Electromagnetic waves > Properties of emission and reflection of electromagnetic waves > Problem 12.1.33

Statement

$12.1.33$ Use the law of conservation of energy to show that in a spherical wave radiated by a point source, the amplitude of the electric field strength and magnetic field induction of the wave decreases inversely with the distance from the source, if the wave energy is not absorbed by the medium.

Solution

The energy per unit area per unit time of a point source is given by the Poynting vector:

\begin{equation}
S = c \epsilon_0 E(r)
\end{equation}

The total power radiated by the source is:

\begin{equation}
P = S \cdot 4 \pi r^2 = 4 \pi c \epsilon_0 E^2(r) r^2
\end{equation}

We see that, except for $E(r)$ and $r$ itself, the other quantities are constants and do not depend on the distance from the source.
This result shows us that, in a non-dissipative medium, the energy in the wave will decrease with the distance from the source.

Answer

\begin{equation}
P = S \cdot 4 \pi r^2 = 4 \pi c \epsilon_0 E^2(r) r^2
\end{equation}