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College Physics for AP® Courses

24.2Production of Electromagnetic Waves

College Physics for AP® Courses24.2 Production of Electromagnetic Waves

Learning Objectives

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

• 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 24.5.

Figure 24.5 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 ($EE size 12{E} {}$) propagates away from the antenna at the speed of light, forming part of an electromagnetic wave.

The electric field ($EE size 12{E} {}$) shown surrounding the wire is produced by the charge distribution on the wire. Both the $EE size 12{E} {}$ 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 ($BB size 12{B} {}$) which propagates outward as well (see Figure 24.6). 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 24.5 reveals the periodic nature of the generator-driven charges oscillating up and down in the antenna and the electric field produced. At time $t=0t=0 size 12{t=0} {}$, 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 $EE size 12{E} {}$-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 $EE size 12{E} {}$-field has moved away at speed $cc size 12{c} {}$.

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$λλ size 12{ left (λ right )} {}$ is proportional to the period of the oscillation and, hence, is smaller for short periods or high frequencies. (As usual, wavelength and frequency$ff size 12{ left (f right )} {}$ 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 24.6. The relationship between $EE size 12{E} {}$ and $BB size 12{B} {}$ is shown at one instant in Figure 24.6 (a). As the current varies, the magnetic field varies in magnitude and direction.

Figure 24.6 (a) The current in the antenna produces the circular magnetic field lines. The current ($II size 12{I} {}$) produces the separation of charge along the wire, which in turn creates the electric field as shown. (b) The electric and magnetic fields ($EE size 12{E} {}$ and $BB size 12{B} {}$) 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 24.6 (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 24.7. 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.

Figure 24.7 A part of the electromagnetic wave sent out from the antenna at one instant in time. The electric and magnetic fields ($EE size 12{E} {}$ and $BB size 12{B} {}$) 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 24.7 to illustrate its basic characteristics.

Making Connections: Self-Propagating Wave

Note that an electromagnetic wave, as shown in Figure 24.7, 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 (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 24.1

From the illustration of the electric field given in Figure 24.8(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.

Exercise 24.2

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

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 $EE size 12{E} {}$-Field and $BB size 12{B} {}$-Field Strengths

There is a relationship between the $EE size 12{E} {}$- and $BB size 12{B} {}$-field strengths in an electromagnetic wave. This can be understood by again considering the antenna just described. The stronger the $EE size 12{E} {}$-field created by a separation of charge, the greater the current and, hence, the greater the $BB size 12{B} {}$-field created.

Since current is directly proportional to voltage (Ohm’s law) and voltage is directly proportional to $EE size 12{E} {}$-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,

$E B = c E B = c size 12{ { {E} over {B} } =c} {}$
24.3

is the ratio of $EE size 12{E} {}$-field strength to $BB size 12{B} {}$-field strength in any electromagnetic wave. This is true at all times and at all locations in space. A simple and elegant result.

Example 24.1

Calculating $BB size 12{B} {}$-Field Strength in an Electromagnetic Wave

What is the maximum strength of the $BB size 12{B} {}$-field in an electromagnetic wave that has a maximum $EE size 12{E} {}$-field strength of $1000 V/m1000 V/m size 12{"1000" {V} slash {m} } {}$?

Strategy

To find the $BB size 12{B} {}$-field strength, we rearrange the above equation to solve for $BB size 12{B} {}$, yielding

$B=Ec.B=Ec. size 12{B= { {E} over {c} } } {}$
24.4

Solution

We are given $EE size 12{E} {}$, and $cc size 12{c} {}$ is the speed of light. Entering these into the expression for $BB size 12{B} {}$ yields

$B=1000 V/m3.00 × 108 m/s= 3.33×10-6 T,B=1000 V/m3.00 × 108 m/s= 3.33×10-6 T, size 12{B = { {"1000 V/m"} over {3 "." "00 " times " 10" rSup { size 8{8} } " m/s"} } =" 3" "." "33" times "10" rSup { size 8{ +- 6} } " T"} {}$
24.5

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

Discussion

The $BB size 12{B} {}$-field strength is less than a tenth of the Earth’s admittedly weak magnetic field. This means that a relatively strong electric field of 1000 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.

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.

Figure 24.9
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