Coaxial Poynting Power Flow
Locate electrical power in the fields between coaxial conductors and integrate its flow.
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Coaxial Poynting Power Flow Theory
Locate electrical power in the fields between coaxial conductors and integrate its flow.
Context
In a slowly varying transverse electromagnetic (TEM) coaxial cable with an inner conductor of radius a and an outer conductor of radius b, the electric field is radial and the magnetic field is azimuthal. The Poynting vector, which describes the direction and magnitude of electromagnetic energy flux, is purely axial.
Instead of just associating power with current in the wires, classical electromagnetism localizes the transported energy entirely within the dielectric fields between the conductors. The energy flux density is given by Sz = Er Hφ.
Worked Example
Consider an ideal TEM coaxial cable with an inner radius a = 1 mm (0.001 m) and an outer radius b = R × a, where the radius ratio R = 4, so b = 4 mm. The cable carries a voltage V = 4 V and a current I = 1 A. We want to find the fields and power flow at a probe radius located halfway logarithmically between the conductors, corresponding to a log-radius fraction u = 0.5.
| Quantity | Calculation | Result |
|---|---|---|
| Probe radius r | r = a × Ru = 0.001 × 40.5 m | 2.0 mm |
| Electric field Er | Er = V / [r ln(b/a)] = 4 / [0.002 × ln(4)] | 1442.6950408889634 V/m |
| Magnetic field Hφ | Hφ = I / (2π r) = 1 / (2π × 0.002) | 79.57747154594766 A/m |
| Axial energy flux Sz | Sz = Er × Hφ = 1442.695 × 79.577 | 114806.02356582128 W/m² |
| Total Integrated field power | V × I = 4 × 1 | 4 W |
| Power inside probe | V × I × u = 4 × 1 × 0.5 | 2.0 W |
Experiment
In the interactive module, you can vary the parameters to observe how they affect the energy flux.
- Change the Radius ratio, Log-radius fraction, Voltage, and Current using the real controls provided in the Coaxial Poynting Power Flow module.
- Observe how the electric and magnetic fields change with radius, and how their cross product (the Poynting vector) varies across the annular cross-section.
- Notice that reversing the sign of the current reverses the direction of power flow, and a zero current condition means no transported power despite the presence of electric energy.
Limitations
This model assumes a perfect-conductor/TEM idealization. It treats the fields as slowly varying (quasi-static) and ignores high-frequency waveguide modes or radiation. It is important to remember that there is no claim that electrons transport power solely in the metal; the power is carried by the electromagnetic fields in the dielectric.
Primary References
See Oersted to Waveguides for a detailed derivation of the fields and Poynting vector in a coaxial cable.
Experience it interactively
Adjust parameters, observe in real time, and build deep intuition with Riano’s interactive Coaxial Poynting Power Flow module.
Try Coaxial Poynting Power Flow on Riano →