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 use concepts from physics, particularly those related to momentum transfer and the properties of light. Let's break this down step by step.

Understanding Light and Momentum

Light, despite being massless, carries momentum. When light strikes a surface, it can exert a force on that surface due to the change in momentum. For a perfectly reflecting surface, the light reflects off the surface, effectively reversing its momentum.

Calculating the Momentum Change

The momentum \( p \) of a photon can be expressed as:

• \( p = \frac{E}{c} \)

where \( E \) is the energy of the photon and \( c \) is the speed of light. The energy \( E \) of the light beam can be related to its intensity \( I \) and the area \( A \) over which it is distributed:

• \( E = I \cdot A \cdot t \)

Here, \( t \) is the time duration for which the light is incident on the sphere. The area \( A \) of the sphere that the light beam interacts with is given by the cross-sectional area of the sphere:

• \( A = \pi R^2 \)

Force Calculation

When light reflects off the sphere, the change in momentum \( \Delta p \) for the light can be calculated as:

• \( \Delta p = 2p \) (since the momentum changes direction)

Thus, the force \( F \) exerted on the sphere can be expressed in terms of the rate of change of momentum:

• \( F = \frac{\Delta p}{\Delta t} \)

Substituting the expression for momentum, we get:

• \( F = \frac{2 \cdot \frac{E}{c}}{\Delta t} \)

Now, substituting \( E \) with \( I \cdot A \cdot t \), we have:

• \( F = \frac{2 \cdot \frac{I \cdot A \cdot t}{c}}{t} \)

This simplifies to:

• \( F = \frac{2 \cdot I \cdot A}{c} \)

Final Expression for the Force

Now substituting \( A = \pi R^2 \) into the equation, we find:

• \( F = \frac{2 \cdot I \cdot \pi R^2}{c} \)

Thus, the force exerted by the beam of light on the perfectly reflecting solid sphere is:

• \( F = \frac{2 \pi R^2 I}{c} \)

Summary

In summary, the force exerted by a parallel beam of light on a perfectly reflecting solid sphere of radius \( R \) with intensity \( I \) is given by the formula:

• \( F = \frac{2 \pi R^2 I}{c} \)

This relationship highlights the fascinating interplay between light and momentum, illustrating how even massless particles can exert forces through their interactions with matter.