A perfectly reflecting solid sphere of radius r is kept in the path of a parallel beam of light of large aperture. If the beam carries on intensity I, find the force exerted by the beam on the sphere.

To determine the force exerted by a parallel beam of light on a perfectly reflecting solid sphere, we can utilize the principles of momentum transfer and the properties of light. When light strikes a surface, it exerts pressure due to the change in momentum of the photons upon reflection. Let's break this down step by step.

Understanding Light Pressure

Light carries momentum, which can be calculated using the formula:

• Momentum (p) = Energy (E) / Speed of Light (c)

For a beam of light with intensity \( I \), the energy per unit area per unit time is given by \( I \). The momentum change upon reflection is twice the momentum of the incoming light, as the light reverses direction.

Calculating the Momentum Change

The force exerted by the light on the sphere can be derived from the momentum change per unit time. The intensity \( I \) can be related to the momentum transfer as follows:

• Power (P) = Intensity (I) × Area (A)
• For a sphere, the area \( A \) that the light beam interacts with is the cross-sectional area, which is \( \pi r^2 \).

Thus, the power incident on the sphere is:

P = I × \( \pi r^2 \)

Force Calculation

The force \( F \) exerted by the beam on the sphere can be calculated using the change in momentum per unit time. Since the light is perfectly reflected, the change in momentum is twice the momentum of the incoming light:

• Change in momentum per second = 2 × (Power / c)

Substituting the expression for power, we have:

F = 2 × (I × \( \pi r^2 \)) / c

Final Expression for the Force

Combining everything, the force exerted by the beam on the perfectly reflecting sphere can be expressed as:

F = (2I × \( \pi r^2 \)) / c

Example Calculation

Let’s say the intensity \( I \) of the beam is 1000 W/m² and the radius \( r \) of the sphere is 0.1 m. The speed of light \( c \) is approximately \( 3 × 10^8 \) m/s. Plugging in these values:

F = (2 × 1000 W/m² × \( \pi (0.1 m)^2 \)) / (3 × 10^8 m/s)

Calculating this gives:

F ≈ (2 × 1000 × \( \pi × 0.01 \)) / (3 × 10^8)

F ≈ (62.83) / (3 × 10^8) ≈ 2.09 × 10^-7 N

This example illustrates how to apply the principles of momentum transfer from light to calculate the force on a reflecting surface. The key takeaway is that the force depends on the intensity of the light and the area of the sphere that intercepts the light beam.