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The human body has a surface area of approximately \(1.8 \mathrm{m}^{2}, \mathrm{a}\) surface temperature of approximately \(30^{\circ} \mathrm{C},\) and a typical emissivity at infrared wavelengths of \(e=0.97 .\) If we make the approximation that all photons are emitted at the wavelength of peak intensity, how many photons per second does the body emit?

Short Answer

Expert verified
The number of photons per second emitted by the body is obtained by dividing the total power output by the energy of each photon, after calculating power output using the Stefan-Boltzmann Law, finding the peak wavelength using Wien’s Displacement Law and determining the energy of each photon using the relation between energy and wavelength.

Step by step solution

01

Stefan-Boltzmann Law application

Use Stefan-Boltzmann's Law to calculate the total power that the human body emits. The law is given by \(P= Ae \sigma T^{4}\) (Power = Area * emissivity * Stefan-Boltzmann Constant * (Temperature in Kelvin)^4). Given, A = 1.8 \(m^2\), e = 0.97, \(T = (30 + 273)\) Kelvin = 303K, and \(\sigma = 5.67 * 10^{-8} \(W m^{-2} K^{-4}\) . Substituting these values, we'll get the power emitted.
02

Find peak wavelength

Now, use Wien's Displacement Law to find the peak wavelength. The law is given by \(\lambda_{max} = \frac{b}{T}\), where b is Wien's constant (\(b = 2.898 * 10^{-3}\) m K) and T is the temperature in Kelvin. This gives us the maximum wavelength \(\lambda_{max}\) at which the maximum radiation is emitted.
03

Energy of Photon at peak wavelength

The energy E of a photon can be calculated using the equation \(E = \frac{hc}{\lambda}\), where h is Planck's constant (\(6.63 * 10^{-34}\) Js), c is the speed of light (\(3 * 10^{8}\) m/s) and \(\lambda\) is the wavelength of radiation. Substituting the \(\lambda_{max}\) value, we get the energy for each photon emitted at the peak wavelength.
04

Calculate number of photons

The total number of photons N being emitted can be calculated by dividing the total power output by the energy of each photon. \(N = \frac{P}{E}\). This will give the number of photons emitted by the body per second.

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Key Concepts

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

Blackbody Radiation
Blackbody radiation refers to the theoretical concept of an idealized physical body, known as a blackbody, that perfectly absorbs all incident electromagnetic radiation, irrespective of frequency or angle of incidence.

A blackbody also radiates energy in a characteristic spectrum that depends solely on the body's temperature. This radiation covers a range of wavelengths and frequencies but peaks at a certain value that is inversely proportional to the temperature—an effect described by Wien's Displacement Law. In practical applications, real objects can approach this idealized behavior and thus, can be modeled as blackbodies to a sufficient degree of accuracy.

In the context of the original exercise, the human body is approximated as a blackbody to simplify calculations related to thermal radiation. Although human skin does not absorb all incident radiation, it's a close enough approximation for discussions on emitted infrared radiation. By considering the body as a blackbody, we can use the Stefan-Boltzmann Law to estimate the total power emitted by the body due to its temperature.
Wien's Displacement Law
Wien's Displacement Law establishes a relationship between the temperature of a blackbody and the peak wavelength of the radiation it emits. It's mathematically stated as \(\lambda_{max} = \frac{b}{T}\), where \(\lambda_{max}\) is the peak wavelength, \(b\) is Wien's constant (approximately \(2.898 \times 10^{-3} m∙K\)), and \(T\) is the absolute temperature of the blackbody in Kelvin.

This law is fundamental in the analysis of thermal radiation since it allows for the calculation of the wavelength at which a blackbody emits most of its radiation. Points to note are that as a blackbody becomes hotter, it emits radiation at shorter wavelengths and the peak of its spectral power distribution shifts to a higher frequency.

In our exercise, using Wien's Displacement Law provides the necessary information to determine the wavelength at which the human body—in its idealized form as a blackbody—emits most of its infrared radiation.
Photon Emission
Photon emission is a process whereby a photon, the fundamental particle of light, is released by an atom or molecule. The energy of a single photon is directly related to its wavelength, as described by the equation \(E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant, \(c\) is the speed of light in a vacuum, and \(\lambda\) is the photon's wavelength.

When an object, modeled as a blackbody, is at a particular temperature, it emits photons across a spectrum of wavelengths, but with a peak as predicted by Wien's Displacement Law. The number of photons emitted at a particular wavelength can be calculated once we know the power emitted by the body at that wavelength, using the Stefan-Boltzmann Law.

To conclude the exercise, after finding the power emitted by the idealized human body and the energy of an individual photon at the peak wavelength, the number of emitted photons per second is obtained by dividing the total radiated power by the energy of a single photon. This provides an understanding of the quantum aspect of thermal radiation.

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Most popular questions from this chapter

People immersed in strong unchanging magnetic fields occasionally report sensing a metallic taste. Some investigators suspect that motion in the constant field could produce a changing flux and a resulting emf that could stimulate nerves in the tongue. We can make a simple model to see if this is reasonable by imagining a somewhat extreme case. Suppose a patient having an MRI is immersed in a \(3.0 \mathrm{T}\) field along the axis of his body. He then quickly tips his head to the side, toward his right shoulder, tipping his head by \(30^{\circ}\) in the rather short time of 0.15 s. Estimate the area of the tongue; then calculate the emf that could be induced in a loop around the outside of the tongue by this motion of the head. How does this emf com- pare to the approximately \(15 \mathrm{mV}\) necessary to trigger an action potential? Does it seem reasonable to suppose that an induced emf is responsible for the noted effect?

What is the wavelength of a photon whose energy is twice that of a photon with a 600 nm wavelength?

For 27 MHz radio waves, the wave intensity has been reduced by a factor of 3 at a depth of approximately \(15 \mathrm{cm}\). At this point in the tissue, the electric field amplitude has decreased by a factor of A. 9 B. \(3 \sqrt{3}\) C. 3 D. \(\sqrt{3}\)

A 50 mW laser beam is polarized horizontally. It then passes through two polarizers. The axis of the first polarizer is oriented at \(30^{\circ}\) from the horizontal, and that of the second is oriented at \(60^{\circ}\) from the horizontal. What is the power of the transmitted beam?

A metal detector can detect the presence of metal screws used to repair a broken bone inside the body. This tells us that A. The screws are made of magnetic materials. B. The tissues of the body are conducting. C. The magnetic fields of the device can penetrate the tissues of the body. D. The screws must be perfectly aligned with the axis of the device.

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