/*! 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 37 The sun emits energy in the form... [FREE SOLUTION] | 91Ó°ÊÓ

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The sun emits energy in the form of electromagnetic waves at a rate of 3.9 \(\times\) 10\(^{26}\) W. This energy is produced by nuclear reactions deep in the sun's interior. (a) Find the intensity of electromagnetic radiation and the radiation pressure on an absorbing object at the surface of the sun (radius \(r = R = 6.96 \times 10^5\) km) and at \(r = R/\)2, in the sun's interior. Ignore any scattering of the waves as they move radially outward from the center of the sun. Compare to the values given in Section 32.4 for sunlight just before it enters the earth's atmosphere. (b) The gas pressure at the sun's surface is about 1.0 \(\times\) 10\(^4\) Pa; at \(r = R/\)2, the gas pressure is calculated from solar models to be about 4.7 \(\times\) 10$^{13} Pa. Comparing with your results in part (a), would you expect that radiation pressure is an important factor in determining the structure of the sun? Why or why not?

Short Answer

Expert verified
The intensity at the sun's surface is approximately \(6.42 \times 10^7 \text{ W/m}^2\).

Step by step solution

01

Calculate Intensity at Sun's Surface

The intensity of electromagnetic radiation emitted by a source is given by the formula: \( I = \frac{P}{A} \), where \( P \) is the power and \( A \) is the area over which the power is spread. At the sun's surface: \( A = 4\pi R^2 \), where \( R = 6.96 \times 10^8 \) m (converted from km). Substitute the values: \[ I = \frac{3.9 \times 10^{26} \text{ W}}{4\pi (6.96 \times 10^8 \text{ m})^2} \approx 6.42 \times 10^7 \text{ W/m}^2 \].

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Key Concepts

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

Radiation Pressure
Radiation pressure results from the impact of electromagnetic waves on a surface. It's the force exerted per unit area when light or other forms of electromagnetic radiation strike an object. In the context of the sun, the radiation pressure can be particularly important when considering how it might affect particles and the structure of the sun itself.
Key Points to Understand Radiation Pressure:
  • Pressure is the result of both the energy of the waves and their momentum.
  • The formula used to calculate radiation pressure depends on whether the radiation is absorbed, reflected, or passes through an object.
  • For a perfectly absorbing surface, radiation pressure can be calculated using the formula: \( P_{rad} = \frac{I}{c} \) where \( I \) is the intensity and \( c \) is the speed of light in a vacuum.
In our original problem, radiation pressure ought to be considered against the gas pressure at various points within the sun.
At the surface, the gas pressure is around \( 1.0 \times 10^4 \) Pa, which greatly exceeds typical radiation pressures. However, deeper in the sun, as pressure from other factors increases massively, the relative contribution of radiation pressure becomes a point of consideration.
While it isn't the dominating force, understanding radiation pressure helps us comprehend the role of electromagnetic radiation in astrophysical processes like those occurring in our sun.
Solar Intensity
Solar intensity refers to the power per unit area received from the sun in the form of electromagnetic radiation. This intensity is crucial for understanding energy distribution throughout the solar system.
Visualizing Solar Intensity:
  • At the sun's surface, intensity is calculated using the formula \( I = \frac{P}{A} \).
  • Here, \( A \) is the surface area of a sphere with radius equivalent to the sun, \( A = 4\pi R^2 \).
  • The calculated intensity at the sun's surface is \( 6.42 \times 10^7 \text{ W/m}^2 \).
This intense radiation diminishes with distance from the sun, as it disperses over an increasingly large area. Hence, the measured intensity of sunlight decreases as it travels to earth, affecting everything from climate systems to solar panel energy calculations on our planet.
Understanding solar intensity is not only key for space sciences but also critical for applications on Earth, including solar energy harnessing and climate modeling.
Nuclear Reactions in the Sun
The sun's tremendous energy is generated by nuclear reactions occurring at its core. These reactions are the heart of solar energy production and involve nuclear fusion processes.
Breaking Down Nuclear Reactions:
  • Nuclear fusion combines lighter atomic nuclei, primarily hydrogen, to form a heavier nucleus, such as helium, releasing energy in the process.
  • The core of the sun reaches extremely high temperatures and pressures, providing the necessary conditions for these reactions to occur efficiently.
  • The energy generated by nuclear reactions travels outward from the core through the radiative and convective zones before being emitted as electromagnetic radiation at the surface.
The energy produced drives not only the sun's luminosity and heat but also influences the entire solar system. Understanding these nuclear processes helps scientists predict the sun’s lifecycle and the fundamental mechanisms of stellar physics.
Ultimately, the study of nuclear reactions in the sun gives us a window into the transformations underpinning the very existence of solar light and warmth that sustains life on Earth.

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Most popular questions from this chapter

Consider each of the electric- and magnetic-field orientations given next. In each case, what is the direction of propagation of the wave? (a) \(\vec{E}\) in the +\(x\)-direction, \(\vec{B}\) in the +\(y\)-direction; (b) \(\vec{E}\) in the -\(y\)-direction, \(\vec{B}\) in the +\(x\)-direction; (c) \(\vec{E}\) in the +\(z\)-direction, \(\vec{B}\) in the -\(x\)-direction; (d) \(\vec{E}\) in the +\(y\)-direction, \(\vec{B}\) in the -\(z\)-direction.

An electromagnetic wave of wavelength 435 nm is traveling in vacuum in the -\(z\)-direction. The electric field has amplitude 2.70 \(\times\) 10\(^{-3}\) V/m and is parallel to the \(x\)-axis. What are (a) the frequency and (b) the magnetic-field amplitude? (c) Write the vector equations for \(\vec{E} (z, t)\) and \(\vec{B} (z, t)\).

He-Ne lasers are often used in physics demonstrations. They produce light of wavelength 633 nm and a power of 0.500 mW spread over a cylindrical beam 1.00 mm in diameter (although these quantities can vary). (a) What is the intensity of this laser beam? (b) What are the maximum values of the electric and magnetic fields? (c) What is the average energy density in the laser beam?

An electromagnetic wave has an electric field given by \(\vec{E} (y, t)\) = (3.10 \(\times\) 10\(^5\) V/m) \(\hat{k}\) cos [ky - (12.65 \(\times\) 10\(^{12}\) rad/s)t]. (a) In which direction is the wave traveling? (b) What is the wavelength of the wave? (c) Write the vector equation for \(\vec{B} (y, t)\).

Scientists are working on a new technique to kill cancer cells by zapping them with ultrahigh-energy (in the range of 10\(^{12}\) W) pulses of light that last for an extremely short time (a few nanoseconds). These short pulses scramble the interior of a cell without causing it to explode, as long pulses would do. We can model a typical such cell as a disk 5.0 \(\mu\)m in diameter, with the pulse lasting for 4.0 ns with an average power of 2.0 \(\times\) 10\(^{12}\) W. We shall assume that the energy is spread uniformly over the faces of 100 cells for each pulse. (a) How much energy is given to the cell during this pulse? (b) What is the intensity (in W/m\(^2\)) delivered to the cell? (c) What are the maximum values of the electric and magnetic fields in the pulse?

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