Question

A pulse of light of duration \(100 n s\) is absorbed completely by a small object initially at rest. Power of the pulse is \(30 \mathrm{~mW}\) and the speed of light is \(3 \times 10^{8} \mathrm{~ms}^{-1}\). The final momentum of the object is (A) \(0.3 \times 10^{-17} \mathrm{~kg} \mathrm{~ms}^{-1}\) (B) \(1.0 \times 10^{-17} \mathrm{~kg} \mathrm{~ms}^{-1}\) (C) \(3.0 \times 10^{-17} \mathrm{~kg} \mathrm{~ms}^{-1}\) (D) \(9.0 \times 10^{-17} \mathrm{~kg} \mathrm{~ms}^{-1}\)

Short answer

The final momentum of the object is not correctly presented in any of the given options, the correct momentum of the object is actually 1 × 10^{-11} kg⋅ms^{-1}.

Step-by-step solution

Calculate the energy of the pulse

Since power is the rate of energy transfer, we can find the energy of the pulse by multiplying the power by the duration the pulse lasts. Thus the energy (E) is given by E = Power × Time. Here, Power = 30 mW = 30 × 10^{-3} W and Time = 100 ns = 100 × 10^{-9} s. So, E = 30 × 10^{-3} W × 100 × 10^{-9} s = 3 × 10^{-3} J.

Calculate the momentum of the object

The momentum (p) of the object after absorbing the light pulse can be found using the relationship between energy (E) and momentum (p) for light, which is given by E = pc, where c is the speed of light. By rearranging the equation, we get p = E / c. We have E = 3 × 10^{-3} J and c = 3 × 10^8 m/s, so the final momentum is p = 3 × 10^{-3} J / 3 × 10^8 m/s = 1 × 10^{-11} kg⋅m/s.

Express the final momentum in the correct units

The final momentum calculated is in kg⋅m/s. However, the options are given in terms of kg⋅ms^{-1}. The correct conversion does not require any change since kg⋅m/s is equivalent to kg⋅ms^{-1}. Thus, the final momentum of the object is 1 × 10^{-11} kg⋅ms^{-1}, which must now be matched with the given options.

Find the correct option

Upon inspection, it seems there was an error in calculation as no option matches our result. Revisiting step 2, we need to ensure we are using the correct units and addressing any calculation errors. Now, reconsidering the calculation of momentum: p = E / c = 3 × 10^{-3} J / (3 × 10^8 m/s) = 1 × 10^{-11} J⋅s/m. For light, energy (E) and mass (m) equivalence is given by E = mc^2, which implies m = E / c^2. Replacing E with the calculated energy p = (E / c^2) × c = E / c = 3 × 10^{-3} J / (3 × 10^8 m/s) = 1 × 10^{-11} kg⋅m/s. Converting J⋅s/m to kg⋅m/s, using the relation 1 J = 1 kg⋅m^2/s^2, we get 1 × 10^{-11} kg⋅m/s is the unchanged correct unit for momentum. Therefore, the correct option that matches our result is not presented among options (A), (B), (C), or (D).

Key concepts

Energy Transfer in Light Absorption

Understanding the concept of energy transfer is crucial when we explore the momentum of objects after absorbing light. Energy transfer is a foundational aspect of physics that defines how energy moves from one system to another. In the case of our exercise, energy is being transferred from the light pulse to the small object. This process happens when the light, carrying energy, is completely absorbed by the object, transferring its energy to the object.

Getting into specifics, the power of the pulse, which is measured in watts (W), represents the rate at which energy is transferred. So, when a pulse with a power of 30 milliwatts (\(30 \times 10^{-3}\text{ W}\)) is completely absorbed, we need to calculate the total energy deposited into the object. This is done by multiplying the power by the pulse duration. In this scenario, a duration of 100 nanoseconds (\(100 \times 10^{-9}\text{ s}\)) is used, giving us the total energy transferred to the object as 3 millijoules (\(3 \times 10^{-3}\text{ J}\)).

It's this energy transfer during the light pulse absorption that changes the object's state, ultimately imparting it with momentum, and hence the exercise leads us to calculate the final momentum of the object post-absorption, based on that energy transfer and the properties of light.

Understanding Pulse Duration in Light Absorption

The pulse duration signifies the length of time for which the pulse of light exists, which is essential for calculating the energy involved in absorption processes. This duration is effectively how long the energy is being imparted onto the absorbent object. In our exercise, the duration is given as 100 nanoseconds (\(100 \times 10^{-9}\text{ s}\)).

Why does pulse duration matter? Because it directly influences the total energy delivered to an object. A longer pulse means more time for energy transfer, resulting in more transferred energy if the power remains constant. On the flip side, a short pulse duration, as in our exercise example, leads to a quick but significant transfer of energy in a brief period. This notion is critical in various technologies, such as in laser physics and fiber optics, where pulse durations can impact the efficiency and functionality of systems.

Speed of Light and Its Relevance to Momentum

The speed of light, typically denoted by the symbol 'c', is a constant value representing the fastest speed at which light travels in a vacuum. It's a fundamental constant of nature with a value of approximately \(3 \times 10^8\text{ m/s}\). This staggering speed plays a pivotal role not just in relativity and astronomy but also in understanding the momentum of objects when they absorb light.

In our exercise, after the small object absorbs the pulse of light, its momentum increases due to the energy transfer mentioned earlier. The final momentum is calculated using a key physics formula that relates energy (E) to momentum (p) through the speed of light: \(E = pc\). When solving for momentum, the equation becomes \(p = E / c\), which allows us to link the transferred energy to the subsequent momentum of the object using the speed of light. Here, by dividing the total energy absorbed (\(3 \times 10^{-3}\text{ J}\)) by the speed of light (\(3 \times 10^8\text{ m/s}\)), we determine the final momentum of the object. Thus, the speed of light is not only crucial for theory but is also practically applied to calculate measurable physical quantities like the momentum post-light absorption.