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Question: A pulse of light of duration \(100ns\) is absorbed completely by a small object initially at rest. P...

A pulse of light of duration 100ns100ns is absorbed completely by a small object initially at rest. Power of the pulse is 30mW30mW and the speed of light is 3×108ms−23 \times {10^8}m{s^{ - 2}} . The final momentum of the object is
A. 0.3×10−17kgms−10.3 \times {10^{ - 17}}kgm{s^{ - 1}}
B. 1.0×10−17kgms−11.0 \times {10^{ - 17}}kgm{s^{ - 1}}
C. 3.0×10−17kgms−13.0 \times {10^{ - 17}}kgm{s^{ - 1}}
D. 9.0×10−17kgms−19.0 \times {10^{ - 17}}kgm{s^{ - 1}}

Explanation

Solution

In this question, we are given that a pulse of is absorbed into an object and we have to find the momentum of the final object. There is movement of photons from the wave pulse to the object so here we can apply the de Broglie formula to find the momentum.

Complete Step by Step Answer:
Let us consider a pulse is absorbed completely by an object initially at rest. Then the force with which the pulse attacks the object is given by the product of the power of the pulse and the duration of the pulse and these values are being given in the question.
Incident Energy E=PtE = Pt ……(1) \ldots \ldots \left( 1 \right)
Where P=30mWP = 30mW is the power and t=100nst = 100ns is the duration of the pulse
Now, we have to find the final momentum of the object. We can use de Broglie equation here, writing the de Broglie wavelength λ=hp\lambda = \dfrac{h}{p}
Energy of the photon be given as E=pcE = pc ……(2) \ldots \ldots \left( 2 \right)
where pp is the momentum of the object after absorbed
solving the two marked equations
Pt=pc ⇒p=Ptc  Pt = pc \\\ \Rightarrow p = \dfrac{{Pt}}{c} \\\
Substituting the given values
By converting the power which is given in megawatt (10−3)\left( {{{10}^{ - 3}}} \right) to watt and time in nanoseconds (10−9)\left( {{{10}^{ - 9}}} \right) to seconds
p=30×10−3×100×10−93×108 ∴p=10−17kgms−1p = \dfrac{{30 \times {{10}^{ - 3}} \times 100 \times {{10}^{ - 9}}}}{{3 \times {{10}^8}}} \\\ \therefore p = {10^{ - 17}}kgm{s^{ - 1}}

Hence, our correct option is B.

Note: When a light wave encounters an object, it can be absorbed, transmitted, reflected, polarized, diffracted depending on the composition of the object or the wavelength of the wave. Absorption occurs when the photons from the incident light cause the object to vibrate.