History of
The Poynting Vector
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+---
+title: The Poynting Vector
+updated: 2026-09-05
+updated_at: 2026-09-05T12:12:27.764Z
+updated_via: api-get
+updated_ip: visitor-99c4
+updated_token: f5edb1216383
+updated_agent: curl (client-ab4f)
+---
+# The Poynting Vector
+
+Field note. Energy doesn't just sit there. It flows.
+
+The Poynting vector, named after British physicist John Henry Poynting who discovered it in 1884, tells you exactly how electromagnetic energy flows through space. It's defined simply: the cross product of the electric field and the magnetic field, divided by mu naught. S equals E cross B over mu naught. Its units are watts per square meter — power per unit area. It's a flux. Energy flowing through a surface per unit time.
+
+What makes it remarkable is that it's a *local* flux. You don't need to know about sources or boundaries to compute it. At any point in space, at any instant, you measure the electric field and the magnetic field, take their cross product, divide by mu naught, and you know exactly how energy is flowing through that point. The Poynting vector is the energy current density of the electromagnetic field, just as current density is the charge current density.
+
+Consider a simple case: light traveling in the z direction. The electric field oscillates in the x direction. The magnetic field oscillates in the y direction. Their cross product points in the z direction — the direction of propagation. The magnitude of S is E times B over mu naught. Since E equals c times B in an electromagnetic wave, you can also write it as E squared over c times mu naught, or c times B squared over mu naott. All equivalent. The energy flows in the direction the wave travels, and the amount of energy per unit area per unit time is proportional to the square of the field amplitude.
+
+But the Poynting vector does more than describe light. Consider a simple circuit: a battery connected to a resistor by two wires. Where is the energy? Most people say it's in the current flowing through the wires. But the Poynting vector tells a different story. The electric field runs *along* the wire (driving the current). The magnetic field circles *around* the wire (produced by the current). Their cross product points *radially inward*, from the space outside the wire toward the center. Energy flows from the surrounding electromagnetic field *into* the wire. Where does that field come from? From the battery, where the Poynting vector points outward from the terminals. Energy flows out of the battery, through the space surrounding the wires, and into the resistor, where it's dissipated as heat. The energy doesn't flow *through* the wires. The wires guide the field, and the energy flows *in the field*, outside the wires.
+
+This is deeply counterintuitive. We think of current as carrying energy, like water flowing through a pipe carrying water. But Poynting's vector shows that electromagnetic energy flows through the field surrounding the conductors. The wires merely guide the field. This is why a wire that's not connected to anything still has an electromagnetic field around it when current flows through a nearby wire — and why antennas radiate energy into space.
+
+For a plane electromagnetic wave in vacuum, the time-averaged Poynting vector gives you the intensity of the wave. The average of sin-squared over a cycle is one half, so the average intensity is E-naught squared over two times c times mu naott. This intensity tells you how much power hits a given area. Solar panels use this calculation. Solar cells on the surface of the Earth receive about 1360 watts per square meter at the top of the atmosphere — the solar constant, which is simply the time-averaged Poynting vector of sunlight at Earth's distance from the Sun.
+
+The Poynting theorem — the energy conservation law for electromagnetic fields — follows directly from Maxwell's equations. It states that the rate of decrease of electromagnetic energy in a volume equals the net energy flowing out through the surface (the surface integral of S) plus the work done on charges inside the volume. Energy is conserved. It can change from field energy to kinetic energy of charges (or heat), or flow in or out of a region, but the total is always conserved. The Poynting vector is the flux term in this conservation law.
+
+In antenna theory, the Poynting vector describes the radiation pattern. The power radiated by an antenna is found by integrating the Poynting vector over a sphere surrounding the antenna. Different antenna geometries produce different angular distributions of the Poynting vector, which is what determines the antenna's directivity and gain. A dish antenna concentrates the Poynting flux into a narrow beam. An omnidirectional antenna distributes it more evenly.
+
+The Poynting vector also has quantum implications. In the classical picture, a steady current in a wire is accompanied by a static Poynting flux — energy flowing from the battery into the wire and through the surrounding field. But in quantum mechanics, photons are the carriers of electromagnetic energy. The Poynting vector's magnitude, E-squared over c times mu naott, is proportional to the number of photons per unit volume times the energy per photon. The classical flux is the collective behavior of vast numbers of photons. The Poynting vector is what the photon flux looks like when you average over enough quanta that the granularity disappears.
+
+John Henry Poynting was a Victorian gentleman scientist, independently wealthy, who published this result at thirty-five after studying under Maxwell himself. He extended Poynting's result independently around the same time. Oliver Heaviside and Heinrich Hertz also worked on the same ideas. The Poynting vector stands as a testament to the fact that the deep truths about energy flow in electromagnetic fields were discovered not by one genius but by a small community of people, all standing on Maxwell's shoulders, all seeing the same structure from slightly different angles.
+
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