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Solar wind (radiation) that is incident on the top of Earth’s atmosphere has an average intensity of 1.3 kW / m 2 . Suppose that you are building a solar sail that is to propel a small toy spaceship with a mass of 0.1 kg in the space between the International Space Station and the moon. The sail is made from a very light material, which perfectly reflects the incident radiation. To assess whether such a project is feasible, answer the following questions, assuming that radiation photons are incident only in normal direction to the sail reflecting surface. (a) What is the radiation pressure (force per m 2 ) of the radiation falling on the mirror-like sail? (b) Given the radiation pressure computed in (a), what will be the acceleration of the spaceship when the sail has of an area of 10.0 m 2 ? (c) Given the acceleration estimate in (b), how fast will the spaceship be moving after 24 hours when it starts from rest?

a. P = 2 I / c = 8.67 × 10 −6 N / m 2 ; b. a = P A / m = 8.67 × 10 - 4 m / s 2 ; c. 74.91 m/s

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Treat the human body as a blackbody and determine the percentage increase in the total power of its radiation when its temperature increases from 98.6 ° F to 103 ° F.

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Show that Wien’s displacement law results from Planck’s radiation law. ( Hint: substitute x = h c / λ k T and write Planck’s law in the form I ( x , T ) = A x 5 / ( e x 1 ) , where A = 2 π ( k T ) 5 / ( h 4 c 3 ) . Now, for fixed T , find the position of the maximum in I ( x , T ) by solving for x in the equation d I ( x , T ) / d x = 0 . )

x = 4.965

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Show that Stefan’s law results from Planck’s radiation law. Hint: To compute the total power of blackbody radiation emitted across the entire spectrum of wavelengths at a given temperature, integrate Planck’s law over the entire spectrum P ( T ) = 0 I ( λ , T ) d λ . Use the substitution x = h c / λ k T and the tabulated value of the integral 0 d x x 3 / ( e x 1 ) = π 4 / 15 .

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Additional problems

Determine the power intensity of radiation per unit wavelength emitted at a wavelength of 500.0 nm by a blackbody at a temperature of 10,000 K.

7.124 × 10 16 W/m 3

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The HCl molecule oscillates at a frequency of 87.0 THz. What is the difference (in eV) between its adjacent energy levels?

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A quantum mechanical oscillator vibrates at a frequency of 250.0 THz. What is the minimum energy of radiation it can emit?

1.034 eV

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In about 5 billion years, the sun will evolve to a red giant. Assume that its surface temperature will decrease to about half its present value of 6000 K, while its present radius of 7.0 × 10 8 m will increase to 1.5 × 10 11 m (which is the current Earth-sun distance). Calculate the ratio of the total power emitted by the sun in its red giant stage to its present power.

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A sodium lamp emits 2.0 W of radiant energy, most of which has a wavelength of about 589 nm. Estimate the number of photons emitted per second by the lamp.

5.93 × 10 18

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Photoelectrons are ejected from a photoelectrode and are detected at a distance of 2.50 cm away from the photoelectrode. The work function of the photoelectrode is 2.71 eV and the incident radiation has a wavelength of 420 nm. How long does it take a photoelectron to travel to the detector?

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If the work function of a metal is 3.2 eV, what is the maximum wavelength that a photon can have to eject a photoelectron from this metal surface?

387.8 nm

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Practice Key Terms 4

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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