5.4 Energy density of an electromagnetic wave

Energy density

The electric and magnetic fields have energy and hence have an energy density. We can see this for a capacitor:The energy stored in a capacitor is $$U=\frac{1}{2}C{V}^2$$ where $C$ is the capacitance and $V$ the potential drop (voltage) across the capacitor. For a parallel plate capacitor $C=\frac{{\epsilon }_{0}A}{d}$ and $V=Ed$ where $A$ is the area of the plates $d$ the distance between them and $E$ the electric field strength.note that $Ad$ is the volumeThus $$U=\frac{1}{2}\frac{{\epsilon }_{0}A}{d}{(Ed)}^2=\frac{1}{2}{\epsilon }_{0}Ad{E}^2$$ So we can write the energy density (Energy per Unit volume) of the field as $$u_E=\frac{U}{Ad}=\frac{1}{2}{\epsilon }_0{E}^2$$ Likewise by calculating the energy stored by a B-field in a current carrying solenoid one can derive: $$u_B=\frac{B^2}{2{\mu }_0}$$ Since we know $E=cB$ $${\displaystyle \begin{array}{c}{u}_E=\frac{1}{2}{\epsilon }_0{E}^2\\ =\frac{1}{2}{\epsilon }_0{c}^2{B}^2\\ =\frac{1}{2}{\epsilon }_0\frac{1}{{\epsilon }_0{\mu }_0}{B}^2\\ =\frac{1}{2}\frac{1}{{\mu }_0}{B}^2\\ =u_B\end{array}}$$ In an EM wave $u=u_E+u_B$ which is $u={\epsilon }_0{E}^2$ or equivalently $u=B^2/{\mu }_0$