Problem 3: Photons Required for Radiation Pressure
Consider a cubical box of volume $V = L^3$ containing isotropic radiation (photons moving randomly in all directions).
1. Momentum Transfer: When a photon of energy $E$ reflects normally off a wall, the change in momentum is $\Delta p = 2p = 2E/c$.
2. Components of Velocity: Photons travel at speed $c$. The velocity components satisfy $c_x^2 + c_y^2 + c_z^2 = c^2$. due to symmetry in isotropic radiation, the average square velocity in any one direction is: $$ \langle c_x^2 \rangle = \frac{c^2}{3} $$
3. Force and Pressure: The force exerted on a wall perpendicular to the x-axis depends on the rate of momentum transfer. $$ F_x = \frac{\Delta p}{\Delta t} = \frac{\text{Power}}{c} $$ Using the equipartition of motion, only $1/3$ of the energy density effectively contributes to pressure on any single face.
Thus, the Radiation Pressure $P$ is related to the Energy Density $u = E_{total}/V$ by: $$ P = \frac{1}{3} u = \frac{1}{3} \frac{E_{total}}{V} $$
Calculation:
Given parameters:
- Wavelength $\lambda = 450 \text{ nm} = 4.5 \times 10^{-7} \text{ m}$
- Pressure $P = 10^5 \text{ Pa}$
- Box edge length $l = 10 \text{ cm} = 0.1 \text{ m}$
The total energy $E_{total}$ is the number of photons ($N$) times the energy of one photon ($hc/\lambda$).
$$ P = \frac{1}{3l^3} \left( N \frac{hc}{\lambda} \right) $$Rearranging to solve for $N$:
$$ N = \frac{3 P l^3 \lambda}{hc} $$Substituting values:
$$ N = \frac{3(10^5)(0.1)^3(4.5 \times 10^{-7})}{(6.67 \times 10^{-34})(3 \times 10^8)} $$