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