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The average intensity of an electromagnetic wave can also be expressed in terms of the magnetic field strength by using the relationship , and the fact that , where is the permeability of free space. Algebraic manipulation produces the relationship
where is the maximum magnetic field strength.
One more expression for in terms of both electric and magnetic field strengths is useful. Substituting the fact that , the previous expression becomes
Whichever of the three preceding equations is most convenient can be used, since they are really just different versions of the same principle: Energy in a wave is related to amplitude squared. Furthermore, since these equations are based on the assumption that the electromagnetic waves are sinusoidal, peak intensity is twice the average; that is, .
On its highest power setting, a certain microwave oven projects 1.00 kW of microwaves onto a 30.0 by 40.0 cm area. (a) What is the intensity in ? (b) Calculate the peak electric field strength in these waves. (c) What is the peak magnetic field strength ?
Strategy
In part (a), we can find intensity from its definition as power per unit area. Once the intensity is known, we can use the equations below to find the field strengths asked for in parts (b) and (c).
Solution for (a)
Entering the given power into the definition of intensity, and noting the area is 0.300 by 0.400 m, yields
Here , so that
Note that the peak intensity is twice the average:
Solution for (b)
To find , we can rearrange the first equation given above for to give
Entering known values gives
Solution for (c)
Perhaps the easiest way to find magnetic field strength, now that the electric field strength is known, is to use the relationship given by
Entering known values gives
Discussion
As before, a relatively strong electric field is accompanied by a relatively weak magnetic field in an electromagnetic wave, since , and is a large number.
An old microwave oven outputs only half the electric field it used to. How much longer does it take to cook things in this microwave oven?
(a)
Describe at least two improvements you could make to a radar set to make it more sensitive (able to detect things at longer ranges). Explain why these would work.
where is the average intensity in , and is the maximum electric field strength of a continuous sinusoidal wave.
and in terms of both electric and magnetic fields as
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