According to Maxwell Equations, how does the light travel straight line?

Question

According to Faraday's law of induction and Ampere's law, a changing magnetic field causes curl of electric field simultaneously; and also a changing electric field causes curl of magnetic field simultaneously. By combining these equations, the equation of electromagnetic wave is derived. I thought I understood completely until I realize I didn't get how the light travels straight line if one causes curl of another. I saw visualization of electromagnetic wave but still it looks like violating faraday law and Ampere law.

Answer 1

From the Maxwell's equation you can derive the wave equation, whose solutions are a propagating wave. Next, we need to define what is meant by propagation direction of the wave, and then look to see if it is, in fact, propagating along a straight line.

In the classic textbook examples, one assumes

1. free-space propagation
2. A single frequency, f, of the wave to simplify the time-dependent derivative term.

From the above, you end up with a simple ordinary differential equation that just depends on the spatial coordinates.

Next, look for plane wave solutions of the form: Exp[-i K x]. So, this gives rise to the notion of phase (or wave) front, and we can assign a wavevector (K) to this front that is orthogonal to E and B, but is parallel to the propagation direction. From here we can see that these plane wave solutions travel in straight lines.

Answer 2

The electromagnetic wave equation allows for plane wave solutions in the form of

$$ \vec{u}=\vec{u_0}\exp(i\vec{k}\cdot\vec{r}-\omega t), $$

and since the wave vector can always be chosen to travel in a single direction (by defining our reference frame conveniently), one can always rewrite $\vec{k}\cdot\vec{r}=kz$ (choosing $\vec{k}=(0,0,k)$). Then, it's clear that the waves propagate in the $z$ direction - in one direction. But these are not the only possible solutions. Other solutions may propagate as spherical waves, in all directions.