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Chapter 13: Problem 9

Show that the mean electric and magnetic energies in a plane electromagnetic wave in vacuo are equal.

Short Answer

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Question: Show that the mean electric and magnetic energies in a plane electromagnetic wave in vacuo are equal. Answer: We can demonstrate that the mean electric and magnetic energies in a plane electromagnetic wave in vacuo are equal by finding the expressions for electric and magnetic energy densities, calculating their mean value over one complete cycle, and then confirming that these mean values are equal. The energy densities for electric and magnetic fields are given by u_E = (1/2)蔚鈧E虏 and u_B = (1/2)B虏/渭鈧. For an electromagnetic wave in vacuo, the ratio between electric and magnetic field amplitudes is constant, E鈧 = cB鈧. After calculating the mean energy densities and simplifying their ratio, we find that mean(u_E)/mean(u_B) = 1, confirming that the mean electric and magnetic energies in a plane electromagnetic wave in vacuo are equal.

Step by step solution

01

1. Electric and magnetic energy densities

The energy densities of the electric and magnetic fields in a plane electromagnetic wave in vacuo are given by: u_E = (1/2)蔚鈧E虏 and u_B = (1/2)B虏/渭鈧, where 蔚鈧 is the vacuum permittivity, E is the electric field amplitude, 渭鈧 is the vacuum permeability, and B is the magnetic field amplitude.
02

2. Electric and magnetic fields in a plane electromagnetic wave

In a plane electromagnetic wave, the electric field E and the magnetic field B oscillate sinusoidally and have a fixed ratio: E = E鈧sin(kx - 蠅t) and B = B鈧sin(kx - 蠅t), where E鈧 and B鈧 are the maximum field amplitudes, k is the wave number, x is the spatial coordinate, 蠅 is the angular frequency, and t is time.
03

3. Relationship between electric and magnetic field amplitudes

For an electromagnetic wave in vacuo, the relationship between electric and magnetic field amplitudes is given by: E鈧 = cB鈧, where c is the speed of light in the vacuum (c 鈮 3脳10鈦 m/s).
04

4. Mean electric and magnetic energy densities

To find the mean energy densities, we'll integrate the energy densities over one complete cycle, divided by the period T. The mean energy densities are then given by: mean(u_E) = (1/T)鈭玼_E dt and mean(u_B) = (1/T)鈭玼_B dt When we calculate the mean electric energy density, we get: mean(u_E) = (1/T)鈭(1/2)蔚鈧(E鈧sin(kx - 蠅t))虏 dt Similarly, the mean magnetic energy density is: mean(u_B) = (1/T)鈭(1/2)(B鈧虏sin虏(kx - 蠅t))/渭鈧 dt
05

5. Calculating the mean energy densities

Using the trigonometric identity sin虏(x) = (1 - cos(2x))/2 and substituting E鈧 = cB鈧, we get the following integrals for the mean energy densities: mean(u_E) = (1/T)鈭(1/2)蔚鈧E鈧虏(1 - cos(2(kx - 蠅t)))/2 dt and mean(u_B) = (1/T)鈭(1/2)B鈧虏(1 - cos(2(kx - 蠅t)))/(2渭鈧) dt The integrals become: mean(u_E) = (蔚鈧E鈧虏/4T)鈭(1 - cos(2(kx - 蠅t))) dt and mean(u_B) = (B鈧虏/4渭鈧T)鈭(1 - cos(2(kx - 蠅t))) dt Since the integrands are the same, we can find the ratio of the mean energy densities by dividing their expressions: mean(u_E)/mean(u_B) = (蔚鈧E鈧虏/4T)/(B鈧虏/4渭鈧T)
06

6. Showing that the mean energy densities are equal

Using E鈧 = cB鈧, we replace E鈧 with cB鈧, and after simplification, we get: mean(u_E)/mean(u_B) = (蔚鈧(cB鈧)虏)/(B鈧虏/渭鈧) Simplifying further, we get: mean(u_E)/mean(u_B) = 蔚鈧渭鈧c虏 = 1 (since c虏 = 1/(蔚鈧渭鈧) in a vacuum) Since the ratio of the mean energy densities is 1, we have shown that the mean electric and magnetic energies in a plane electromagnetic wave in vacuo are equal.

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Key Concepts

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

Electromagnetic Wave Theory
Electromagnetic waves are ripples of energy that travel through the vacuum of space at the speed of light, characterized by oscillating electric and magnetic fields perpendicular to each other and to the direction of propagation. These fields are created by accelerating charges and do not require a medium to travel through, differentiating them from mechanical waves.

The energy carried by an electromagnetic wave is distributed between its electric and magnetic fields. This energy distribution is crucial to applications ranging from radio broadcasting to microwave heating. Understanding how energy is stored in these fields opens the door to innovations in communication, medical technology, and many other fields.
Electric Field Amplitude
The electric field amplitude, denoted as E or E鈧 for its maximum value, represents the highest strength of the electric field in an electromagnetic wave. It dictates the amount of force that would be exerted on a charge placed within the field. The energy density of the electric field is proportional to the square of this amplitude, signifying that even small increases in electric field amplitude can lead to significant amounts of energy stored within the field.

This concept helps explain phenomena such as the intensity of light: higher electric field amplitude corresponds to brighter light, and in the context of antennas, it means more powerful transmission of signals. For students, grasping the implication of E's amplitude on the energy density is key to understanding wave behaviors.
Magnetic Field Amplitude
In tandem with its electric counterpart, the magnetic field amplitude, symbolized by B or B鈧 at its peak, indicates the maximum magnitude of the magnetic field in the wave. This value, while often less intuitive than its electric sibling, is integral to the magnetic aspect of energy storage in an electromagnetic wave. In a vacuum, the energy density of the magnetic field, just like the electric component, is proportional to the square of the magnetic field amplitude.

The concept that a magnetic field stores energy and does work is a leap from the classic iron-filings visual of magnetic lines of force. The realization that a changing magnetic field is fundamental to the operation of generators and motors underlies much of modern technology. When students understand the magnetic field's role alongside the electric field, they unlock a complete picture of electromagnetic wave dynamics.

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Most popular questions from this chapter

Estimate the E-field at \(1 \mathrm{~m}\) from a \(30 \mathrm{~W}\) lamp, assuming that the energy is radiated equally in all directions and that no losses occur by conduction or convection of heat.

Derive the expression (7.27) for the magnetic B-field due to a moving charge by using the displacement current.

A spherical cavity of radius \(a\) is found in an LIH dielectric of relative permittivity \(c_{r}\). If the electric field far from the cavity is uniform and of magnitude \(E\), what is the field in the cavity itself?

A straight wire of negligible cross-section is effectively infinite in length but has a break of length \(2 l\) between points \(\mathrm{A}\) and \(\mathrm{B}\) on it. All parts of the wire catry a steady conduction current \(I\) so that charge accumulates at the points A and \(\mathrm{B}\). Find the B-field at a point a distance \(R\) along the perpendicular bisector of AB. Use both the current element formula (Biot-Savart law) and the circuital law and check that the same result is obtained.

Show that the intrinsic impedance of free space, defined as \(E / H\) for a plane wave in sacwo, has the value \(376.6 \Omega\).
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