Balance momentum carried by radiation

The vacuum Poynting vector is energy flow per unit area per unit time. Divide it by speed squared to obtain field momentum per unit volume. For a traveling plane wave, momentum moves at c, so its flux in the propagation direction is intensity divided by c. One quantity is a volume density and the other is a force per area. Confusing them leaves a missing speed factor and turns pressure into the wrong physical dimensions.

For a finite object, the electromagnetic momentum balance includes the change of field momentum stored inside a surrounding volume and momentum crossing its surface. In a steady beam, the time-averaged storage term vanishes and the difference between incoming and outgoing momentum flux gives the force. For a switched pulse, integrate over the complete interaction and include any residual stored fields. A local intensity alone cannot determine force without the target's scattering and absorption behavior.

At normal incidence, label energy fractions absorbed, specularly reflected, and transmitted as A, R, and T. They sum to one for a passive target in the stated steady situation. Each absorbed unit of incident energy transfers momentum equal to that energy divided by c. A reflected unit reverses direction and transfers twice as much momentum. A transmitted unit continuing forward in vacuum leaves the same momentum as it brought and transfers none in this idealization.

The resulting pressure is the sum of absorption and twice reflection times incident intensity divided by c. The formula assumes normal incidence, an effectively stationary target, and forward transmission in the same exterior vacuum. Oblique incidence requires projections of both crossing rate and momentum, while diffraction or diffuse scattering requires integrating outgoing directions. For a moving mirror, Doppler shifts alter the energy balance. Do not transplant the stationary normal-incidence expression into those different geometries.

Radiation emitted by a source carries momentum direction by direction. Integrate the angular power weighted by its outward unit vector and divide by c to obtain the momentum-loss rate. A stationary fixed-axis electric dipole pattern is symmetric under reversal of direction, so opposite angular contributions cancel. It radiates energy while carrying zero net vector momentum in that frame. Nonzero total power alone therefore does not determine a recoil force or a preferred recoil direction.

For a normally incident pulse, total energy replaces intensity times area times duration in the impulse balance. Pressure during the pulse depends on its temporal shape, while the integrated impulse depends on the total momentum difference. The vector lab can illustrate addition and reversal of a directed momentum component, but it contains only one vector. Reflection coefficients, heating, forces, and the integration of an angular emission pattern must be calculated on paper rather than inferred from nonexistent readouts.