The third wave restates the right hand curl rule that a current loop produces a B field perpendicular to the loop; a loop in the anti-clockwise sense produces a B field toward the observer.
From the previous post "Magnetic Genie",
∇×Y=−4π∇ρe−4πc2∂je∂t
If c is upwards, then ic is to the left. Since ∇×Y is in anti-clockwise, then ∇ρe and ∂je∂t are both in the clockwise sense because of the negative sign. This directions are consistent with the right hand rule for finding magnetic field due to a current loop.
A rotating disc with a certain non uniform charge distribution will generate a EMW when rotated...
∇×Y=−4π∇ρe
The presence of ρe creates an E field,
∇.E=4πρe
A rotating Y generates a changing B field and a rotating E field,
−∇×E=−i1c∂B∂t+Y
So,
−∇(∇.E)+∇2E=−i1c∂∇×B∂t+∇×Y
−∇(4πρe)+∇2E=1c∂(−i∇×B)∂t−4π∇ρe
But a rotating B in turn generates a changing E field,
−i∇×B=1c∂E∂t
And we have
∇2E=1c∂∂t{1c∂E∂t}
∇2E=1c2∂2E∂t2
An E field wave!
Now consider a flat coil spiral disc or a loop carrying currents...
∇×Y=−4π∇ρe−4πc2∂je∂t
we will now set −4π∇ρe=0 and consider only the flow of je
∇×Y=−4πc2∂je∂t
We know that je creates a changing E field and around je a circular B field,
−i∇×B=1c∂E∂t+4πcje
Differentiating wrt t,
∂∂t{−i∇×B}=1c∂2E∂t2+4πc∂je∂t --- (*)
The vortex of Y creates a changing B field and an E field along it circular path,
−∇×E=−i1c∂B∂t+Y
−∇(∇.E)+∇2E=−i1c∂∇×B∂t+∇×Y
But ∇.E=4πρe
−∇(4πρe)+∇2E=1c∂(−i∇×B)∂t−4πc2∂je∂t
Substitute (*) into the above and since we set ∇ρe=0,
∇2E=1c2∂2E∂t2
And we have a wave.
In both cases we can independently create a EMW, which is expected from superposition.
