24.4 Energy in electromagnetic waves  (Page 2/6)

The average intensity of an electromagnetic wave $I_{\text{ave}}$ can also be expressed in terms of the magnetic field strength by using the relationship $B=E/c$ , and the fact that ${\epsilon }_0=1/{\mu }_0{c}^2$ , where ${\mu }_0$ is the permeability of free space. Algebraic manipulation produces the relationship

where $B_0$ is the maximum magnetic field strength.

One more expression for $I_{\text{ave}}$ in terms of both electric and magnetic field strengths is useful. Substituting the fact that $c\cdot B_0=E_0$ , 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, $I_0=2I_{\text{ave}}$ .

Test prep for ap courses

Section summary

• The energy carried by any wave is proportional to its amplitude squared. For electromagnetic waves, this means intensity can be expressed as
  $I_{\text{ave}}=\frac{{\mathrm{c\epsilon }}_0{E}_0^2}{2},$

  where $I_{\text{ave}}$ is the average intensity in ${\text{W/m}}^2$ , and $E_0$ is the maximum electric field strength of a continuous sinusoidal wave.

• This can also be expressed in terms of the maximum magnetic field strength $B_0$ as
  $I_{\text{ave}}=\frac{{\text{cB}}_0^2}{{\mathrm{2\mu }}_0}$

  and in terms of both electric and magnetic fields as

  $I_{\text{ave}}=\frac{E_0{B}_0}{{\mathrm{2\mu }}_0}.$
• The three expressions for $I_{\text{ave}}$ are all equivalent.