Problem 2 - Master Resource Book in JEE Main Physics

Question

A parallel beam of light is incident normally on a plane surface absorbing \(40 \%\) of the light and reflecting the rest. If the incident beam carries \(60 \mathrm{~W}\) of power, the force exerted by it on the surface is (a) \(3.2 \times 10^{-8} \mathrm{~N}\) (b) \(3.2 \times 10^{-7} \mathrm{~N}\) (c) \(5.12 \times 10^{-7} \mathrm{~N}\) (d) \(5.12 \times 10^{-8} \mathrm{~N}\)

Step-by-Step Solution

Answer

The force exerted by the light on the surface is \(3.2 \times 10^{-7} \text{ N}\).

Step 1: Analyze the Problem

We have a light beam with a power of 60 W incident normally on a plane surface that absorbs 40% of the light and reflects the rest. We need to calculate the force exerted by the light on the surface using this information.

Step 2: Calculate Reflected Power

Since the surface absorbs 40% of the light, it reflects 60%. The reflected power, \( P_{reflected} \), can be calculated as: \( P_{reflected} = (60\% \times 60 \, \text{W}) = 0.6 \times 60 = 36 \, \text{W} \).

Step 3: Use Conservation of Momentum

The force exerted by the light can be calculated using the change in momentum. Change in momentum for light is calculated based on the reflection and absorption properties. The formula for force due to light when reflecting and absorbing is \( F = \frac{2P_{reflected} + P_{absorbed}}{c} \), where \( c \) is the speed of light \( 3 \times 10^8 \text{ m/s} \).

Step 4: Substitute into Formula

Reflected force is due to the change in momentum, so we use twice the reflected portion: Force \( F = \frac{2 \times 36 + (60 - 36)}{3 \times 10^8} \).

Step 5: Simplifying Calculations

Simplify the expression to find the force: \( F = \frac{2 \times 36 + 24}{3 \times 10^8} = \frac{72 + 24}{3 \times 10^8} = \frac{96}{3 \times 10^8} \).

Step 6: Solve for the Force

Calculate the force exerted: \( F = \frac{96}{3 \times 10^8} = 3.2 \times 10^{-7} \text{ N} \).

Key Concepts

Reflection of light

Light can behave like a particle and reflect off surfaces, similar to how a ball bounces off a wall. When a beam of light hits a smooth object, part of it is reflected back in the original direction. This is known as the reflection of light. Reflection occurs in two main types:

• Specular reflection: where the light is reflected at a specific angle, like from a mirror.
• Diffuse reflection: where the light is scattered in various directions, usually from rough surfaces.

In our problem, 60% of the light is reflected. This reflected light is responsible for a change in momentum, which contributes to the force on a surface. When light reflects, it changes direction but remains at the same speed, which implies a momentum shift.

Absorption of light

When light energy enters a surface and is not re-emitted, it is absorbed. During absorption of light, the energy is transformed, often into heat, within the material. For this exercise, the material absorbs 40% of the incident light.

• The absorbed light does not contribute to reflection.
• However, it does contribute to the overall momentum change of the light.

Absorption alters the energy distribution and pressure due to light, which must be considered when calculating forces exerted by light.

Conservation of momentum

The conservation of momentum principle holds that the momentum of a closed system remains constant if it is not acted on by external forces. In the context of light, when it hits a surface and either reflects or gets absorbed, it changes momentum.

• Reflected light changes momentum but in a direction opposite to absorption.
• Both absorbed and reflected parts contribute to the exerted force due to their momentum change.

The force by the light can be calculated with this principle, encapsulated in a specific formula; for a mix of reflection and absorption, we use:\[F = \frac{2P_{reflected} + P_{absorbed}}{c}\]This formula perfectly illustrates the conservation of momentum by considering changes from both absorbing and reflecting portions of light energy.

Photon momentum

Photons, though massless, carry momentum. The momentum of a single photon is directly proportional to its energy and inversely proportional to the speed of light \( c \). The formula is:\[p = \frac{E}{c}\]Where:

• \( p \) is the momentum
• \( E \) is the energy of the photon
• \( c \) is the speed of light

In a beam of light, we consider the collective momentum of many photons. Therefore, when light strikes a surface, even without a tangible mass, it influences force through cumulative photon momentum. By understanding and using photon momentum, we can predict how much force a light beam exerts when it is partially absorbed or reflected. This is crucial in engineering and physics when working with light and electromagnetic waves.