Representation of electromagnetic wave?

Question

In Representation of electromagnetic waves we show both electric and magnetic fields perpendicular to each other and having zero magnitude at one point and maximum at other point. This representation and understanding of EM waves violating law of conservation of energy...

Answer 1

The law of conservation of energy does not specify that the energy in each point in space should remain constant. If it was the case, not much would be happening in the world.

EM waves, as other types of waves, carry energy through the space and, in the process, the energy in each point on its path is constantly changing. However, at any given point in time, the total energy at all points in space will be equal to the energy generated by the source of the wave by that time.

So the representation of an EM wave you describe, although obviously simplified, does not violate the law of conservation of energy.

Answer 2

The Poynting vector $\mathbf S = \frac{1}{\mu_0}\mathbf E \times \mathbf B$ is a measure of the energy flux density (energy per unit area per unit time).

Consider a plane polarised electromagnetic wave travelling in the positive z-direction such that $\mathbf S(z,t) = \frac{1}{\mu_0}E_{\rm x}(z,t) B _{\rm y}(z,t) \hat z$.

At position $z=z_0$ at time $t=t_0$ the electric and magnetic fields are zero so

$S(z_0,t_0)=0$
and just before and just after that position
$S(z_0-\Delta z,t_0)=\frac{1}{\mu_0}E_{\rm x}(z_0-\Delta z,t_0) B _{\rm y}(z_0-\Delta z,t_0) $
$S(z_0+\Delta z,t_0)=\frac{1}{\mu_0}E_{\rm x}(z_0+\Delta z,t_0) B _{\rm y}(z_0+\Delta z,t_0) $.

And this is where you feel that energy is not being conserved as energy is being transported at $z=z_0-\Delta z$, not being transported at $z=z_0$ and again being transported at $z=z_0+\Delta z$.

However see what happens at those three positions a little later at time $t= t_0 + \Delta t$

$S(z_0-\Delta z,t_0+\Delta t)=\frac{1}{\mu_0}E_{\rm x}(z_0-\Delta z,t_0+\Delta t) B _{\rm y}(z_0-\Delta z,t_0+\Delta t) $
$S(z_0,t_0+ \Delta t)=\frac{1}{\mu_0}E_{\rm x}(z_0,t_0+\Delta t) B _{\rm y}(z_0,t_0+\Delta t) $
$S(z_0+\Delta z,t_0)=0$

with $E_{\rm x}(z_0-\Delta z,t_0) B _{\rm y}(z_0-\Delta z,t_0) = E_{\rm x}(z_0,t_0+\Delta t) B _{\rm y}(z_0,t_0+\Delta t) $

So the energy which was being transported at position $z=z_0-\Delta z$ at time $t=t_0$ is now being transported at position $z=z_0$ at time $t=t_0+\Delta t$ and the zero energy being transported which was at $z=z_0$ and $t=t_0$ is now at $z=z_0+\Delta z$ at time $t=t_0+\Delta t$.

Think of it as a line of people who alternatively have a box (energy) and no box.
Those with a box pass it to their right which means that some will transfer a box (energy) whilst others will not transfer a box but the number of boxes (energy) stays constant.

Answer 3

What is mixing here are EM radiation, radio waves and photons. Light from a thermic source is not a wave. It is a stream of photons, emitted by excited atoms. Not any characteristic of a wave could be observed.

What Heinrich Hertz observed were radio waves, be this from a spark generator or a radio generator. In this cases in a antenna rod electrons get accelerated forth and back synchronously and by this emitting photons in some periodicity. Of cours at some moments the emission of photons is zero. This happens at every moment the electrons change their acceleration from negative to positive and from positive to negative. The Pointing vector is good for counting of the energy from the electric and the magnetic field components of the radio wave.

All the accelerated electrons in the antenna rod emit photons with the same phase of their electric and magnetic field components. To say that the photons are coherent is not precise because there are emitted photons of different wavelengths (even dependent from the length of the rod in relation to the frequency of the wave generator, so a radar could emit X-rays and infrared radiation at the same time).

The interesting thing is that in the near field of the antenna the electric and the magnetic field components are exactly shifted by 90°. From this one could conclude that the emitted photons are moving forwards in the same manner.