Poynting Vector Calculator
The Poynting vector S = E × H is the power per unit area carried by an electromagnetic field, in W/m². Its direction is the direction of energy flow. Enter any two of S, E and H and the third is solved. Add the angle between the fields, the wave impedance, the amplitude type and an area to see average power density, energy density, radiation pressure and total power.
S = E × H|S| = E · H · sin θ⟨S⟩ = ½ E₀H₀Step-by-step solution
Companion quantities
Electromagnetic wave
E (teal) and H (amber) oscillate at right angles. E × H points along the direction of travel, so S is along the propagation axis.
Power density versus E field (plane wave)
Reference data
| E field | H field | |S| (W/m²) | Comparison |
|---|
| Situation | Power density | Notes |
|---|---|---|
| Solar constant above the atmosphere | 1361 W/m² | Total across all wavelengths |
| Sunlight at the ground, clear sky | about 1000 W/m² | Peak at noon |
| Microwave oven leakage limit | 50 W/m² (5 mW/cm²) | Regulatory limit near the door |
| RF exposure reference level, public | 10 W/m² | Above about 2 GHz, ICNIRP guideline |
| RF exposure reference level, workers | 50 W/m² | Above about 2 GHz, ICNIRP guideline |
| Medium-wave broadcast signal (1 mV/m) | 2.7 × 10⁻⁹ W/m² | Typical usable field strength |
Values are approximate and for orientation only. For regulatory limits, check the current standard for the frequency range concerned.
FAQ
What does the Poynting vector tell us?
It gives the direction and rate of energy flow per unit area in an electromagnetic field. Integrating S over a surface gives the total power passing through it. For a plane wave, S points in the direction the wave travels.
Why is the average half of the product of the peaks?
For a sinusoidal wave the instantaneous value E·H varies as sin²(ωt), averaging ½. So with peak amplitudes E₀ and H₀ the time-averaged power density is ½E₀H₀. With RMS values the average is simply Erms·Hrms.
What is the relation between E and H in free space?
For a plane wave in free space E/H = η₀ ≈ 376.73 Ω, so H = E / 376.73. Then S = E²/η₀ = η₀H². Near an antenna or inside a waveguide the ratio differs from η₀, which is why the wave-impedance check is shown.
Does light carry momentum?
Yes. An absorbing surface feels a radiation pressure ⟨S⟩/c, and a perfect reflector feels twice that. Sunlight at 1000 W/m² gives about 3.3 µPa on an absorber, which is why solar sails are so large.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| S | Poynting vector (power per unit area) | W/m² |
| E | Electric field vector | V/m |
| H | Magnetic field vector | A/m |
What it means
The Poynting vector S = E × H represents the instantaneous power flow per unit area in an electromagnetic field. Its direction gives the direction of energy flow, and its magnitude is the power density (W/m²). It is named after John Henry Poynting. This vector is crucial in understanding the propagation of electromagnetic waves, the flow of energy in transmission lines, and the operation of antennas. In a plane wave, S = E² / η₀, where η₀ is the intrinsic impedance of free space (about 377Ω). The Poynting vector also helps explain how energy is transferred from a source to a load in circuits, showing that energy flows in the space around the conductors, not inside them. Example: In a uniform plane wave with electric field amplitude 100 V/m, the power density is S = E²/377 = 10000/377 ≈ 26.5 W/m². This is the intensity of the wave.
Worked example
Poynting Vector – Practical Example
Real‑World| Parameter | Value |
|---|---|
| E | 100 V/m |
| H | 0.25 A/m |
| Formula | S = E × H |
Common mistakes
- Cross product: S = E × H – direction is given by the right‑hand rule.
- Units: W/m² (power per unit area).
- Instantaneous: The Poynting vector gives the instantaneous power flow.
- Average: For sinusoidal fields, the average power is the real part of the complex Poynting vector.
- Conservation: The divergence of S is related to energy dissipation and field energy change.
Applications
The Poynting vector S = E × H represents the directional energy flux density of an electromagnetic field, indicating the magnitude and direction of power flow per unit area. This is fundamental for understanding wave propagation, power transmission, and radiation. Engineers use it to calculate the power radiated by antennas, to analyse waveguide transmission, and to design microwave circuits. By determining the Poynting vector, they can quantify the energy transfer in electromagnetic systems and ensure that components can handle the power flow. This concept is essential for electromagnetics and telecommunications.
- Antenna radiation pattern and power density calculations
- Waveguide and transmission line power analysis
- Microwave circuit design and power handling
- Electromagnetic compatibility and interference studies
- Educational understanding of energy flow in fields
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