Consider normal incidence of a monochromatic beam of photons of power $P$ on a flat surface. Of the incident beam, 10% gets absorbed, 10% gets transmitted, and the rest is reflected by the flat surface. If $c$ is the speed of light, what is the force exerted on the flat surface by the beam?
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For radiation pressure force calculations, use the formula $F = (1 + R - T)\frac{P}{c}$, where $R$ is the reflection coefficient, and $T$ is the transmission coefficient.
Here, $R=0.8$ and $T=0.1$, so $F = (1 + 0.8 - 0.1)\frac{P}{c} = 1.7\frac{P}{c}$.
Step 1: Understanding the Question:
A light beam of power $P$ carrying momentum is incident normally on a flat surface.
The beam undergoes three distinct interactions: absorption (10%), transmission (10%), and reflection (80%).
We need to calculate the total force exerted on the surface, which is equal to the rate of transfer of momentum to the surface. Step 2: Key Formula or Approach:
The momentum carried per second by a light beam of power $P$ is:
\[ p_{sec} = \frac{P}{c} \]
For each component:
- Absorbed light: transfers all its momentum to the surface: $\Delta p = p_i$.
- Transmitted light: passes through without transferring any momentum: $\Delta p = 0$.
- Reflected light: bounces back, reversing direction, transferring twice its initial momentum: $\Delta p = 2p_i$. Step 3: Detailed Explanation:
• Let us calculate the force contribution from each part of the beam:
• 1. Absorbed portion (10%):
The force exerted by absorption is:
\[ F_{abs} = 0.10 \times \frac{P}{c} \]
• 2. Transmitted portion (10%):
Since this portion passes through the surface without any momentum change relative to the surface:
\[ F_{trans} = 0 \]
• 3. Reflected portion (80%):
The force exerted by reflection is:
\[ F_{ref} = 2 \times 0.80 \times \frac{P}{c} = 1.60 \frac{P}{c} \]
• The total force $F$ on the surface is the sum of these contributions:
\[ F = F_{abs} + F_{trans} + F_{ref} \]
\[ F = 0.10 \frac{P}{c} + 0 + 1.60 \frac{P}{c} = 1.7 \frac{P}{c} \]
Step 4: Final Answer:
The total force exerted on the surface by the beam is $1.7 \frac{P}{c}$.
