Problem 80 - University Physics with Modern Physics

Question

Consider a beam of monochromatic light with intensity \(I\) incident on a perfectly absorbing surface oriented perpendicular to the beam. Use the photon concept to show that the radiation pressure exerted by the light on the surface is given by \(I / c\) .

Step-by-Step Solution

Answer

The radiation pressure exerted by the light is \( \frac{I}{c} \).

Step 1: Understand Radiation Pressure

Radiation pressure is the pressure exerted upon any surface exposed to electromagnetic radiation. This pressure is a result of the momentum transferred to the surface by the photons in the light beam.

Step 2: Use the Photon Concept

Photons are particles of light, each carrying energy and momentum. For light of intensity \(I\), the photon rate (number of photons hitting the surface per second per unit area) can be calculated using the energy each photon carries. Since energy is inversely proportional to wavelength, each photon's energy \(E\) is \(E = hu\), where \(h\) is Planck's constant and \(u\) is frequency.

Step 3: Relate Intensity to Photon Number

The intensity \(I\) is the energy per unit area per unit time. Therefore, the number of photons per unit area per unit time, \(N\), is \( N = \frac{I}{hu} \).

Step 4: Calculate Momentum Transferred

Each photon carries a momentum \(p = \frac{hu}{c}\). For a perfectly absorbing surface, the momentum transferred to the surface is equal to the momentum of the photons hitting the surface.

Step 5: Derive the Radiation Pressure

The force exerted by the photons on the surface is equal to the change in momentum per unit time, which is \(F = N \cdot \frac{hu}{c} = \frac{I}{hu} \cdot \frac{hu}{c} = \frac{I}{c}\). Since pressure is force per unit area, the radiation pressure \(P\) is \(P = \frac{F}{A} = \frac{I}{c}\).

Key Concepts

Photon Energy

Photon energy is a fundamental concept when discussing light and its interactions. Photons are tiny particles of light, and each one carries a specific amount of energy. The energy of a photon is directly related to its frequency. The formula for photon energy is given by

• \( E = h u \),

where \( E \) is the photon energy, \( h \) is Planck's constant, and \( u \) is the frequency of the light. This relationship indicates that higher frequency light, which corresponds to shorter wavelengths, contains more energy per photon.
Understanding photon energy is crucial for analyzing light interactions, including radiation pressure, as it helps describe how light particles transfer their energy to surfaces upon impact.

Momentum Transfer

Momentum transfer occurs when photons strike a surface. Although photons have no mass, they do possess momentum, which is a product of their energy. The momentum \( p \) of a photon is given by the equation:

• \( p = \frac{E}{c} = \frac{h u}{c} \),

where \( c \) is the speed of light. When photons impact a surface, they exert a force by transferring momentum to the surface.
For a perfectly absorbing surface, this momentum transfer effectively stops the photon, allowing us to calculate a force corresponding to the radiation pressure. The total force depends on the number of photons hitting the surface and is responsible for the observable "push" or pressure that light can exert.
Understanding momentum transfer is essential for grasping the concept of radiation pressure.

Electromagnetic Radiation

Electromagnetic radiation is a form of energy that is emitted and absorbed by charged particles. It consists of waves of electric and magnetic fields oscillating at specific frequencies. Light is a familiar form of electromagnetic radiation, which travels as photons at the speed of light, \( c \).

• All electromagnetic waves carry energy and momentum regardless of their frequency or wavelength.

The behavior and properties of electromagnetic radiation are critical when discussing how light interacts with matter, such as when calculating radiation pressure.
As we investigate interactions like momentum transfer and energy exchange, electromagnetic radiation serves as the framework that explains these phenomena. Each photon represents a quantized packet of this radiant energy, illustrating the dual wave-particle nature of light.

Intensity of Light

Intensity of light refers to the amount of energy a light beam carries per unit area, per unit time. It is typically measured in watts per square meter (W/m²). The intensity of light affects how much energy and momentum are transferred when photons strike a surface, impacting the extent of radiation pressure exerted.
Mathematically, intensity \( I \) is expressed as:

• \( I = \frac{P}{A} \), where \( P \) is power and \( A \) is the area.

This formula shows that for a given power, increasing the area over which the energy is distributed will decrease the intensity, and vice versa. In the context of radiation pressure, intensity is crucial as it directly contributes to the number of photons impacting the surface per unit time, thereby determining the total radiation pressure according to \( P = \frac{I}{c} \).
Understanding the intensity of light helps us deduce how energetic and forceful a light beam is upon impact.