Question

To show how a propagating triangle electromagnetic wave can satisfy Maxwell's equations if the wave travels at speed c.

Light, radiant heat (infrared radiation), X rays, and radio waves are all examples of traveling electromagnetic waves. Electromagnetic waves consist of mutually compatible combinations of electric and magnetic fields ("mutually compatible" in the sense that changes in the electric field generate the magnetic field, and vice versa).
The simplest form for a traveling electromagnetic wave is a plane wave. One particularly simple form for a plane wave is known as a "triangle wave," in which the electric and magnetic fields are linear in position and time (rather than sinusoidal). In this problem we will investigate a triangle wave traveling in the x direction whose electric field is in the y direction. This wave is linearly polarized along the y axis; in other words, the electric field is always directed along the y axis. Its electric and magnetic fields are given by the following expressions:
Ey(x,t)=E0(x−vt)/a and Bz(x,t)=B0(x−vt)/a,
where E0, B0, and a are constants. The constant a, which has dimensions of length, is introduced so that the constants E0 and B0 have dimensions of electric and magnetic field respectively. This wave is pictured in the figure at time t=0. (Figure 1) Note that we have only drawn the field vectors along the x axis. In fact, this idealized wave fills all space, but the field vectors only vary in the x direction.
We expect this wave to satisfy Maxwell's equations. For it to do so, we will find that the following must be true:
The amplitude of the electric field must be directly proportional to the amplitude of the magnetic field.
The wave must travel at a particular velocity (namely, the speed of light).
Part A
What is the propagation velocity of the electromagnetic wave whose electric and magnetic fields are given by the expressions in the introduction? Express in terms of v and the unit vectors. The answer will NOT involve c; we have not yet shown that this wave travels at the speed of light.
Part B
To use Faraday's law for this problem, you will need to construct a suitable loop, around which you will integrate the electric field. In which plane should the loop lie to get a nonzero electric field line integral and a nonzero magnetic flux?
a) xy
b) yz
c) zx
Part F
To use the Amprere-Maxwell law you will once again need to contract a suitable loop, but this time you will integrate the magnetic field line integral and hence nonzero electric flux?
a) xy
b) yz
c)zx

Answer 1

The propagation velocity of the electromagnetic wave is v = ac. The loop for Faraday's law should lie in the xy plane, and the loop for Ampere-Maxwell law should lie in the yz plane.

Step-by-step explanation:

Part A: The propagation velocity of the electromagnetic wave can be determined from the given expressions for the electric and magnetic fields. We can see that both fields have the form (x - vt)/a. This means that for the wave to propagate at speed c, we must have (v/a) = c. Therefore, the propagation velocity is v = ac.

Part B: To use Faraday's law, we need to construct a loop that will enclose an area where the electric field lines change. In this case, the electric field is in the y direction, so the loop should lie in the xy plane (option a).

Part F: To use the Ampere-Maxwell law, we need to construct a loop that will enclose an area where the magnetic field lines change. In this case, the magnetic field is in the z direction, so the loop should lie in the yz plane (option b).