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Chapter 6: Problem 29

A diode laser emits at a wavelength of \(987 \mathrm{nm}\). (a) In what portion of the electromagnetic spectrum is this radiation found? (b) All of its output energy is absorbed in a detector that measures a total energy of \(0.52 \mathrm{~J}\) over a period of \(32 \mathrm{~s}\). How many photons per second are being emitted by the laser?

Short Answer

Expert verified
The given radiation with a wavelength of 987 nm falls within the near-infrared (NIR) range of the electromagnetic spectrum. The laser emits approximately 8.09 × 10^{16} photons per second.

Step by step solution

01

Identify the portion of the electromagnetic spectrum

The given wavelength for the diode laser is 987 nm. We can use the wavelength to find the portion of the electromagnetic spectrum it belongs to. By referring to the electromagnetic spectrum ranges, we can identify that a 987 nm wavelength falls within the near-infrared (NIR) range.
02

Calculate the energy of a single photon

To calculate the energy of a single photon, we can use the formula: Energy (E) = \( \cfrac{hc}{λ} \) where h = Planck's constant (6.63 × 10^{-34} Js), c = speed of light (3 × 10^8 m/s), λ = wavelength (987 nm or 987 × 10^{-9} m). Plugging the values into the equation, we get: E = \( \cfrac{6.63 × 10^{-34} Js × 3 × 10^8 m/s}{987 × 10^{-9} m} \) Calculating this value, we obtain: E ≈ 2.01 × 10^{-19} J
03

Calculate the total number of photons

Now, we can find the total number of photons using the absorbed energy and the energy of a single photon. The absorbed energy is given as 0.52 J and the energy of a single photon is 2.01 × 10^{-19} J. Total number of photons (N) = \( \cfrac{Total~energy~absorbed}{Energy~of~a~single~photon} \) N = \( \cfrac{0.52 J}{2.01 × 10^{-19} J} \) Calculating this value, we obtain: N ≈ 2.59 × 10^{18}
04

Calculate photons emitted per second

We can find the number of photons emitted per second by dividing the total number of photons by the duration of time (32 seconds). Photons emitted per second = \( \cfrac{Total~number~of~photons}{Time~duration} \) Photons emitted per second = \( \cfrac{2.59 × 10^{18}}{32 s} \) Calculating this value, we obtain: Photons emitted per second ≈ 8.09 × 10^{16} Hence, the laser emits approximately 8.09 × 10^{16} photons per second.

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Most popular questions from this chapter

(a) Using Equation \(6.5,\) calculate the energy of an electron in the hydrogen atom when \(n=2\) and when \(n=6 .\) Calculate the wavelength of the radiation released when an electron moves from \(n=6\) to \(n=2 .\) (b) Is this line in the visible region of the electromagnetic spectrum? If so, what color is it?

(a) A red laser pointer emits light with a wavelength of \(650 \mathrm{nm}\). What is the frequency of this light? (b) What is the energy of one of these photons? (c) The laser pointer emits light because electrons in the material are excited (by a battery) from their ground state to an upper excited state. When the electrons return to the ground state, they lose the excess energy in the form of \(650 \mathrm{nm}\) photons. What is the energy gap between the ground state and excited state in the laser material?

Consider a fictitious one-dimensional system with one electron. The wave function for the electron, drawn at the top of the next page, is \(\psi(x)=\sin x\) from \(x=0\) to \(x=2 \pi\). Sketch the probability density, \(\psi^{2}(x),\) from \(x=0\) to \(x=2 \pi\). (b) At what value or values of \(x\) will there be the greatest probability of finding the electron? (c) What is the probability that the electron will be found at \(x=\pi ?\) What is such a point in a wave function called? [Section 6.5\(]\)

The Lyman series of emission lines of the hydrogen atom are those for which \(n_{f}=1 .\) (a) Determine the region of the electromagnetic spectrum in which the lines of the Lyman series are observed. (b) Calculate the wavelengths of the first three lines in the Lyman series- those for which \(n_{i}=2,3,\) and 4 .

Which of the following represent impossible combinations of \(n\) and \(l:(\) a \() 1 p,(\) b \() 4 s,(c) 5 f,(\) d) \(2 d ?\)
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