91Ó°ÊÓ

The electric field is a fundamental concept in electromagnetism. It describes the force that a charged particle experiences in space. In an electromagnetic wave, the electric field oscillates perpendicular to the direction of wave propagation. This means if a wave moves in the +z direction, as in the given exercise, the electric field can be directed along the x-axis (or -x-axis).

To visualize it, imagine holding a pencil representing the propagation direction. If you extend your fingers perpendicular to the pencil, that's the direction the electric field might point. The electric field's magnitude tells us the strength of this push or pull a charged particle would feel. In this exercise, the electric field has a magnitude of 4.00 V/m. This indicates the potential energy difference that exists per unit charge in that field.

Keep in mind, the electric field and magnetic field together form the electromagnetic wave, and they are always at right angles to each other and to their direction of motion.

The magnetic field is another crucial component of electromagnetic waves. Alongside the electric field, it helps in transferring the energy in these waves. In a traveling electromagnetic wave, the magnetic field oscillates perpendicular to the electric field and the direction of the wave's travel.

From the exercise, the electromagnetic wave is moving along the +z axis, and the electric field is in the +x direction. The magnetic field (B) must, therefore, be in the y-direction to maintain perpendicularity and agree with the right-handed coordinate system. The relation of B with E is governed by the speed of light, given by the formula:

• \[ c = \frac{E}{B} \]

Rearranging this allows us to find B:

• \[ B = \frac{E}{c} \]

When we plug in the values, B becomes approximately 1.33 x 10^(-8) T, indicating a rather small but significant component of the wave.

The right-handed coordinate system is a helpful method to determine directions in a three-dimensional space, especially when dealing with vectors. It is called so because it uses the orientation of your right hand to deduce directions. For electromagnetic waves, this system is imperative for finding the magnetic field direction when the electric field and the direction of wave propagation are known.

If you curl the fingers of your right hand from the positive x direction (where the electric field is in the exercise) to the positive z direction (the direction of wave propagation), your thumb will point in the direction of the positive y-axis where the magnetic field exists. This right-hand rule ensures consistency and allows us to visualize how the fields relate without error. This is why the magnetic field is concluded to be in either the +y or -y direction, fitting snuggly into this spatial orientation.

The speed of light is a universal constant (denoted as \(c\)), and it plays a crucial role in the nature of electromagnetic waves. Light travels in a vacuum at approximately 3.00 x 10^8 m/s, a speed that ties together the electric and magnetic fields in these waves.

In the context of this exercise, understanding the speed of light helps in calculating the magnetic field's magnitude. As mentioned earlier, the relationship between the electric field (E) and the magnetic field (B) is given by:

• \[ c = \frac{E}{B} \]

This equation shows the direct proportionality of E and B with light's speed, illustrating how tightly they are interlinked. Faster or slower speeds of these fields wouldn't comply with the rigid framework of electromagnetism defined by the speed of light, highlighting its essential role in consistently determining the behavior of electromagnetic waves.