# How is the combination of electric & magnetic waves (electromagnetic wave) illustrated as a single wave?

This is the common presentation of an electromagnetic wave:

But it can also be shown as a single wave with a single defined wavelength & amplitude :

I know that the wavelength shown above represents that of E & B fields (which are equal); so what about the amplitude?

The first diagram shows the values of E(t) & B(t) versus the direction of propagation; what does the second diagram show? (Does it show anything at all or is it only for simplification?)

The electromagnetic field contains transverse electric and magnetic fields. In your picture, the electric field is vibrating in a single plane. This confinement of electric field in a single plane is known as polarization. So the first picture depicts a polarized electromagnetic wave. The state of an electromagnetic field can be described by the state of polarization alone, as it is one of the degrees of freedom associated with an electromagnetic wave.

The polarization direction is given by the direction of electric field. Hence the second picture is drawn with the electric field intensity against time (or distance).

How the polarization (or the electric field) alone could give enough information about the electromagnetic wave?

They are electromagnetic waves and are hence always under the influence of Maxwell's equations. According to these four set of equations, the oscillating electric field (which you can create by oscillating charges) create oscillating magnetic fields. These oscillating magnetic fields in turn produce oscillating electric fields. This is how the wave propagates. Hence the electric and magnetic field strengths of an electromagnetic wave are connected and is given by

$$\frac{E}{B}=c$$

where $c$ is the speed of light in vacuum. At every instant, the ratio of the electric field to the magnetic field in an electromagnetic wave equals the speed of light ( provided our wave is travelling through vacuum; otherwise it will be $v$, the velocity of light through a particular medium). This came as a consistency requirement for the solutions to Maxwell's equations to be valid. In vectorial form, the above result is given as

\begin{align} \vec{E}(z,t)&=E_0\cos(kz-\omega t+\delta)\hat{x}\\ \vec{B}(z,t)&=B_0\cos(kz-\omega t+\delta)\hat{y}=\frac{1}{c}E_0\cos(kz-\omega t+\delta)\hat{y} \end{align}

where we have assumed that the wave propagates in the $z$ direction, the electric field oscillates in the $x$ direction (direction of polarization), and the magnetic field oscillates in the $y$ direction.

Hence once known the strength of the electric field and the polarization direction, there are information hidden regarding it's velocity in that medium and the magnetic field. Hence by this way you can represent the three dimensional monochromatic wave using a two dimensional diagram connecting the electric field strength and time (or distance). It's just a matter of simplification.

• Thank you for the answer. Please edit the x^ in the B equation; it should be y^. – S.Shahsiah Nov 8 '16 at 16:18