What Can You Use Instead of the Time-Averaged Poynting Vector?

The Poynting vector describes the instantaneous flow of electromagnetic energy. If you want a different quantity, first decide what you are trying to measure: steady power flow, total energy delivered by a pulse, or the changing flow at each moment. These are related, but they are not interchangeable.

In vacuum, the instantaneous Poynting vector is $\mathbf{S}(t)=\mathbf{E}(t)\times\mathbf{H}(t)=\mathbf{E}(t)\times\mathbf{B}(t)/\mu_0$. Its units are watts per square metre. In a material, the macroscopic expression is usually written $\mathbf{E}\times\mathbf{H}$.

Choose the quantity that matches your goal

For a periodic field with period $T$, the cycle average is

$$\langle\mathbf{S}\rangle=\frac{1}{T}\int_{t_0}^{t_0+T}\mathbf{S}(t)\,dt.$$

This gives the net average energy-flow density over a cycle. It is the usual choice for a continuous sinusoidal wave when the rapid oscillations are not being measured individually.

For a short pulse or other nonperiodic field, use the time integral instead:

$$\mathbf{F}=\int_{-\infty}^{\infty}\mathbf{S}(t)\,dt.$$

This time-integrated Poynting vector has units of joules per square metre and gives the net vector energy transport per unit area over the chosen time interval. For a surface with unit normal $\hat{\mathbf{n}}$, the signed net energy crossing per unit area is $\int_{-\infty}^{\infty}\mathbf{S}(t)\cdot\hat{\mathbf{n}}\,dt$. For a pulse traveling in a fixed direction through a plane normal to that direction, this quantity is often called energy fluence. Positive and negative values represent net flow in opposite directions.

If you need to know how the flow changes during the pulse, retain $\mathbf{S}(t)$ rather than averaging it. If you only need a scalar measure of field strength, an average energy density or an RMS field may be useful—but neither is automatically the same as energy transported through a surface.

Harmonic fields: calculate the average with phasors

For sinusoidal fields, phasors avoid integrating the oscillations explicitly. Suppose the real fields are written using peak-amplitude phasors as $\mathbf{E}(t)=\operatorname{Re}(\tilde{\mathbf{E}}e^{-i\omega t})$ and $\mathbf{H}(t)=\operatorname{Re}(\tilde{\mathbf{H}}e^{-i\omega t})$. Then

$$\langle\mathbf{S}\rangle=\frac{1}{2}\operatorname{Re}\left(\tilde{\mathbf{E}}\times\tilde{\mathbf{H}}^{*}\right).$$

The star denotes complex conjugation. The factor of $1/2$ applies when the phasors represent peak amplitudes. If you use RMS phasors instead, omit that factor. When it is nonzero, the average vector points in the direction of net energy flow, and its magnitude is the average power crossing a unit area perpendicular to that flow.

Some texts also define a complex Poynting vector, $\frac{1}{2}\tilde{\mathbf{E}}\times\tilde{\mathbf{H}}^{*}$. Its real part gives average active power flow. Its imaginary part is associated with reactive power: energy that is exchanged locally with fields or stored and returned, rather than carried away as net average power. The imaginary part is not a substitute for the average Poynting vector when you want the net transported power.

Worked example: a sinusoidal plane wave

Consider a plane wave in vacuum traveling in the positive $z$ direction, with peak electric-field amplitude $E_0$ and electric field along $x$. The magnetic-field amplitude is $H_0=E_0/Z_0$, where $Z_0$ is the impedance of vacuum. Because the fields are in phase, the instantaneous flux is

$$\mathbf{S}(t)=\frac{E_0^2}{Z_0}\cos^2(\omega t-kz)\,\hat{\mathbf{z}}.$$

The average of $\cos^2$ over one cycle is $1/2$, so

$$\langle\mathbf{S}\rangle=\frac{E_0^2}{2Z_0}\,\hat{\mathbf{z}}.$$

For example, if $E_0=10\,\mathrm{V/m}$ and $Z_0\approx377\,\Omega$, the average flux is about $0.133\,\mathrm{W/m^2}$ in the positive $z$ direction. This is an average power per unit area, not the total energy of a finite pulse. To find that energy per area, integrate the pulse’s instantaneous flux over its duration.

Why averaging the magnitude can give the wrong answer

The average vector and the average magnitude answer different questions. The vector average accounts for direction, so flow in opposite directions can cancel. Averaging $|\mathbf{S}|$ discards that directional information and measures the average size of the instantaneous flux instead.

A standing wave illustrates the difference. It can have oscillating local energy flow even though its cycle-averaged net flow is zero. In that case, $\langle\mathbf{S}\rangle=\mathbf{0}$, while $\langle|\mathbf{S}|\rangle$ can be positive. If your question is how much net energy crosses a surface over time, use the signed normal component and integrate it; do not use the average magnitude as a replacement.

Common mistakes

• Confusing average power with total energy. The cycle average has units of power per area. Integrating flux over time gives energy per area.
• Dropping direction. A scalar average of $|\mathbf{S}|$ cannot distinguish energy moving forward from energy moving backward.
• Using the wrong phasor convention. Peak phasors require the factor $1/2$ in the average; RMS phasors do not.
• Treating the complex vector as wholly transported power. Its real part represents average active flow; the imaginary part describes reactive behavior.
• Assuming the time average is always appropriate. For pulses, transients, and measurements resolved within a cycle, use the instantaneous flux or the relevant time integral.