Energy and Momentum of EM Waves: Formulas and Pressure

Electromagnetic waves carry energy and momentum even though they have no mass. Sunlight warms your skin because its waves deliver energy, and the tiny push of that light is called radiation pressure.

Understanding energy and momentum of EM waves explains solar heating, laser cooling, satellite effects and why comet tails point away from the Sun. It also connects field strength to the power a receiver can collect.

Key Concepts

Energy density u is energy per unit volume, measured in J/m^3. For an EM wave in vacuum the electric and magnetic parts contribute equally on average. Intensity I is power per unit area, measured in W/m^2. Momentum density is momentum per unit volume carried by the wave. Radiation pressure is the force per unit area exerted when the wave is absorbed or reflected. A perfectly absorbing surface feels pressure I / c, while a perfectly reflecting surface feels 2I / c because momentum reverses.

Formulas

u = (1/2) ε0 E^2 + B^2/(2 μ0) in J/m^3. For a plane wave in vacuum this simplifies using E = c B. Average intensity I = (1/2) c ε0 E0^2 where E0 is amplitude in V/m. Momentum of a photon p = E_photon / c = h / λ in kg m/s. Radiation pressure for absorption P = I / c in pascals (Pa), and for perfect reflection P = 2I / c. These assume normal incidence on a flat surface in vacuum.

Worked Example

Sunlight at Earth has intensity about 1000 W/m^2 on a clear day at normal incidence. Find pressure on a black absorbing panel and force on 2 m^2. Step 1: P = I / c = 1000 / (3 × 10^8) = 3.33 × 10^-6 Pa. Step 2: Force F = P × A = 3.33 × 10^-6 × 2 = 6.66 × 10^-6 N. Step 3: If the panel were a perfect mirror, pressure doubles to 6.66 × 10^-6 Pa and force to 1.33 × 10^-5 N. The force is tiny, but over large solar sails and long times it can change a spacecraft path.

Common Misconceptions

• Light has no mass, so students think it has no momentum. EM waves carry momentum proportional to energy divided by c.
• Radiation pressure is not the same as air pressure from heating. It exists even in vacuum.
• Reflecting light does not reduce pressure. Perfect reflection gives twice the momentum change of absorption.

Real-World Applications

• Solar sails use radiation pressure for slow, fuel-free spacecraft propulsion.
• Optical tweezers trap tiny particles using momentum transfer from focused laser beams.
• Accurate satellite orbit corrections include the small push of sunlight.

EM waves transport energy at intensity I and carry momentum that produces pressure I/c on absorption and 2I/c on reflection. The effect is small but real and useful from solar power to precision instruments.