Modern Physics for Scientists and Engineers
4th Edition
ISBN: 9781133103721
Author: Stephen T. Thornton, Andrew Rex
Publisher: Cengage Learning
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Chapter 3, Problem 22P
To determine
The wavelength for which the spectral distribution calculated by Planck and Rayleigh-Jeans results differ by
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Students have asked these similar questions
For a temperature of 5800 K (the sun's surface temperature), find the wavelength for
which the spectral distribution calculated by the Planck and Rayleigh-Jeans results differ
by 5%.
Hint: To match the required condition consider:
IR-J = 1.05 I
For the thermal radiation from an ideal blackbody radiator with a surface temperature of 2000 K, let Ic represent the intensity per unit wavelength according to the classical expression for the spectral radiancy and IP represent the corresponding intensity per unit wavelength according to the Planck expression.What is the ratio Ic/IP for a wavelength of (a) 400 nm (at the blue end of the visible spectrum) and (b) 200 mm (in the far infrared)? (c) Does the classical expression agree with the Planck expression in the shorter wavelength range or the longer wavelength range?
A blackbody is an object with a radiation spectrum that is dependent solely on its tempera-
ture. A blackbody spectrum (or spectral radiancy curve) is described by the Planck Radiation
Law.
(a)
i. Sketch the spectral radiancy curves for blackbodies with temperatures of T = 4000 K
and T = 6000 K, respectively. Describe the main differences between the two
curves in terms of the appropriate physical laws defined as a function of tempera-
ture.
ii. What is the wavelength at peak intensity for each blackbody? State the part of
the electromagnetic spectrum to which each wavelength belongs.
(b) Use the Planck Radiation Law to determine the power radiated per unit area between
the wavelengths A 500 nanometres and λ = 503 nanometres for the T 6000 K
blackbody. What fraction of the blackbody's radiancy lies in this wavelength range?
=
Chapter 3 Solutions
Modern Physics for Scientists and Engineers
Ch. 3 - Prob. 1QCh. 3 - Prob. 2QCh. 3 - Prob. 3QCh. 3 - Prob. 4QCh. 3 - Prob. 5QCh. 3 - Prob. 6QCh. 3 - Prob. 7QCh. 3 - Prob. 8QCh. 3 - Prob. 9QCh. 3 - In the experiment of Example 3.2, how could you...
Ch. 3 - Prob. 11QCh. 3 - Prob. 12QCh. 3 - Prob. 13QCh. 3 - Prob. 14QCh. 3 - Prob. 15QCh. 3 - Prob. 16QCh. 3 - Prob. 17QCh. 3 - Prob. 18QCh. 3 - Prob. 19QCh. 3 - Prob. 20QCh. 3 - Prob. 21QCh. 3 - Prob. 22QCh. 3 - Prob. 23QCh. 3 - Prob. 24QCh. 3 - Prob. 25QCh. 3 - Prob. 26QCh. 3 - Prob. 1PCh. 3 - Prob. 2PCh. 3 - Across what potential difference does an electron...Ch. 3 - Prob. 4PCh. 3 - Prob. 5PCh. 3 - Prob. 6PCh. 3 - Prob. 7PCh. 3 - Prob. 8PCh. 3 - Prob. 9PCh. 3 - Prob. 10PCh. 3 - Prob. 11PCh. 3 - Prob. 12PCh. 3 - Prob. 13PCh. 3 - Prob. 14PCh. 3 - Prob. 15PCh. 3 - Prob. 16PCh. 3 - Calculate max for blackbody radiation for (a)...Ch. 3 - Prob. 18PCh. 3 - Prob. 19PCh. 3 - Prob. 20PCh. 3 - White dwarf stars have been observed with a...Ch. 3 - Prob. 22PCh. 3 - Prob. 23PCh. 3 - Prob. 24PCh. 3 - Prob. 25PCh. 3 - Prob. 26PCh. 3 - Prob. 27PCh. 3 - Prob. 32PCh. 3 - Prob. 33PCh. 3 - Prob. 34PCh. 3 - Prob. 35PCh. 3 - Prob. 36PCh. 3 - Prob. 37PCh. 3 - Prob. 38PCh. 3 - Prob. 39PCh. 3 - Prob. 40PCh. 3 - Prob. 41PCh. 3 - Prob. 42PCh. 3 - Prob. 43PCh. 3 - Prob. 44PCh. 3 - Prob. 45PCh. 3 - Prob. 46PCh. 3 - Prob. 47PCh. 3 - Prob. 48PCh. 3 - Prob. 49PCh. 3 - Prob. 50PCh. 3 - Prob. 52PCh. 3 - Prob. 53PCh. 3 - Prob. 54PCh. 3 - Prob. 55PCh. 3 - Prob. 56PCh. 3 - Prob. 57PCh. 3 - Prob. 58PCh. 3 - Prob. 59PCh. 3 - Prob. 60PCh. 3 - Prob. 61PCh. 3 - Prob. 62PCh. 3 - Prob. 63PCh. 3 - Prob. 64PCh. 3 - Prob. 65PCh. 3 - Prob. 66PCh. 3 - Prob. 67PCh. 3 - Prob. 68PCh. 3 - The Fermi Gamma-ray Space Telescope, launched in...Ch. 3 - Prob. 70P
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- Consider a black body of surface area 22.0 cm² and temperature 5700 K. (a) How much power does it radiate? 131675.5 W (b) At what wavelength does it radiate most intensely? 508.421 nm (c) Find the spectral power per wavelength at this wavelength. Remember that the Planck intensity is "intensity per unit wavelength", with units of W/m³, and "power per unit wavelength" is equal to that intensity times the surface area, with units of W/m 131.5775 Your response differs significantly from the correct answer. Rework your solution from the beginning and check each step carefully. W/marrow_forwardPrior to Planck’s derivation of the distribution law for black-body radiation, Wien found empirically a closely related distribution function which is very nearly but not exactly in agreement with the experimental results, namely ρ(λ,T) = (a/λ5)e−b/λkT. This formula shows small deviations from Planck’s at long wavelengths. (a) Find a form of the Planck distribution which is appropriate for short wavelengths (Hint: consider the behaviour of the term ehc/λkT - 1 in this limit). (b) Compare your expression from (a) with Wien’s empirical formula and hence determine the constants a and b. (c) Integrate Wien’s empirical expression for ρ(λ,T) over all wavelengths and show that the result is consistent with the Stefan–Boltzmann law (Hint: to compute the integral use the substitution x = hc/λkT and then refer to the Resource section). (d) Show that Wien’s empirical expression is consistent with Wien’s law.arrow_forwardDetermine lm , the wavelength at the peak of the Planck distribution, and the corresponding frequency ƒ, at these temperatures: (a) 3.00 K; (b) 300 K; (c) 3000 K.arrow_forward
- For a black body, the temperature and the wavelength of the emission maximum, Amax, are related by Wein's Law, expressed as: T/°C λmax/nm Values of Amax from a small pinhole in an electrically heated container were determined at a series of temperatures. The results are given below. Deduce the value of Planck's constant. 1000 2181 c = 3.00 x 108 m/s 1500 1600 λmaxT = 2000 1240 k= 1.38 x 10-34 J-S hc 4.965k 2500 1035 3000 878 3500 763arrow_forwardThe moon has a mass of 7.35 * 1022 kg, and the length of a sidereal day is 27.3 days. (a) Estimate the de Broglie wavelength of the moon in its orbit around the earth. (b) Using Mearth for the mass of the earth and Mmoon for the mass of the moon, we can use Newton’s law of gravitation to determine the radius of the moon’s orbit in terms of an integer-valued quantum number m as Rm = m2amoon, where amoon is the analog of the Bohr radius for the earth–moon gravitational system. Determine amoon in terms of Newton’s constant G, Planck’s constant h, and the masses Mearth and Mmoon. (c) The mass of the earth is Mearth = 5.97 * 1024 kg. Estimate the numerical value of amoon. (d) The radius of the moon’s orbit is 3.84 * 108 m. Estimate the moon’s quantum number m. (e) The quantized energy levels of the moon are given by E = -E0/m2. Estimate the quantum ground-state energy E0 of the moon.arrow_forwardPlanck’s constant has the value h = 6.626 × 10–34 joule-seconds (J-s), and the speed of light is c = 3 × 108 m/s. Using these values, calculate the wavelength carried by photons emitted with an energy of 1.1 × 10-19 J. Pick the closest value:arrow_forward
- The wavelength λmax at which the Planck distribution is a maximum can be found by solving dρ(λ,T)/dT = 0. Differentiate ρ(λ,T) with respect to T and show that the condition for the maximum can be expressed as xex − 5(ex − 1) = 0, where x = hc/λkT. There are no analytical solutions to this equation, but a numerical approach gives x = 4.965 as a solution. Use this result to confirm Wien’s law, that λmaxT is a constant, deduce an expression for the constant, and compare it to the value quoted in the text.arrow_forwardA cavity radiator has its maximum spectral radiance at a wavelength of 6.8×10 -7 m. If the body is heated so that T/T 0 = 2.2, at what wavelength (in nm) will the spectral radiance have its new maximum value? Wien's constant b = 2.897 x 10 3 m K. Answer:arrow_forwardRadiation has been detected from space that is characteristic of an ideal radiator at T = 2.728 K. (This radiation is a relic of the Big Bang at the beginning of the universe.) For this temperature, at what wavelength does the Planck distribution peak? In what part of the elec- tromagnetic spectrum is this wavelength?arrow_forward
- Calculate the de Broglie wavelength of proton, if it is moving with speed of 2 × 105 m/s. Mass of proton (m) = 1.67 x 10-27 kg. Planck's × constant = 6.625 × 10-34 Js.arrow_forwardThrough what potential difference ΔVΔV must electrons be accelerated (from rest) so that they will have the same wavelength as an x-ray of wavelength 0.130 nmnm? Use 6.626×10−34 J⋅sJ⋅s for Planck's constant, 9.109×10−31 kgkg for the mass of an electron, and 1.602×10−19 CC for the charge on an electron. Express your answer using three significant figures. =89.0 V Through what potential difference ΔVΔV must electrons be accelerated so they will have the same energy as the x-ray in Part A? Use 6.626×10−34 J⋅sJ⋅s for Planck's constant, 3.00×108 m/sm/s for the speed of light in a vacuum, and 1.602×10−19 CC for the charge on an electron. Express your answer using three significant figures. Second question is what I need help on! Thanks!arrow_forwardPlanck's radiation law can be written ux = 8лhc 1 25 eßhc/2-1 Show that the wavelength corresponding to the maximum energy density of the radiation fulfills the condition λmax T = . constant What is this constant? (This result is known as Wien's transition law.) Tip: you can solve the constant approximation by e.g. iterating an equation of the form Xn = 5 (1-e¯Xn-1) with a suitable initial value x1.arrow_forward
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