If a parallel beam of light having intensity I is incident normally on a perfectly reflecting surface, the force exerted on the surface, equal F. When the surface is held at an angle `theta`, the force is

Options

A. `2F tan theta`
B. `F cos theta`
C. `2F cos^(2) theta`
D. 2F

Answer

To solve the problem of finding the force exerted on a perfectly reflecting surface when a parallel beam of light with intensity \( I \) is incident at an angle \( \theta \), we can follow these steps:

### Step 1: Understand the Force at Normal Incidence
When light is incident normally on a perfectly reflecting surface, the force \( F \) exerted on the surface is given by the formula:
\[ F = \frac{2I}{c} A \]
where \( A \) is the area of the surface and \( c \) is the speed of light. The factor of 2 comes from the change in momentum of the photons upon reflection.

### Step 2: Analyze the Incident Light at Angle \( \theta \)
When the light is incident at an angle \( \theta \) to the normal, the effective area \( A_{\perpendicular} \) that the light interacts with is reduced. The perpendicular area can be expressed as:
\[ A_{\perpendicular} = A \cos \theta \]

### Step 3: Calculate the Change in Momentum for One Photon
The momentum \( p \) of a single photon is given by:
\[ p = \frac{h}{\lambda} \]
where \( h \) is Planck's constant and \( \lambda \) is the wavelength of the light. When the photon reflects off the surface, the change in momentum \( \Delta p \) for one photon is:
\[ \Delta p = 2p \cos \theta = 2 \left(\frac{h}{\lambda}\right) \cos \theta \]

### Step 4: Determine the Number of Photons Striking the Surface per Second
The number of photons \( n \) striking the surface per second can be calculated using the intensity:
\[ n = \frac{I A_{\perpendicular}}{\frac{hc}{\lambda}} = \frac{I A \cos \theta \lambda}{hc} \]

### Step 5: Calculate the Total Change in Momentum per Second
The total change in momentum per second (which corresponds to the force) is:
\[ \text{Total Change in Momentum} = n \cdot \Delta p = \left(\frac{I A \cos \theta \lambda}{hc}\right) \cdot \left(2 \frac{h}{\lambda} \cos \theta\right) \]
This simplifies to:
\[ \text{Total Change in Momentum} = \frac{2I A \cos^2 \theta}{c} \]

### Step 6: Final Expression for the Force
Thus, the force \( F' \) exerted on the surface when the light is incident at an angle \( \theta \) is:
\[ F' = \frac{2I A \cos^2 \theta}{c} \]

### Step 7: Relate to the Original Force \( F \)
From the normal incidence case, we know:
\[ F = \frac{2IA}{c} \]
Thus, we can express \( F' \) in terms of \( F \):
\[ F' = F \cos^2 \theta \]

### Final Answer
The force exerted on the surface when the light is incident at an angle \( \theta \) is:
\[ F' = F \cos^2 \theta \]