/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 A laser used in retinal surgery ... [FREE SOLUTION] | 91影视

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Chapter 12: Problem 15

A laser used in retinal surgery emits photons of wavelength \(632.8 \mathrm{~nm}\) in pulses of \(0.5 \mathrm{~W}\) that last \(2 \times 10^{-2} \mathrm{~s}\). The number of photons in each pulse is nearest to (A) \(10^{10}\) (B) \(10^{15}\) (C) \(10^{20}\) (D) \(10^{25}\) (E) \(10^{30}\)

Short Answer

Expert verified
Answer: (B) \(10^{15}\)

Step by step solution

01

Convert the wavelength to meters

Given the wavelength of the photons is \(632.8~\mathrm{nm}\), we need to convert this to meters so that we can use it in our calculations. There are \(1\times10^9\) nanometer in 1 meter. Therefore, the wavelength in meters is: $$\text{Wavelength} = 632.8\times10^{-9}~\mathrm{m}$$
02

Calculate the energy of a single photon

To calculate the energy of a single photon, we can use the Planck's constant \((h)\) and the speed of light \((c)\), with the following formula: $$E_{photon} = \frac{hc}{\lambda}$$ Where \(E_{photon}\) is the energy of a single photon and \(\lambda\) is the wavelength. The speed of light \((c)\) is approximately \(3\times10^{8}~\mathrm{m/s}\), and the Planck's constant \((h)\) is approximately \(6.626\times 10^{-34}~\mathrm{Js}\). Plug in the values to get the energy of a single photon: $$ E_{photon} = \frac{(6.626\times 10^{-34}~\mathrm{Js})(3\times10^{8}~\mathrm{m/s})}{632.8\times10^{-9}~\mathrm{m}}$$
03

Calculate the total energy of the pulse

The total energy of the pulse can be calculated using the power of the laser and the duration of the pulse: $$E_{total} = \text{Power} \times \text{Duration}$$ Given the power of the laser is \(0.5~\mathrm{W}\) and the duration of the laser pulse is \(2 \times 10^{-2}~\mathrm{s}\). Plug in these values to get the total energy of the pulse: $$E_{total} = (0.5~\mathrm{W}) (2\times10^{-2}~\mathrm{s})$$
04

Calculate the number of photons

To find the number of photons, divide the total energy of the pulse by the energy of a single photon: $$N = \frac{E_{total}}{E_{photon}}$$ Plugging in the values derived in steps 2 and 3: $$N= \frac{(0.5~\mathrm{W})(2\times10^{-2}~\mathrm{s})}{(\frac{(6.626\times 10^{-34}~\mathrm{Js})(3\times10^{8}~\mathrm{m/s})}{632.8\times10^{-9}~\mathrm{m}})}$$ This corresponds to: (A) \(10^{10}\)\ (B) \(10^{15}\)\ (C) \(10^{20}\)\ (D) \(10^{25}\)\ (E) \(10^{30}\) Calculating the value of \(N\) using the above equation, we get \(N \approx 10^{15}\). Therefore, the answer is (B) \(10^{15}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Laser Physics
Laser physics is the branch of optics that deals with the understanding and application of lasers. Lasers, which stand for Light Amplification by Stimulated Emission of Radiation, emit light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. Unlike ordinary light sources, which emit a broad spectrum of light, lasers emit light in a specific, coherent wavelength, making them highly focused and precise.

This coherent light is particularly useful in various applications such as retinal surgery, where lasers must be precise enough to target very small areas without damaging surrounding tissue. In our exercise, the laser emits photons at a precise wavelength of 632.8 nm. The laser's precision and the ability to generate high intensity in a directed beam are quintessentially what makes laser physics an essential field in modern medicine and technology.
Photon Energy Calculation
The energy of a photon is calculated based on its wavelength or frequency. This energy determination is critical in fields such as quantum mechanics, astronomy, and any science involving electromagnetic radiation. For example, in the exercise, to determine the energy of photons emitted by a laser for retinal surgery, we use the formula:

\(E_{photon} = \frac{hc}{\lambda}\)

where \(E_{photon}\) is the energy of a single photon, \(h\) is Planck's constant, \(c\) is the speed of light, and \(\lambda\) is the photon's wavelength. This formula underpins the quantum mechanical relationship between energy and electromagnetic waves, illustrating that light has both wave-like and particle-like properties, a central concept in quantum physics.
Planck's Constant Usage
Planck's constant (\(h\)) is a fundamental constant in physics that plays a crucial role in quantum mechanics. It is the proportionality constant that relates the energy (\(E\)) of a photon to its frequency (\(u\)): \(E = hu\). It's also involved in the equation for calculating photon energy when the wavelength is known, as seen in the practice problem.

Planck's constant has a value of approximately \(6.626 \times 10^{-34}~\mathrm{Js}\), which indicates the very small energy scales typical in quantum processes. It forms the bedrock for much of our understanding of atomic and subatomic processes, including the behavior of photons in laser light. Moreover, Planck's constant usage not only includes energy calculations but also operations such as quantifying action in physical systems, establishing a limit to the precision with which certain pairs of physical properties, known as complementary variables, like position and momentum, can be known simultaneously 鈥� a concept known as the Heisenberg uncertainty principle.

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Most popular questions from this chapter

An atomic nucleus is induced to break into two pieces, during which process energy is released. One can say with certainty that (A) the original atomic mass was greater than that of iron. (B) the fragments will each have an atomic mass less than that of iron. (C) the masses of the fragments will add up to be less than the mass of the original nucleus. (D) (A) and (C) (E) None of the above

The fusion reaction \({ }_1^2 \mathrm{H}+{ }_1^3 \mathrm{H} \rightarrow{ }_2^4 \mathrm{He}+\mathrm{n}\) releases about \(17.5 \mathrm{MeV}\). The available energy is \(14.1 \mathrm{MeV}\) given to the neutron. The world's energy consumption in 2013 has been estimated as \(4 \times 10^{20} \mathrm{~J}\). About one out of every 6420 hydrogen atoms in the earth's oceans is deuterium. The number of kilograms of water that would be enough to supply the world's annual energy needs is most nearly (A) \(10^8 \mathrm{~kg}\) (B) \(10^9 \mathrm{~kg}\) (C) \(10^{10} \mathrm{~kg}\) (D) \(10^{11} \mathrm{~kg}\) (E) \(10^{12} \mathrm{~kg}\)

Meteorites created in the early solar system contaned aluminum-26, which is a radioactive isotope of aluminum with a half-life of \(7.2 \times 10^5 \mathrm{yrs}\). Aluminum-26 decays first into an excited state of magnesium-26 via the reaction \({ }_{13}^{26} \mathrm{Al} \rightarrow{ }_{12}^{26} \mathrm{Mg}^*+\mathrm{e}^{+}\), where the \(e^{+}\)has energy \(2.99 \mathrm{MeV}\). (The \(e^{+}\)is a positron; see previous problem. The asterisk (*) indicates "excited.") The \({ }_{12}^{26} \mathrm{Mg}^*\) then decays into the stable isotope magnesium- 26 via the reaction \({ }_{12}^{26} \mathrm{Mg}^* \rightarrow{ }_{12}^{26} \mathrm{Mg}+\gamma\). The \(\gamma\) has energy \(1.8 \mathrm{MeV}\). a) If you were asked to calculate the de Broglie wavelength of the positron, would it be permissible to use Newtonian physics? Justify your answer. b) What is the wavelength of the photon emitted when the excited magnesium-26 decays into its ground state? What is its momentum? c) \({ }_{12}^{26} \mathrm{Mg}\) has an atomic mass of \(25.9826 \mathrm{u}\). What is the speed of the recoiling nucleus when the photon is emitted? d) What is the nucleus' kinetic energy in electron volts? e) Precise measurements indicate that for a certain meteorite \(A\) the present ratio \({ }^{26} \mathrm{Mg} /{ }^{27} \mathrm{Al}=5 \times 10^{-5}\), where \({ }^{27} \mathrm{Al}\) is the common, stable isotope of aluminum. For a meteorite \(B\) the ratio is \({ }^{26} \mathrm{Mg} /{ }^{27} \mathrm{Al}=1.55 \times 10^{-7}\). Assuming that the different ratios are due to the difference in the meteorites' times of creation, how much older is meteorite \(B\) than \(A\) ?

Which of the following statements is true? The existence of the de Broglie wavelength \(\lambda_{d B}\) implies (A) that matter particles should undergo interference. (B) that matter waves travel at the speed of light. (C) that the frequency of matter waves is \(c / \lambda_{d B}\), where \(c\) is the speed of the particle. (D) that matter waves are given off by accelerating charges. (E) that matter waves are polarized.

Of the following scientists, who did not contribute to the development of quantum mechanics? (A) Niels Bohr (B) Max Planck (C) Albert Einstein (D) James Clerk Maxwell (E) John Nicholson
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