# What does it physically mean to take the curl of the curl of a field (wave equation derivation)?

What does it physically mean to take the curl of the curl of a field in the derivation of the electromagnetic wave equation from Maxwell's equations, as presented here, on Wikipedia? Why was it a logical first step to take?

• The curl of the curl is a mathematical move with no physica context. It was a logical context because then the laplacian of the electric and magnetic fields could be isolated to derive the wave equations. – user70720 Mar 22 '15 at 23:54
• The divergence of a curl is zero, and the divergence and the curl are the two most common differential operators for vector fields, so it's not crazy to try it and see what you get. – Timaeus Mar 22 '15 at 23:55
• Ahh. I wondered if it might have been as simple as that, but thought it worth checking if there was significance I was missing. Thank you both! – perilousGourd Mar 22 '15 at 23:58

## 1 Answer

From the point of view of electromagnetism I can see it the following non-mathematical way.

$curl$ or $rot$ differential operators calculates magnitude of "vorticity" of the field, i.e. how much it is "spinning" or changing in rotation.

Now, "vorticity" of the "vortex" is how much this "vortex" spinning again. Take, for example, long spring and connect its ends as torus. Here you will measure "vorticity" of the "vortex".

This model presumably can be applied to the $E$ and $B$ field in electromagnetism to demonstrate its interdependence and nature of EMR in individual $E$ and $B$ field components.