Magnet Field and Electric field representation image misunderstanding

Question

I have a question about the similar images below that I found on the internet about Magnet field vs Electric field. Electric fields should be generated by changing magnetic fields.

Question: Why is in this and similar images, the greatest value of the Magnetic field corresponds to the greatest value of the Electric field ?

I am asking, because the changing Magnetic field is absent at maximum amplitude of Magnetic field, so Electric field must be zero. And the maximum value of the Electric field must be at the point where changing of the Magnetic field is maximum, at zero value.

For me Electric field is derivative of Magnetic field.

I think Electric field must be shifted by 90 degrees relative to Magnetic field along the direction of radiation propagation.

Like generating electrical field ( voltage ) in magnet and moving copper coil: quicker movements - bigger voltage ( current ).

I found similar question, but there is no answer for me. EM Waves, Kinks, and the interaction of Electric and Magnetic Fields

Thank you in advance.

Answer 1

This is possible because the electric field is not directly proportional to the rate of change of the magnetic field (and vice versa). Rather, the curl of the electric field is proportional to the rate of change of the magnetic field: $$\nabla \times \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}$$ $$\nabla \times \mathbf{B}=\frac{1}{c^2}\frac{\partial \mathbf{E}}{\partial t}$$ (in a source-free region).

Edit Using the definition of curl, we can find the curl of $E$ and $B$ in the image you posted: $$\nabla\times \mathbf{E}=\frac{\partial E_z}{\partial y}\hat{\mathbf{x}} \text{, and } \nabla\times \mathbf{B}=-\frac{\partial B_x}{\partial y}\hat{\mathbf{z}}.$$ One more thing to note: your image does not show variations in time. It's a snapshot of the field at a moment in time, so there's nothing to indicate the rate of change of the fields with respect to time.

Answer 2

The conventional wisdom about electric and magnetic fields generating each other is, to some extent, completely wrong (but the nevertheless, useful, if you don't think about it too deeply).

If you look at the formulation of electromagnetism known as Jefimenko's Equations:

$${\bf E}({\bf r}, t)=\frac 1 {4\pi\epsilon_0}\int\Big[ \frac{{\bf r}-{\bf r}'}{|{\bf r}-{\bf r}'|^3}\rho({\bf r}',t_r)+ \frac{{\bf r}-{\bf r}'}{|{\bf r}-{\bf r}'|^2}\frac 1 c\frac{\partial\rho({\bf r}',t_r)}{\partial t}- \frac 1 {|{\bf r}-{\bf r}'|}\frac 1 {c^2}\frac{\partial{\bf J}({\bf r}',t_r)}{\partial t} \Big]d^3{\bf r}'$$

$${\bf B}({\bf r}, t)=-\frac {\mu_0} {4\pi}\int\Big[ \frac{{\bf r}-{\bf r}'}{|{\bf r}-{\bf r}'|^3}\times {\bf J}({\bf r}',t_r)+ \frac{{\bf r}-{\bf r}'}{|{\bf r}-{\bf r}'|^2}\times \frac 1 c\frac{\partial{\bf J}({\bf r}',t_r)}{\partial t} \Big]d^3{\bf r}'$$

you will see the the only things that generates an electric field at ${\bf r}, t$ are charge density, changing charge density, and changing current, and all occurring at a different location, ${\bf r}'$, in the past:

$$t_r = t-\frac {|{\bf r}-{\bf r}'|} c $$

Likewise, a magnetic field is caused by current, and changing current.

It's just that the dynamic electric and magnetic field are generated in such away that the time derivative of one is proportional to the curl of the other, even if the sources existed millions of light years aways, millions of years ago.

In a plane wave (propagating in the $z$-direction), that means:

$$ \frac{\partial {\bf E}}{\partial t} \propto \frac{\partial {\bf B}}{\partial z}$$

and

$$ \frac{\partial {\bf B}}{\partial t} \propto \frac{\partial {\bf E}}{\partial z}$$

That is, they are in-phase.