## Illustration 32.4: Electromagnetic Waves, E x B

Please wait for the animation to completely load.

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_{o} sin (k z - ω t), E_{y} = 0, E_{z} = 0,

B_{x} = 0, B_{y} = B_{o} sin (k z - ω t), and B_{z} = 0,

where E_{o} = c B_{o} 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 **E** and **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 **E**.

Finally, there is an important relationship between **E** and **B** and the direction of propagation. The direction of travel is determined by **E** x **B**. In other words, **E** and **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.

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