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Learning Objectives

By the end of this section, you will be able to do the following:

Describe the electric and magnetic waves as they move out from a source, such as an AC generator
Explain the mathematical relationship between the magnetic field strength and the electrical field strength
Calculate the maximum strength of the magnetic field in an electromagnetic wave, given the maximum electric field strength

The information presented in this section supports the following AP® learning objectives and science practices:

6.A.1.1 The student is able to use a visual representation to construct an explanation of the distinction between transverse and longitudinal waves by focusing on the vibration that generates the wave. (S.P. 6.2)
6.A.1.2 The student is able to describe representations of transverse and longitudinal waves. (S.P. 1.2)
6.A.2.2 The student is able to contrast mechanical and electromagnetic waves in terms of the need for a medium in wave propagation. (S.P. 6.4, 7.2)
6.B.3.1 The student is able to construct an equation relating the wavelength and amplitude of a wave from a graphical representation of the electric or magnetic field value as a function of position at a given time instant and vice versa, or construct an equation relating the frequency or period and amplitude of a wave from a graphical representation of the electric or magnetic field value at a given position as a function of time and vice versa. (S.P. 1.4)
6.F.2.1 The student is able to describe representations and models of electromagnetic waves that explain the transmission of energy when no medium is present. (S.P. 1.1)

We can get a good understanding of electromagnetic waves (EM) by considering how they are produced. Whenever a current varies, associated electric and magnetic fields vary, moving out from the source like waves. Perhaps the easiest situation to visualize is a varying current in a long straight wire, produced by an AC generator at its center, as illustrated in Figure 7.5.

A long straight gray wire with an A C generator at its center, functioning as a broadcast antenna for electromagnetic waves, is shown. The wave distributions at four different times are shown in four different parts. Part a of the diagram shows a long straight gray wire with an A C generator at its center. The time is marked t equals zero. The bottom part of the antenna is positive and the upper end of the antenna is negative. An electric field E acting upward is shown by an upward arrow. Part b of the di

Figure 7.5 This long straight gray wire with an AC generator at its center becomes a broadcast antenna for electromagnetic waves. Shown here are the charge distributions at four different times. The electric field

(E)

propagates away from the antenna at the speed of light, forming part of an electromagnetic wave.

The electric field $(E)$ shown surrounding the wire is produced by the charge distribution on the wire. Both the $E$ and the charge distribution vary as the current changes. The changing field propagates outward at the speed of light.

There is an associated magnetic field $(B)$ which propagates outward as well (see Figure 7.6). The electric and magnetic fields are closely related and propagate as an electromagnetic wave. This is what happens in broadcast antennae such as those in radio and TV stations.

Closer examination of the one complete cycle shown in Figure 7.5 reveals the periodic nature of the generator-driven charges oscillating up and down in the antenna and the electric field produced. At time $t = 0$ , there is the maximum separation of charge, with negative charges at the top and positive charges at the bottom, producing the maximum magnitude of the electric field, or $E,$ field, in the upward direction. One-fourth of a cycle later, there is no charge separation and the field next to the antenna is zero, while the maximum $E -$ field has moved away at speed $c .$

As the process continues, the charge separation reverses and the field reaches its maximum downward value, returns to zero, and rises to its maximum upward value at the end of one complete cycle. The outgoing wave has an amplitude proportional to the maximum separation of charge. Its wavelength $(λ)$ is proportional to the period of the oscillation and, hence, is smaller for short periods or high frequencies. As usual, wavelength and frequency $(f)$ are inversely proportional.

Electric and Magnetic Waves: Moving Together

Following Ampere’s law, current in the antenna produces a magnetic field, as shown in Figure 7.6. The relationship between $E$ and $B$ is shown at one instant in Figure 7.6 (a). As the current varies, the magnetic field varies in magnitude and direction.

Part a of the diagram shows a long straight gray wire with an A C generator at its center, functioning as a broadcast antenna. The antenna has a current I flowing vertically upward. The bottom end of the antenna is negative and the upper end of the antenna is positive. An electric field is shown to act vertically downward. The magnetic field lines B produced in the antenna are circular in direction around the wire. Part b of the diagram shows a long straight gray wire with an A C generator at its center,

Figure 7.6 (a) The current in the antenna produces the circular magnetic field lines. The current

(I)

produces the separation of charge along the wire, which in turn creates the electric field as shown. (b) The electric and magnetic fields

(E

and

B)

near the wire are perpendicular; they are shown here for one point in space. (c) The magnetic field varies with current and propagates away from the antenna at the speed of light.

The magnetic field lines also propagate away from the antenna at the speed of light, forming the other part of the electromagnetic wave, as seen in Figure 7.6 (b). The magnetic part of the wave has the same period and wavelength as the electric part, since they are both produced by the same movement and separation of charges in the antenna.

The electric and magnetic waves are shown together at one instant in time in Figure 7.7. The electric and magnetic fields produced by a long straight wire antenna are exactly in phase. Note that they are perpendicular to one another and to the direction of propagation, making this a transverse wave.

A part of the electromagnetic wave sent out from the antenna at one instant in time is shown. The wave is shown with the variation of two components, E and B, moving with velocity c. E is a sine wave in one plane with small arrows showing the vibrations of particles in the plane. B is a sine wave in a plane perpendicular to the E wave. The B wave has arrows to show the vibrations of particles in the plane. The waves are shown intersecting each other at the junction of the planes because E and B are perpen

Figure 7.7 A part of the electromagnetic wave sent out from the antenna at one instant in time. The electric and magnetic fields

(E

and

B)

are in phase, and they are perpendicular to one another and the direction of propagation. For clarity, the waves are shown only along one direction, but they propagate out in other directions too.

Electromagnetic waves generally propagate out from a source in all directions, sometimes forming a complex radiation pattern. A linear antenna like this one will not radiate parallel to its length, for example. The wave is shown in one direction from the antenna in Figure 7.7 to illustrate its basic characteristics.

Making Connections: Self-Propagating Wave

Note that an electromagnetic wave, as shown in Figure 7.7, is the result of a changing electric field causing a changing magnetic field, which causes a changing electric field, and so on. Therefore, unlike other waves, an electromagnetic wave is self-propagating, even in a vacuum, or empty space. It does not need a medium to travel through. This is unlike mechanical waves, which do need a medium. The classic standing wave on a string, for example, does not exist without the string. Similarly, sound waves travel by molecules colliding with their neighbors. If there is no matter, sound waves cannot travel.

Instead of the AC generator, the antenna can also be driven by an AC circuit. In fact, charges radiate whenever they are accelerated. But while a current in a circuit needs a complete path, an antenna has a varying charge distribution forming a standing wave, driven by the AC. The dimensions of the antenna are critical for determining the frequency of the radiated electromagnetic waves. This is a resonant phenomenon and when we tune radios or TV, we vary electrical properties to achieve appropriate resonant conditions in the antenna.

Applying the Science Practices: Wave Properties and Graphs

Exercise 1

From the illustration of the electric field given in Figure 7.8(a) of an electromagnetic wave at some instant in time, please state what the amplitude and wavelength of the given waveform are. Then write down the equation for this particular wave.

The amplitude is 60 V/m, while the wavelength in this case is 1.5 m. The equation is $E = (60 \frac{V}{m}) cos (\frac{4}{3} π x) .$

Exercise 2

Now, consider another electromagnetic wave for which the electric field at a particular location is given over time by $E ? = ? (30 \frac{V}{m}) ? \sin ? (4.0 p \times 10^{6} t) .$ What are the amplitude, frequency, and period? Finally, draw and label an appropriate graph for this electric field.

The amplitude is 30 V/m, while the frequency is 2.0 MHz and hence the period is 5.0 × 10⁻⁷ s. The graph should be similar to that in Figure 7.8(b).

Graphs of two trigonometric functions: in (a) it is the graph of E=(60 V/M)cos(4/3πx), while in (b) it is the graph of E=(30 V/M)sin(4.0π−10 6 t).

Figure 7.8 Two waveforms which describe two electromagnetic waves. Notice that (a) describes a wave in space, while (b) describes a wave at a particular point in space over time.

Receiving Electromagnetic Waves

Electromagnetic waves carry energy away from their source, similar to a sound wave carrying energy away from a standing wave on a guitar string. An antenna for receiving EM signals works in reverse. And like antennas that produce EM waves, receiver antennas are specially designed to resonate at particular frequencies.

An incoming electromagnetic wave accelerates electrons in the antenna, setting up a standing wave. If the radio or TV is switched on, electrical components pick up and amplify the signal formed by the accelerating electrons. The signal is then converted to audio and/or video format. Sometimes big receiver dishes are used to focus the signal onto an antenna.

In fact, charges radiate whenever they are accelerated. When designing circuits, we often assume that energy does not quickly escape AC circuits, and mostly this is true. A broadcast antenna is specially designed to enhance the rate of electromagnetic radiation, and shielding is necessary to keep the radiation close to zero. Some familiar phenomena are based on the production of electromagnetic waves by varying currents. Your microwave oven, for example, sends electromagnetic waves, called microwaves, from a concealed antenna that has an oscillating current imposed on it.

Relating

Relating $E -$ Field and $B -$ Field Strengths

There is a relationship between the $E -$ and $B -$ field strengths in an electromagnetic wave. This can be understood by again considering the antenna just described. The stronger the $E -$ field created by a separation of charge, the greater the current and, hence, the greater the $B -$ field created.

Since current is directly proportional to voltage, Ohm's law, and voltage is directly proportional to $E -$ field strength, the two should be directly proportional. It can be shown that the magnitudes of the fields do have a constant ratio, equal to the speed of light. That is,

7.3

\frac{E}{B} = c

is the ratio of $E -$ field strength to $B -$ field strength in any electromagnetic wave. This is true at all times and at all locations in space. A simple and elegant result.

Example 7.1 Calculating $B$ -Field Strength in an Electromagnetic Wave

What is the maximum strength of the $B -$ field in an electromagnetic wave that has a maximum $E -$ field strength of $1,000 V/ m$ ?

Strategy

To find the $B -$ field strength, we rearrange the above equation to solve for $B,$ yielding

7.4

B = \frac{E}{c} .

Solution

We are given $E,$ and $c$ is the speed of light. Entering these into the expression for $B$ yields

7.5

B = \frac{1,000 V/m}{3.00 \times 10^{8} m/s} = 3 . 33 \times 10^{- 6} T,

Where T stands for Tesla, a measure of magnetic field strength.

Discussion

The $B$ -field strength is less than a tenth of Earth’s admittedly weak magnetic field. This means that a relatively strong electric field of 1,000 V/m is accompanied by a relatively weak magnetic field. Note that as this wave spreads out, say with distance from an antenna, its field strengths become progressively weaker.

The result of this example is consistent with the statement made in the module Maxwell’s Equations: Electromagnetic Waves Predicted and Observed that changing electric fields create relatively weak magnetic fields. They can be detected in electromagnetic waves, however, by taking advantage of the phenomenon of resonance, as Hertz did. A system with the same natural frequency as the electromagnetic wave can be made to oscillate. All radio and TV receivers use this principle to pick up and then amplify weak electromagnetic waves, while rejecting all others not at their resonant frequency.

Take-Home Experiment: Antennas

For your TV or radio at home, identify the antenna, and sketch its shape. If you don’t have cable, you might have an outdoor or indoor TV antenna. Estimate its size. If the TV signal is between 60 and 216 MHz for basic channels, then what is the wavelength of those EM waves?

Try tuning the radio and note the small range of frequencies at which a reasonable signal for that station is received. This is easier with digital readout. If you have a car with a radio and extendable antenna, note the quality of reception as the length of the antenna is changed.

PhET Explorations: Radio Waves and Electromagnetic Fields

Broadcast radio waves from KPhET. Wiggle the transmitter electron manually or have it oscillate automatically. Display the field as a curve or vectors. The strip chart shows the electron positions at the transmitter and at the receiver.

This icon links to a P H E T Interactive activity when clicked.

Figure 7.9 Radio Waves and Electromagnetic Fields