KAHIBARO
Table of Contents
Energy flow in electromagnetic fields
The Poynting vector describes how electromagnetic energy moves through space. Electric and magnetic fields do not only exert forces, they also store and transport energy. The Poynting vector gives the direction of that transport and the rate of energy flow per unit area.
It is defined as
$$
\mathbf{S} = \frac{1}{\mu_0}\,\mathbf{E} \times \mathbf{B}
$$
in vacuum, where $\mathbf{E}$ is the electric field, $\mathbf{B}$ is the magnetic field, and $\mu_0$ is the permeability of free space.
The direction of $\mathbf{S}$ is perpendicular to both $\mathbf{E}$ and $\mathbf{B}$, following the right hand rule for the cross product. Its magnitude tells us how much electromagnetic energy passes each second through a unit area placed perpendicular to the flow.
The Poynting vector in vacuum is
$$
\mathbf{S} = \frac{1}{\mu_0}\,\mathbf{E} \times \mathbf{B}
$$
Its direction is the direction of electromagnetic energy transport.
Its unit is
$$
\mathrm{W/m^2}
$$
which means power per unit area.
Physical meaning
To understand the Poynting vector, imagine a beam of light moving through space. Light is an electromagnetic wave, so it has electric and magnetic fields oscillating in time. Even though the fields change, the wave carries energy from its source to another place. The Poynting vector points along the direction in which that energy travels.
If electromagnetic energy crosses a surface of area $A$, then the power passing through the surface is related to $\mathbf{S}$. For a surface whose area vector is $d\mathbf{A}$, the small power crossing it is
$$
dP = \mathbf{S} \cdot d\mathbf{A}
$$
and the total power is
$$
P = \int \mathbf{S} \cdot d\mathbf{A}
$$
This dot product shows that only the component of $\mathbf{S}$ perpendicular to the surface contributes to energy transfer through that surface.
Units of the Poynting vector
Since the Poynting vector measures energy flow per unit time per unit area, its unit is watt per square meter. This can also be seen from the formula. The result matches the idea of intensity for electromagnetic radiation.
| Quantity | Meaning | SI unit |
|---|---|---|
| $\mathbf{S}$ | Energy flow per unit area per unit time | $\mathrm{W/m^2}$ |
| $P$ | Power crossing a surface | $\mathrm{W}$ |
| $d\mathbf{A}$ | Oriented surface element | $\mathrm{m^2}$ |
Direction of energy flow
Because $\mathbf{S}$ is a cross product, the energy flow is perpendicular to both fields. In an electromagnetic wave in vacuum, the electric field, magnetic field, and direction of propagation are all mutually perpendicular.
If the wave travels in the $x$ direction, with $\mathbf{E}$ in the $y$ direction and $\mathbf{B}$ in the $z$ direction, then
$$
\mathbf{S} \propto \mathbf{E} \times \mathbf{B}
$$
points in the $x$ direction.
Poynting vector in electromagnetic waves
For a plane electromagnetic wave in vacuum, the fields satisfy
$$
E = cB
$$
where $c$ is the speed of light. Then the magnitude of the Poynting vector is
$$
S = \frac{1}{\mu_0}EB
$$
Using $B = E/c$, this becomes
$$
S = \frac{E^2}{\mu_0 c}
$$
Since
$$
\frac{1}{\mu_0 c} = \varepsilon_0 c
$$
we may also write
$$
S = \varepsilon_0 c E^2
$$
or, in terms of the magnetic field alone,
$$
S = \frac{c}{\mu_0} B^2
$$
These formulas are useful for electromagnetic waves such as radio waves, visible light, and X rays.
For a plane electromagnetic wave in vacuum,
$$
S = \frac{1}{\mu_0}EB = \varepsilon_0 c E^2 = \frac{c}{\mu_0}B^2
$$
and the energy flows in the direction of wave propagation.
Instantaneous and average energy flow
In an electromagnetic wave, the fields usually oscillate with time, so the Poynting vector also oscillates. Its value at a particular moment is the instantaneous energy flux.
For sinusoidal waves, the average value over one full cycle is often more useful. If $E_0$ and $B_0$ are the peak values, then the average magnitude of the Poynting vector is
$$
\langle S \rangle = \frac{1}{2\mu_0} E_0 B_0
$$
Using $E_0 = cB_0$, this can be written as
$$
\langle S \rangle = \frac{1}{2}\varepsilon_0 c E_0^2
$$
This average value is the intensity of the wave.
For a sinusoidal electromagnetic wave, the time averaged Poynting vector magnitude is
$$
\langle S \rangle = \frac{1}{2}\varepsilon_0 c E_0^2
$$
This average corresponds to the wave intensity.
Example of interpretation
Suppose sunlight arrives at a surface with average intensity $1000\ \mathrm{W/m^2}$. This means the average magnitude of the Poynting vector is about
$$
\langle S \rangle = 1000\ \mathrm{W/m^2}
$$
If the surface area is $2\ \mathrm{m^2}$ and it faces the Sun directly, then the power received is
$$
P = SA = 1000 \times 2 = 2000\ \mathrm{W}
$$
If the surface is tilted, then less power passes through it because the dot product with the area vector becomes smaller.
Relation to energy conservation
The Poynting vector is part of the energy conservation law for electromagnetism. Electromagnetic fields contain energy, and that energy can move from one region to another. The Poynting vector tells us how the energy flows.
A full conservation statement combines field energy density, energy flow, and work done on charges. At a beginner level, the key idea is simple: when electromagnetic energy leaves one region and enters another, the Poynting vector tracks that transfer.
Common situations
In circuits, energy does not travel only inside the wire. The electric and magnetic fields around the wire can carry energy from the source to the load, and the Poynting vector describes that flow.
In electromagnetic radiation, such as light from a lamp or radio waves from an antenna, the Poynting vector points away from the source and shows how energy spreads through space.
| Situation | Meaning of $\mathbf{S}$ |
|---|---|
| Light beam | Energy carried in direction of beam |
| Radio wave | Energy transmitted from antenna |
| Surface in sunlight | Power received per unit area |
| Region around a circuit | Energy flow through surrounding fields |
Summary formulas
The most important formulas for the Poynting vector are collected here.
Definition in vacuum:
$$
\mathbf{S} = \frac{1}{\mu_0}\,\mathbf{E} \times \mathbf{B}
$$
Power through a surface:
$$
P = \int \mathbf{S} \cdot d\mathbf{A}
$$
Plane wave magnitude:
$$
S = \frac{1}{\mu_0}EB = \varepsilon_0 c E^2 = \frac{c}{\mu_0}B^2
$$
Time average for a sinusoidal wave:
$$
\langle S \rangle = \frac{1}{2}\varepsilon_0 c E_0^2
$$
Final intuition
The Poynting vector is the electromagnetic version of a flow arrow for energy. It tells us where the energy is going, and how much passes through each square meter each second. In a light wave, it points in the direction the light travels. In general, wherever electric and magnetic fields exist together, the Poynting vector helps describe the transport of electromagnetic energy.