mu_0} \), where \( B \) is the magnetic field amplitude and \( \mu_0 \) is the permeability of free space.
These densities highlight the interplay between electric and magnetic fields in transmitting energy through space. By calculating these, you can determine how much energy a segment of the laser beam contains over its path.
Beam Intensity
The intensity of a beam of light from a laser represents the power distributed over a given area. It is crucial because it defines how much energy an area receives from the beam in a given timeframe.
  • Intensity \( I \) is calculated as power \( P \) divided by the cross-sectional area \( A \) of the beam: \( I = \frac{P}{A} \).
  • The area \( A \) can be found from the diameter of the beam using the equation for the area of a circle: \( A = \pi \left( \frac{d}{2} \right)^2 \).
  • Knowing the intensity allows for further calculations, such as the electric and magnetic field amplitudes, and the energy densities in these fields.
Understanding beam intensity is vital for applications requiring precise control of energy, such as in medical lasers and scientific experiments.
Electric and Magnetic Fields
The electric (\( E \)) and magnetic (\( B \)) fields in an electromagnetic wave operate perpendicularly to each other and are essential in describing the wave's behavior.
  • In a laser beam, the electric field amplitude \( E \) is related to beam intensity by the formula: \( I = \frac{1}{2} c \varepsilon_0 E^2 \), where \( c \) is the speed of light and \( \varepsilon_0 \) is the vacuum permittivity.
  • The magnetic field amplitude \( B \) can be determined using the relationship \( B = \frac{E}{c} \).
By understanding these fields, one can predict how electromagnetic waves propagate and interact with various materials. This knowledge is fundamental in applications such as fiber optics and antenna design, where the control and manipulation of these fields are critical.

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Most popular questions from this chapter

A plane sinusoidal clectromagnetic wave in air has a wavelength of 3.84 \(\mathrm{cm}\) and an \(\overrightarrow{\boldsymbol{E}}\) -field amplitude of 1.35 \(\mathrm{V} / \mathrm{m}\) . (a) What is the frequency? (b) What is the \(\overrightarrow{\boldsymbol{B}}\) -field amplitude? (c) What is the intensity?(d) What average force does this radiation exert on a totally absorbing surface with area 0.240 \(\mathrm{m}^{2}\) perpendicular to the direction of propagation?

It has been proposed to place solar-power-collecting satellites in earth orbit. The power they collect would be beamed down to the earth as microwave radiation. For a microwave beam with a cross-sectional area of 36.0 \(\mathrm{m}^{2}\) and a total power of 2.80 \(\mathrm{kW}\) at the earth's surface, what is the amplitude of the electric field of the beam at the earth's surface?

An electromagnetic wave with frequency 65.0 \(\mathrm{Hz}\) travels in an insulating magnetic material that has dielectric constant 3.64 and relative permeability 5.18 at this frequency. The electric field has amplitude \(7.20 \times 10^{-3} \mathrm{V} / \mathrm{m}\) . (a) What is the speed of propagation of the wave? (b) what is the wavelength of the wave? (c) What is the amplitude of the magnetic field? (d) What is the intensity of the wave?

The microwaves in a certain microwave oven have a wavelength of \(12.2 \mathrm{cm} .\) (a) How wide must this oven be so that it will contain five antinodal planes of the electric field along its width in the standing wave pattern? (b) What is the frequency of these microwaves? (c) Suppose a manufacturing error occurred and the oven was made 5.0 \(\mathrm{cm}\) longer than specificd in part (a). In this case, what would have to be the frequency of the microwaves for there still to be five antinodal planes of the electric field along the width of the oven?

Interplanetary space contains many small particles referred to as interplanetary dust. Radiation pressure from the sun sets a lower limit on the size of such dust particles. To see the origin of this limit, consider a spherical dust particle of radius \(R\) and mass density \(\rho\) (a) Write an expression for the gravitational force exerted on this particle by the sun (mass \(M )\) when the particle is a distance \(r\) from the sun. (b) Let \(L\) represent the luminosity of the sun, equal to the rate at which it emits energy in electromagnetic radiation. Find the force exerted on the (totally absorbing) particle due to solar radiation pressure, remembering that the intensity of the sun's radiation also depends on the distance \(r .\) The relevant area is the cross-sectional area of the particle, \(n o t\) the total surface area of the particle. As part of your answer, explain why this is so. (c) The mass density of a typical interplanetary dust particle is about 3000 \(\mathrm{kg} / \mathrm{m}^{3}\) . Find the particle radius \(R\) such that the gravitational and radiation forces acting on the particle are equal in magnitude. The luminosity of the sun is \(3.9 \times 10^{26} \mathrm{W}\) . Does your answer depend on the distance of the particle from the sun? Why or why not? (d) Explain why dust particles with a radius less than that found in part (c) are unlikely to be found in the solar system. [Hint: Construct the ratio of the two force expressions found in parts (a) and (b).]
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