31.6.2 Radiation Pressure

Electromagnetic waves transfer not only energy but also momentum. An object gains momentum when it absorbs electromagnetic waves, much as a ball at rest gains momentum when struck by a ball in motion.

Suppose we shine a beam of light on an object that completely absorbs the light energy. If the object absorbs energy during a time interval $Δ t$ , its momentum changes by

Δ p = \frac{energy absorbed}{c}

This is a consequence of Maxwell’s theory, which we’ll state without proof.

The figure illustrates a photograph of conception of a future spacecraft powered by radiation pressure from the sun. — Artist’s conception of a future spacecraft powered by radiation pressure from the sun.

The momentum change implies that the light is exerting a force on the object. Newton’s second law, in terms of momentum, is $F = Δ p / Δ t$ . The radiation force due to the beam of light is

F = \frac{Δ p}{Δ t} = \frac{(energy absorbed) / Δ t}{c} = \frac{P}{c}

where P is the power (joules per second) of the light.

It’s more interesting to consider the force exerted on an object per unit area, which is called the radiation pressure $p_{rad}$ . The radiation pressure on an object that absorbs all the light is

p_{rad} = \frac{F}{A} = \frac{P / A}{c} = \frac{I}{c}

(31.42)

where I is the intensity of the light wave. The subscript on $p_{rad}$ is important in this context to distinguish the radiation pressure from the momentum p.

EXAMPLE 31.5 Solar sailing

A low-cost way of sending spacecraft to other planets would be to use the radiation pressure on a solar sail. The intensity of the sun’s electromagnetic radiation at distances near the earth’s orbit is about $1300 W / m^{2}$ . What size sail would be needed to accelerate a 10,000 kg spacecraft toward Mars at $0.010 m / s^{2} ?$

MODEL Assume that the solar sail is perfectly absorbing.

SOLVE The force that will create a $0.010 m / s^{2}$ acceleration is $F = m a = 100 N$ . We can use Equation 31.42 to find the sail area that, by absorbing light, will receive a 100 N force from the sun:

A = \frac{c F}{I} = \frac{(3.00 \times 10^{8} m / s) (100 N)}{1300 W / m^{2}} = 2.3 \times 10^{7} m^{2}

REVIEW If the sail is a square, it would need to be $4.8 km \times 4.8 km$ , or roughly $3 mi \times 3 mi$ . This is large, but not entirely out of the question with thin films that can be unrolled in space. But how will the crew return from Mars?