# Electromagnetic Waves

## Energy in Electromagnetic Waves


### Learning Objectives

By the end of this section, you will be able to:
1. Explain how the energy and amplitude of an electromagnetic wave are related.
2. Given its power output and the heating area, calculate the intensity of a microwave oven’s electromagnetic field, as well as its peak electric and magnetic field strengths
Anyone who has used a microwave oven knows there is energy in electromagnetic waves. Sometimes this energy is obvious, such as in the warmth of the summer sun. Other times it is subtle, such as the unfelt energy of gamma rays, which can destroy living cells.
Electromagnetic waves can bring energy into a system by virtue of their electric and magnetic fields. These fields can exert forces and move charges in the system and, thus, do work on them. If the frequency of the electromagnetic wave is the same as the natural frequencies of the system (such as microwaves at the resonant frequency of water molecules), the transfer of energy is much more efficient.
But there is energy in an electromagnetic wave, whether it is absorbed or not. Once created, the fields carry energy away from a source. If absorbed, the field strengths are diminished and anything left travels on. Clearly, the larger the strength of the electric and magnetic fields, the more work they can do and the greater the energy the electromagnetic wave carries. 
A wave’s energy is proportional to its amplitude squared ( or ). This is true for waves on guitar strings, for water waves, and for sound waves, where amplitude is proportional to pressure. In electromagnetic waves, the amplitude is the maximum field strength of the electric and magnetic fields. (See .) 
Thus the energy carried and the intensity  of an electromagnetic wave is proportional to  and . In fact, for a continuous sinusoidal electromagnetic wave, the average intensity  is given by
where  is the speed of light,  is the permittivity of free space, and  is the maximum electric field strength; intensity, as always, is power per unit area (here in ).
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, .

### Test Prep for AP Courses


### Section Summary

1. The energy carried by any wave is proportional to its amplitude squared. For electromagnetic waves, this means intensity can be expressed as
    where 
    
2. This can also be expressed in terms of the maximum magnetic field strength  as
    and in terms of both electric and magnetic fields as
    
3. The three expressions for  are all equivalent.

### Problems & Exercises
