EM Wave Solutions
3D E, B, Poynting vector. Builds on Maxwell's Equations.
An electromagnetic wave is traveling in the +z direction. The electric field oscillates in the x-direction. You want to know where the energy is going.
In which direction does electromagnetic energy flow?
Energy flows in the z-direction — along the Poynting vector S = (1/μ₀)(E × B)! Even though E points in x and B points in y, their cross product points in z, which is exactly the direction the wave propagates. The magnitude |S| = E₀B₀/(2μ₀) gives the intensity (power per unit area). The fields oscillate sideways, but the energy marches straight ahead.
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Watch E (red), B (blue), and S (yellow) vectors as the wave propagates. E and B oscillate perpendicular to each other. The Poynting vector S always points in the direction of travel. Toggle between 3D view and field-line view.
S = (1/μ₀)(E × B)
Energy flows perpendicular to both E and B. The Poynting vector is the electromagnetic equivalent of ‘which way is the energy going?’ — and its magnitude tells you how much.
The Sun’s Poynting vector at Earth’s distance gives the solar constant: ~1361 W/m². This is the power per area that drives all weather, photosynthesis, and solar panels.
The Poynting vector field around a transmitting antenna reveals the radiation pattern — where energy goes and where it doesn’t. Dish antennas focus S into a narrow beam.
Tightly focused laser beams can trap microscopic particles. The gradient of the Poynting vector creates forces that push particles toward the beam focus — a Nobel Prize-winning technique.
The Poynting vector carries momentum p = S/c² per unit volume. A reflective sail doubles the momentum transfer. Japan’s IKAROS spacecraft demonstrated solar sailing in 2010.
“E and B dance sideways, but the energy marches straight ahead. The Poynting vector is the compass of electromagnetic energy flow.”
EM Wave Solution Lab
Visualize 3D electromagnetic wave solutions. See E, B, and the Poynting vector in real time. Explore linear, circular, and elliptical polarization. Examine energy density and momentum.
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Start with a linearly polarized wave and watch E, B, and S. Rotate the view to see all three vectors in 3D. Now switch to circular polarization — E and B rotate as the wave propagates, but S still points straight ahead! Check the energy density display: half the energy is always in E and half in B. Try increasing the amplitude and watch the Poynting vector (energy flow) grow as E².
EM waves are self-sustaining oscillations of E and B fields. Energy flows along S = E × B/μ₀, always perpendicular to both fields. The wave carries energy, momentum, and angular momentum (for circular polarization) — all encoded in the field geometry.