Question

A parallel beam of light is incident normally on a plane surface absorbing 50% of the light and reflecting the rest. If the incident beam carries $90\text{ W}$ of power, the force exerted by it on the surface is ( $\text{C} =$ velocity of light in air $= 3 \times 10^8\text{ m/s}$ )}

• A. $4.5 \times 10^{-7}\text{ N}$
• B. $1.5 \times 10^{-7}\text{ N}$
• C. $3 \times 10^{-7}\text{ N}$
• D. $9 \times 10^{-7}\text{ N}$

Hint:

Perfect absorber ($r=0$) $\rightarrow F=P/c$. Perfect reflector ($r=1$) $\rightarrow F=2P/c$. For anything in between, use the $(1+r)$ factor.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
Light carries momentum. When it strikes a surface, it transfers some or all of its momentum to the surface, exerting a radiation pressure and, consequently, a force.
The momentum of a photon is $p = E/c$. The total force exerted depends on how much light is absorbed vs. reflected.
Absorbed photons transfer momentum $p$. Reflected photons transfer momentum $2p$ (because their direction is reversed).
Step 2: Key Formula or Approach:
For a surface that absorbs a fraction $a$ and reflects a fraction $r$ (such that $a+r=1$), the force exerted by a beam of power $P$ is:
\[ F = F_{\text{absorbed}} + F_{\text{reflected}} \]
\[ F = a\left(\frac{P}{c}\right) + r\left(\frac{2P}{c}\right) \]
Alternatively, since $a = 1 - r$, this simplifies to:
\[ F = \frac{P}{c}(1 - r) + \frac{2Pr}{c} = \frac{P}{c}(1 + r) \]
Step 3: Detailed Explanation:
Given values:
Power, $P = 90\text{ W}$
Speed of light, $c = 3 \times 10^8\text{ m/s}$
Reflection fraction, $r = 50% = 0.5$ (since it absorbs 50%, it reflects the remaining 50%).
Substitute these values into the formula:
\[ F = \frac{P}{c} (1 + r) \]
\[ F = \frac{90}{3 \times 10^8} (1 + 0.5) \]
\[ F = \left( 30 \times 10^{-8} \right) \times 1.5 \]
\[ F = 45 \times 10^{-8}\text{ N} \]
Adjusting into standard scientific notation:
\[ F = 4.5 \times 10^{-7}\text{ N} \]
Step 4: Final Answer:
The force exerted on the surface is $4.5 \times 10^{-7}\text{ N}$.