7.1.1: Illustrations

Illustration 1: Creation of Electromagnetic Waves

This Illustration shows the electric field lines due to a positive charge. Initially the charge is not moving. The slider can be used to vary the speed of the charge. When translation mode is selected, the slider controls the instantaneous velocity. When oscillation mode is selected, the slider sets the maximum speed.

How are the changing fields in an electromagnetic wave created? Electromagnetic waves such as heat, light, and radio waves are created by a charge that is accelerating. The magnitude of the electric field is related to the acceleration of the charge.

Play the animation in translation mode and move the velocity slider. Notice how the electric field lines form a disturbance that moves away from the charge when the velocity changes. Because a changing electric field will give rise to a changing magnetic field, and a changing magnetic field will cause a changing electric field, a traveling electromagnetic wave is created. Move the slider in translation mode and note that abrupt changes in the velocity produce very complex wave patterns.

Play the animation in oscillation mode and notice the sinusoidal appearance of the electromagnetic disturbance.

You may wonder where the energy that goes into the electromagnetic wave comes from. Is energy being created from nothing? The answer is no, of course. A charge will not oscillate on its own; it must be driven by some force. For example, the charge might be part of an AC current, oscillating as the voltage source oscillates. Although some energy is being radiated from the charge, energy is being put into the charge to keep it oscillating.

Illustration authored by Melissa Dancy and Wolfgang Christian.

Illustration 2: Wave Crests

In the 19^th century it was discovered that a moving charge produces an electromagnetic wave. Rapidly oscillating charged particles (such as electrons in an atom) produce visible light, while slowly oscillating charges (such as those in an antenna) produce radio waves. Although waves with different frequencies produce different effects when they interact with matter, their propagation through space is quite similar. These similarities are the subject of this Illustration. Restart.

Electromagnetic waves have regions of high- and low-field strengths that are analogous to the high and low pressure regions of a sound wave. Analogies between electromagnetic waves and sound waves can be useful, but they should not be pushed too far. This Illustration shows one such analogy. An oscillating charge within the back circle produces a wave, and this wave is seen to propagate away from the source. The wave crests and troughs moving away from the source represent regions of strong electric field. The troughs are regions where the electric field is also strong, but the field is pointing in the opposite direction from the field at the crests. As waves propagate away from the source, their amplitude decreases. The red wave traveling to the right illustrates this.

Electromagnetic waves are different from sound waves, and this Illustration does little to point out this difference. Sound requires a medium for propagation, whereas electromagnetic waves do not need a medium: They can propagate in a vacuum. Furthermore, electric fields cannot propagate energy without a complementary magnetic field. The magnetic field associated with an electromagnetic wave is perpendicular to the electric field and is not shown. But the wavelength, frequency, and amplitude of the electric field are correctly illustrated and provide clues to the following questions:

• Do the frequency and period of the electromagnetic wave depend on the distance from the source?
• How does electric field amplitude depend on the distance from the source?

Illustration 3: Electromagnetic Plane Waves

Electromagnetic waves (such as radio waves) can sometimes be approximated as plane waves if the observer is located far from the source. But what, exactly, does this plane wave look like? Before you begin, we should point out that plane waves are (like point masses) an idealization. Typical electromagnetic waves are not plane waves, not because they are curved (although they usually do have some curvature), but because they contain many frequencies and because they originate from more than one source. Although radio waves approximate a plane wave, visible light usually does not, unless it is produced by a laser. Because waves can be constructed by adding together multifrequency plane waves, understanding this Illustration is a good place to start. Restart.

The animation shows a plane electromagnetic wave's electric field. The magnetic field is not shown. Click-drag inside the large panel before you play the animation. What do you see? The lines pointing away from the \(z\) axis represent the electric field as measured along the axis. Move the slider. The slider controls the position (given in meters) of the transparent square. The transparent square represents the plane (hence the name plane wave) in which you are viewing the electric field in the right panel. Use the slider to estimate the wavelength. Play the animation. Note the animation time (given in nanoseconds) in the right-hand panel. What is the frequency of the wave? In what portion of the electromagnetic spectrum is the wave? Since the period is \(6.68\times 10^{-8}\text{ s}\), the frequency is one over this or about \(1.5\times 10^{7}\text{ Hz}\), or \(15\text{ MHz}\). Since \(c = 3\times 10^{8}\text{ m/s}=\lambda f\), this means that \(\lambda = c/f = 20\text{ m}\), which is a radio wave.

The vectors along the \(z\) axis show the electric field along this path. What does the electric field in the \(xy\) plane look like for a particular value of \(z\)? Remember it is a plane wave. Move the square and notice that all points within the square have the same electric field, hence the name electromagnetic plane wave.

Notice that the wave equation for a pressure wave, \(P(x, t) = A \sin(k x -\omega t)\), traveling in the \(x\) direction could be changed to describe this electromagnetic plane wave (traveling in the \(z\) direction) as \(\mathbf{E}(z, t) = E_{\text{max}} \sin (k x -\omega t)\mathbf{i}\). Why does the electric field vector have a component in the \(x\) direction but not in the \(z\) direction? Maxwell's equations tell us that the electromagnetic wave is a transverse wave. Therefore, unlike the pressure wave, the electromagnetic wave cannot have a component in the direction of propagation.

Note that \(k = 2π/\lambda\) and \(\omega = 2\pi f\) so that \(v = \omega /k =\lambda f\), where \(v\) is the wave speed, \(\lambda\) is the wavelength, and \(f\) is the frequency.

Illustration authored by Melissa Dancy and Wolfgang Christian.

Illustration 4: Electromagnetic Waves, \(E\times B\)

This Illustration shows a representation of a simple sinusoidal electromagnetic wave that includes both the electric and magnetic fields versus \(z\) (the direction of travel). When the wave is traveling in the positive \(z\) direction, these two fields can be written as

\[E_{x} = E_{0} \sin (k z -\omega t),\quad E_{y} = 0,\quad E_{z} = 0,\nonumber\]

\[B_{x} = 0,\quad B_{y} = B_{0}\sin (k z -\omega t),\quad\text{and}\quad B_{z} = 0,\nonumber\]

where \(E_{0} = c B_{0}\) and \(c\) is the speed of light in MKS units.

The electric field is measured along the \(z\) axis and drawn as a red line; the magnetic field is drawn as a green line. The length of these lines represents each field's magnitude. Although we have not drawn an arrowhead, each line represents a vector whose direction is away from the \(z\) axis along the line. Click-drag to the right or left to rotate the animation about the \(z\) axis. Click-drag up or down to rotate in the \(xy\) plane.

Many students misunderstand what is represented in this Illustration and think that the fields extend in the \(x\) and \(y\) directions in the same way a wave on a rope would. In other words, students often believe that the fields only extend a finite distance in the \(xy\) plane like the wave on a string. The representation actually tells you a field's strength only at different points along the \(z\) axis. However, with an electromagnetic plane wave traveling in the \(z\) direction, the field is uniform in the \(xy\) plane. You may want to review the field representations in Illustration 32.3 to clarify this important concept.

Here we draw electric and magnetic vectors of equal length at points along the \(z\) axis. This is a bit of a misrepresentation. Electric and magnetic fields are measured in different units and their numeric values are, in fact, not equal in the MKS system. However, the energy carried by the electric field is equal to the energy carried by the magnetic field, and so most textbooks draw the vectors with equal lengths.

This Illustration also points out an important relationship between electric and magnetic fields. The \(\mathbf{E}\) and \(\mathbf{B}\) fields in an electromagnetic wave are in phase. Because one of these fields determines the other, we often only discuss one of them. We usually choose \(\mathbf{E}\).

Finally, there is an important relationship between \(\mathbf{E}\) and \(\mathbf{B}\) and the direction of propagation. The direction of travel is determined by \(\mathbf{E}\times\mathbf{B}\). In other words, \(\mathbf{E}\) and \(\mathbf{B}\) are perpendicular and the direction of propagation is given by the right-hand rule. You should confirm this cross-product relationship for yourself in the two animations.

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.