Electromagnetic energy and momentum

The densities carried over from electrostatics and magnetostatics are u_E = ½ε₀E² and u_B = B²/(2μ₀), each in J m⁻³, and both apply point by point to any field, not only inside a capacitor or a solenoid. In a plane wave in vacuum the two are locked together, because E = cB with c = 1/√(ε₀μ₀). Substitute: u_B = B²/(2μ₀) = E²/(2μ₀c²) = ½ε₀E² = u_E. The fields are in phase, so the equality holds at every instant, not merely on average, and the total is u = ε₀E² = B²/μ₀. Sunlight above the atmosphere has E₀ = 1.01 × 10³ V m⁻¹, so the peak density is ε₀E₀² = 9.07 × 10⁻⁶ J m⁻³ and the cycle average is half of it, 4.54 × 10⁻⁶ J m⁻³. That is under 5 μJ in a cubic metre of full sunlight, which is why the energy matters only once you count how fast it moves through.

Define S⃗ = (1/μ₀) E⃗ × B⃗, in W m⁻²: the power crossing unit area, pointing along the flow. For a wave travelling along x̂ with E⃗ = E ŷ and B⃗ = B ẑ, the cross product gives S⃗ = (EB/μ₀) x̂, so the energy travels the way the wave does and the triad E⃗, B⃗, propagation is right-handed. Its size follows from the energy density: with B = E/c and c² = 1/(ε₀μ₀), S = EB/μ₀ = E²/(μ₀c) = ε₀cE² = uc, which is what you get by pushing a slab of density u past a plane at speed c. S⃗ is not limited to waves. Take a wire of radius a and length L carrying a steady current i with V across it: at the surface E = V/L runs along the wire and B = μ₀i/(2πa) wraps around it, so S⃗ points radially inward with magnitude Vi/(2πaL). Multiply by the side area 2πaL and the inflow is Vi — the dissipated power enters through the surface of the wire, not along the copper.

For E = E₀cos(kx − ωt), S = ε₀cE₀²cos²(kx − ωt), so S oscillates at twice the field frequency and never goes negative. No detector follows 5 × 10¹⁴ Hz, so what is measured is the average, and ⟨cos²⟩ = ½ gives I = ⟨S⟩ = ½ε₀cE₀² = E₀B₀/(2μ₀) = cε₀Eᵣₘₛ², with Eᵣₘₛ = E₀/√2. Run it backwards to size the fields behind a familiar intensity. The Sun radiates 3.85 × 10²⁶ W, and at r = 1.50 × 10¹¹ m the inverse-square spread gives I = 3.85 × 10²⁶ ÷ (4π(1.50 × 10¹¹)²) = 1.36 × 10³ W m⁻². Then E₀ = √(2I/(ε₀c)) = 1.01 × 10³ V m⁻¹ and B₀ = E₀/c = 3.38 μT — an order of magnitude below the Earth's static field, and an electric field far below the 3 × 10⁶ V m⁻¹ that breaks down air. A 5.0 mW pointer with a 1.0 mm beam concentrates more: I = 5.0 × 10⁻³ ÷ 7.85 × 10⁻⁷ = 6.4 × 10³ W m⁻², giving E₀ = 2.2 × 10³ V m⁻¹.

Maxwell's equations give a wave that brings momentum p = U/c along its direction with every energy U it delivers, so the momentum density is g = u/c = S/c², in kg m⁻² s⁻¹, and momentum crosses unit area at the rate S/c. The mechanism shows up in the force on a charge sitting in the beam. The electric field drives the charge along E⃗, giving it a velocity component parallel to E⃗; the magnetic force qv⃗ × B⃗ acting on that motion then points along the propagation direction, and it keeps pointing that way through both halves of the cycle, because v⃗ and B⃗ reverse together. Absorbing a wave therefore leaves the absorber with net forward momentum, not only with heat. For sunlight, g = 1.36 × 10³ ÷ (3.00 × 10⁸)² = 1.5 × 10⁻¹⁴ kg m⁻² s⁻¹ — nothing to a wall, but decisive for a micrometre dust grain, whose area-to-mass ratio lets radiation pressure compete with the Sun's gravity.

Momentum arriving per second per unit area is a pressure. A perfectly absorbing surface facing the beam takes I/c, and since I/c is also the mean energy density, the absorbing pressure equals ⟨u⟩. A perfect mirror at normal incidence returns the wave with reversed momentum, so it takes twice as much, 2I/c. Sunlight at Earth gives 1.36 × 10³ ÷ (3.00 × 10⁸) = 4.5 μPa absorbed and 9.1 μPa reflected, against 1.0 × 10⁵ Pa of atmosphere — smaller by a factor of 2 × 10¹⁰, which is why the effect was not measured until Lebedev, then Nichols and Hull, worked in evacuated chambers around 1900. Scale the area and it stops being negligible. A square-kilometre sail, perfectly reflecting and face-on at 1 AU, feels 2IA/c = 9.1 N. At an areal density of 10 g m⁻² that sail masses 1.0 × 10⁴ kg, so its acceleration is 9.1 × 10⁻⁴ m s⁻² — about 78 m s⁻¹ of speed gained per day, with no propellant spent.

Every relation above assumes a monochromatic plane wave in vacuum, with E = cB in phase and both fields transverse, meeting a flat target head-on. Loosen them one at a time. Turn the mirror through θ and it both intercepts less power and returns the momentum at an angle, so the pressure on its surface falls to 2I cos²θ/c — a quarter of the head-on value at θ = 60°. Real surfaces are neither black nor perfect mirrors: with specular reflectance R the pressure is (1 + R)I/c, and diffuse scattering redistributes the returned momentum rather than reversing it, so the factor sits below 2. Time-averaging needs many cycles inside the measurement window, which is free at 5 × 10¹⁴ Hz but not for a femtosecond pulse. A finite beam is only locally plane, so I varies across the profile. And inside a dielectric the split of energy and momentum between field and medium is still argued over — Abraham and Minkowski wrote different answers — which is why these results are quoted for vacuum.