Introduction: Electromagnetic Energy Flow and the Poynting Vector

The Poynting vector describes how electromagnetic energy flows through space. Whenever an electric field E and a magnetic field B coexist, they can transport energy from one region to another. The Poynting vector points in the direction of that energy flow and has a magnitude equal to the power crossing a unit area, measured in watts per square metre (W/m²).

This calculator lets you input the magnitudes of the electric and magnetic fields and the angle between them. It then returns the magnitude of the Poynting vector, i.e., the electromagnetic energy flux density. Below you will find the core formulas, a step‑by‑step explanation of how to interpret the result, a worked example, a comparison table, and a brief discussion of limitations and assumptions.

Poynting Vector Formula and Units

In classical electrodynamics, the Poynting vector $S$ is defined as

$S = \frac{1}{μ_{0}} E \times B$

where:

$E$ is the electric field (in volts per metre, V/m),
$B$ is the magnetic flux density (in tesla, T),
$μ_{0}$ is the permeability of free space (vacuum permeability),
$\times$ denotes the vector cross product.

The magnitude of the Poynting vector is

$S = \frac{E B \sin θ}{μ 0}$

Here $E$ and $B$ are the field magnitudes, and $θ$ is the angle between the directions of $E$ and $B$ (between 0° and 180°). The calculator uses exactly this magnitude formula.

The same relationship can be expressed using MathML as:

S = \frac{E B \sin θ}{μ}) 0

In SI units, the permeability of free space is approximately

$μ 0 \approx 4 π \times 10^{- 7} H/m \approx 1.25663706 \times 10^{- 6} N/A 2 .$

With these units, the result for $S$ is in watts per square metre (W/m²), representing power per unit area carried by the electromagnetic field.

How the Calculator Uses the Formula

The calculator implements the magnitude formula for the Poynting vector in vacuum:

You enter the electric field magnitude E in V/m.
You enter the magnetic field magnitude B in T.
You enter the angle between E and B, in degrees, between 0° and 180°.
The calculator converts the angle to radians internally and evaluates the sine.
It multiplies E, B, and sin(θ), and then divides by $μ_{0}$ .
The final value is displayed as the magnitude of the Poynting vector in W/m².

When $θ = 90 °$ (fields perpendicular), the sine is 1 and the formula simplifies to

$S = \frac{E B}{μ} .$

This perpendicular case is especially important for plane electromagnetic waves in free space, where $E$ and $B$ are mutually perpendicular and also perpendicular to the direction of propagation.

Interpreting the Poynting Vector Result

The numerical result from the calculator tells you how much electromagnetic power passes through each square metre perpendicular to the direction of energy flow.

Units: The value is in W/m². For example, a value of 500 W/m² means that 500 joules of energy pass through each square metre every second.
Direction: Although the calculator reports only the magnitude, the full vector points in the direction of $\vec{E} \times \vec{B}$ . You can determine the direction using the right‑hand rule: point your index finger along $E$ , your middle finger along $B$ , and your thumb points along $S$ .
Dependence on angle: The magnitude scales with $\sin θ$ . It is zero when $θ = 0^{°}$ or 180° (fields parallel or antiparallel) and maximal when the fields are perpendicular.

In practice, typical magnitudes can range from a few W/m² for weak radio fields, to about 1000 W/m² for sunlight at Earth’s surface, to extremely large values near powerful microwave or laser sources.

Worked Example: Approximate Sunlight at Earth’s Surface

As a concrete example, consider electromagnetic radiation from the Sun at the top of Earth’s atmosphere. The average solar irradiance (the solar constant) is about 1360 W/m². For simplicity, suppose we approximate this with our Poynting vector formula and assume perpendicular fields ( $θ = 90 °$ ).

Step 1: Choose a representative electric field amplitude

For a plane wave in vacuum, the electric and magnetic field amplitudes are related by

$E = c B,$

where $c \approx 3.00 \times 10^{8} m/s$ is the speed of light. If we start from a rough electric field amplitude of $E \approx 780 V/m$ (a commonly cited order of magnitude for sunlight in space), then the corresponding magnetic field is

$B = \frac{E}{c} \approx \frac{780}{3.00 \times 10^{8}} \approx 2.6 \times 10^{- 6} T .$

Step 2: Compute the Poynting vector magnitude

With $θ = 90 °$ , we have $\sin θ = 1$ and

$S = \frac{E B}{μ}_0 .$

Substituting the values:

$S \approx \frac{780 \times 2.6 \times 10^{- 6}}{1.2566 \times 10^{- 6}} .$

The numerator is

$780 \times 2.6 \times 10^{- 6} \approx 2028 \times 10^{- 6} \approx 2.03 \times 10^{- 3},$

and the denominator is $1.2566 \times 10^{- 6}$ . The ratio is approximately

$S \approx \frac{2.03 \times 10^{- 3}}{1.2566 \times 10^{- 6}} \approx 1.6 \times 10^{3} W/m^{2},$

so about 1600 W/m², which is in the same order of magnitude as the known solar constant of about 1360 W/m².

Step 3: Interpretation

This result tells us that, in space just outside the atmosphere, sunlight carries on the order of a thousand watts of power per square metre. This is why solar panels exposed directly to the Sun can generate significant electrical power per unit area.

Comparison: Special Cases of the Poynting Vector

The simple cross‑product formula for the Poynting vector can be applied in several idealised situations. The table below compares three common cases that are useful for interpreting the calculator output.

Scenario	Field Relationship	Angle $θ$	Approximate Formula for $S$	Typical Magnitude (W/m²)
Plane wave in vacuum (perpendicular fields)	$E = c B$ , uniform, $E ⊥ B$	90°	$S = \frac{E B}{μ_{0}} = \frac{E^{2}}{μ_{0} c}$	$\sim 10^{3}$ for sunlight near Earth
Fields not exactly perpendicular	Uniform magnitudes, arbitrary orientation	0°–180°	$S = \frac{E B sin θ}{μ 0}$	Between 0 and the perpendicular‑field value
Parallel or antiparallel fields (idealised)	$E ∥ B$ or $E ↑ ↓ B$	0° or 180°	$S = 0$ from the cross‑product magnitude	Zero in this ideal limit; in reality, other effects may matter

In many practical teaching examples, a plane wave in free space is assumed, which corresponds to the first row. The calculator is directly suited to that case when you set the angle to 90° and provide compatible values of E and B (for instance, using $E = c B$ ).

Assumptions and Limitations

While the cross‑product formula for the Poynting vector is general in Maxwell’s theory, using it as implemented in this calculator rests on several simplifying assumptions. Being aware of these helps you avoid misinterpreting results.

Key assumptions

Vacuum permeability: The calculation uses the permeability of free space, $μ_{0}$ . This is appropriate for electromagnetic waves in vacuum or air at ordinary conditions, where the magnetic properties are very close to those of free space.
SI units: Inputs are assumed to be in SI units: V/m for electric field, T for magnetic field, and degrees for the angle. The output is in W/m².
Uniform fields: The formula implicitly assumes that the electric and magnetic fields are uniform over the area of interest, so that a single pair of magnitudes and a single angle are meaningful.
Classical regime: The description is based on classical electrodynamics. Quantum effects (such as photon counting) are not included.

When the simple formula may not fully apply

Complex media: In lossy, dispersive, or strongly magnetic materials, the relationship between $E$ , $B$ , and energy flow can be more complicated. Effective permeabilities and permittivities may be frequency‑dependent.
Near‑field regions: Close to antennas, coils, or resonant structures, the fields can be reactive and not purely radiative. In such near‑field zones, the time‑averaged Poynting vector does not always correspond directly to net power transported away.
Waveguides and cavities: In guided‑wave structures (waveguides, resonators, optical fibres), boundary conditions and modes shape the field distribution. Local values of the Poynting vector can vary significantly across the cross‑section.
Strong sources and nonlinear media: Extremely intense fields can lead to nonlinear effects in materials, again altering the simple linear relationships used in the basic formula.

In these more advanced situations, the calculator can still provide intuition, but precise engineering or research work typically requires solving Maxwell’s equations with appropriate boundary conditions and material models.

Summary

The Poynting vector encapsulates the flow of electromagnetic energy through space. By combining the magnitudes of the electric and magnetic fields and the angle between them, you can estimate the power per unit area being carried by an electromagnetic field configuration. The calculator on this page implements the standard magnitude formula using vacuum permeability and SI units, making it a practical tool for students, educators, and engineers who need quick estimates of energy flux in idealised conditions.

Poynting Vector Calculator

Introduction: Electromagnetic Energy Flow and the Poynting Vector

Poynting Vector Formula and Units

How the Calculator Uses the Formula

Interpreting the Poynting Vector Result

Worked Example: Approximate Sunlight at Earth’s Surface

Step 1: Choose a representative electric field amplitude

Step 2: Compute the Poynting vector magnitude

Step 3: Interpretation

Comparison: Special Cases of the Poynting Vector

Assumptions and Limitations

Key assumptions

When the simple formula may not fully apply

Summary

Embed this calculator

Poynting Vector Calculator

Introduction: Electromagnetic Energy Flow and the Poynting Vector

Poynting Vector Formula and Units

How the Calculator Uses the Formula

Interpreting the Poynting Vector Result

Worked Example: Approximate Sunlight at Earth’s Surface

Step 1: Choose a representative electric field amplitude

Step 2: Compute the Poynting vector magnitude

Step 3: Interpretation

Comparison: Special Cases of the Poynting Vector

Assumptions and Limitations

Key assumptions

When the simple formula may not fully apply

Summary

Embed this calculator

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