# How does an electromagnetic wave move?

Somewhere I found the explanation that the EM fields create and destroy each other during the oscillation (I suppose by Faraday's law) and this makes the wave "move". I can't imagine this because unitary vectors in E,B and k directions are a right hand ordered set of vectors and by the fact of E and B are in phase.

So, this is my question, what is the reason because the wave moves?

You've got half of the answer right when mentioning Faraday's Law of Induction, which tells us how electric fields can be generated from time-varying magnetic fields. The other half of the answer involves Ampère's Law with Maxwell's correction, which tells us how magnetic fields can be generated either from an electric current or time-varying electric fields. This coupled interaction between the electric and magnetic fields allows the EM waves to propagate.

The previously stated claim is typically proven mathematically in most standard textbooks written on classical electromagnetism, and so such derivations can readily be found in the appropriate literature. I will give one such derivation here, adapted from the Wikipedia article that you can and should read in order to learn more about EM radiation in fuller detail:

Consider Maxwell's Equations in the microscopic form (in SI units),

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} \tag{1}$$

$$\nabla \cdot \mathbf{B} = 0 \tag{2}$$

$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \tag{3}$$

$$\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \tag{4}$$

with (3) and (4) being, respectively, the Maxwell-Faraday Equation and the Maxwell-Ampère equation.

In free space (i.e. in a location that contains no electrical charges or currents), the equations take the form,

$$\nabla \cdot \mathbf{E} = 0 \tag{5}$$

$$\nabla \cdot \mathbf{B} = 0 \tag{6}$$

$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \tag{7}$$

$$\nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \tag{8}$$

Taking the curl of (7), we end up with,

\begin{align} \nabla \times \left(\nabla \times \mathbf{E}\right) & = \nabla \times \left(-\frac{\partial \mathbf{B}}{\partial t}\right) \\ & = - \frac{\partial}{\partial t} \left(\nabla \times \mathbf{B}\right) \\ & = - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} \end{align} \tag{9}

where we have made a substitution in the last step using (8).

From vector calculus, for a vector field $\mathbf{F}$, $\nabla \times \left(\nabla \times \mathbf{F}\right) = \nabla \left(\nabla \cdot \mathbf{F} \right) - \nabla^2 \mathbf{F}$, where $\nabla^2$ is the vector Laplacian operator, is an identity. Using this identity to rewrite (9),

$$\nabla \left(\nabla \cdot \mathbf{E} \right) - \nabla^2 \mathbf{E} = - \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} \tag{10}$$

But since we're in free space, by equation (5) the first term on the left-hand side of (10) vanishes, and after redefining $\mu_0 \epsilon_0 = 1/c^2$ we're left with,

$$\nabla^2 \mathbf{E} - \frac{1}{c^2} \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0 \tag{11}$$

which is clearly the wave equation with the electric field $\mathbf{E}$ as the dependent function.

Taking the curl of (8) and performing an almost identical procedure, the wave equation for the magnetic field $\mathbf{B}$ can also be derived,

$$\nabla^2 \mathbf{B} - \frac{1}{c^2} \frac{\partial^2 \mathbf{B}}{\partial t^2} = 0 \tag{12}$$

The fact that Maxwell's Equations in free space directly lead to these wave equations shows that the coupling between electric and magnetic fields described by Faraday and Ampère directly allows for the forward propagation of electromagnetic waves.

• Yes I know, but what is the physically meaning? I already read a fragment of the wiki article and it says that both not create each other, they just are there oscillating in the same way. Are there a physical reason for the wave movement? Dec 18 '17 at 14:13
• Where does it say that the electric and magnetic fields of the wave are not creating and sustaining each other?
– JM1
Dec 18 '17 at 14:26
• You can search: "..This relationship between the two occurs without either type field causing the other; rather, they occur together in the same way that time and space changes occur together and are interlinked in special relativity." Dec 18 '17 at 16:30
• I see. What that passage is hinting at is that electric fields and magnetic fields don't actually exist; there is only the electromagnetic field, the components of which we commonly refer to as the electric "field" and the magnetic "field". In that case, it is in fact misleading to say that the electric field and the magnetic field of an EM wave create each other.
– JM1
Dec 19 '17 at 3:19
• However, it is perfectly valid to say that the electric and magnetic components of the EM field are coupled to each other, and this coupling means that a local disturbance in the EM field (e.g. the acceleration of a charged particle) cannot influence the behavior of all other locations of the EM field instantaneously; the disturbance instead can only influence the field in the local neighborhood, and so the disturbance will be observed as a propagating EM wave, similar to how a guitarist plucking a guitar string will result in a wave that propagates down its length.
– JM1
Dec 19 '17 at 3:25

An electromagnetic wave is a change in the electromagnetic field of an object. The change in its electromagnetic field can only propagate at the speed of light. If the sun disappeared, it would take 499 seconds for the change in the gravitational field to propagate to earth and earth to go off at a trajectory; at the same time, earth would go dark due to the change in the electromagnetic field also reaching it at the same time. The further away from the source of the electromagnetic field, the weaker the field gets, so eventually the electromagnetic wave becomes too weak to detect. When an electromagnetic field is not changing, it does not induce a current in an antenna because the antenna is a capacitor, and current only flows when the voltage (charge of the capacitor) is changing, which is why voltage leads current by 90° – it is an AC circuit.

The change in the electromagnetic field strength clearly propagates as a wave outwards at the speed of light when you think about it. The field itself is made of photons.