15.9: Electromagnetic Waves
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Maxwell predicted the existence of electromagnetic waves, and these were generated experimentally by Hertz shortly afterwards. In addition, the predicted speed of the waves was 3×108m s−1, the same as the measured speed of light, showing that light is an electromagnetic wave.
In an isotropic, homogeneous, nonconducting, uncharged medium, where the permittivity and permeability are scalar quantities, Maxwell's equations can be written
∇⋅E=ρ.
∇⋅H=0.
∇×H=ϵ˙E.
∇×E=−μ˙H.
These equations involve E, H, and t. Let us see if we can eliminate E and hence find an equation in just H and t.
Take the curl of equation 15.9.3, and make use of equation 15.6.4:
graddivH−∇2H=ϵ∂∂tcurlE
Substitute for div H and curlE from equations 15.9.2 and 15.9.4 to obtain
∇2H=ϵμ¨H
This is the equation in terms of just H and t that we sought.
Comparison with equation 15.1.2 shows that this is a wave of speed 1/√ϵμ (Verify that this has the dimensions of speed.)
In a similar manner the reader should easily be able to eliminate B to derive the equation
∇2E=ϵμ¨E
In a vacuum, the speed is 1/√ϵoμo. With μo=4π×10−7H m−1 and ϵo=8.854×10−12F m−1, this comes to 2.998×108m s−1.
Can we eliminate t from the equations, and hence obtain a relation between just E and H? If you do, you will obtain
EH=√μϵ,
which, in a vacuum, or free space, becomes
EH=√μoϵo=377Ω,
which is the impedance of a vacuum, or of free space. Since the SI units of E and H are, respectively V m-1 and A m-1, it is easy to verify that the units of impedance are V A-1, or Ω.