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12.7 Implication of Maxwell's Equations

If we take the curl of both sides of the first equation we get

B +E /ct = j / c

and substitute using the third, we get

B +2B/(ct)2 =j / c

The double cross product identity applied to the operator "del" yields

B  = (B) - ()B

Applying this identity and the second equation we get

(-2/(ct)2)B = -j / c

The analogous sequence of steps starting with the third equation gives us                         

( -2 / (ct)2)E = + (1/c) j / ct

In the absence of charge and current, the right hand sides of these equations vanishes, and both E and B obey the homogeneous wave equation. Hertz wrote down these equations and suggested that electromagnetic waves could move through empty space, and that oscillating currents in a wire should be able to produce such waves.

Marconi tested this hypothesis by setting up an oscillating circuit connected to an antenna on land, with another on a ship off shore. He did indeed send waves from one "receivable" by the other. He had invented the radio. Within thirty years radios were everywhere, electronics had developed great sophistication, and television was invented.
All of this came from Hertz' theoretical observations based on Maxwell's equations. All of which could barely be imagined without the divergence theorem and Stokes' theorem, which played a significant role in their discovery.