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Chapter 17: Problem 106

The rate at which radiant energy from the sun reaches the earth's upper atmosphere is about 1.50 kW/m\(^2\). The distance from the earth to the sun is \(1.50 \times 10{^1}{^1} m\), and the radius of the sun is \(6.96 \times 10{^8} m\). (a) What is the rate of radiation of energy per unit area from the sun's surface? (b) If the sun radiates as an ideal blackbody, what is the temperature of its surface?

Short Answer

Expert verified
(a) 6.33 脳 10鈦� W/m虏; (b) Approximately 5780 K.

Step by step solution

01

Calculate Total Power Emitted by the Sun

Using the given radiant energy rate at Earth's upper atmosphere, we first calculate the total power emitted by the Sun. The formula to use is \[ P = I \cdot 4\pi R^2 \]where \( I \) is the solar constant (1.50 kW/m虏), and \( R \) is the distance from the Earth to the Sun (\( 1.50 \times 10^{11} \) m).Substituting the values, we get:\[ P = 1.50 \times 10^3 \times 4\pi \times (1.50 \times 10^{11})^2 \]Calculating this, \[ P = 3.96 \times 10^{26} \text{ W} \]where \( P \) is the total power emitted by the Sun.
02

Calculate Rate of Radiation Per Unit Area (Intensity) at Sun's Surface

Having calculated the total power emitted, we now find the intensity at the Sun's surface. The formula is:\[ I_{sun} = \frac{P}{4\pi r^2} \]where \( r \) is the radius of the Sun (\( 6.96 \times 10^8 \) m).Substitute \( P = 3.96 \times 10^{26} \text{ W} \) and \( r \):\[ I_{sun} = \frac{3.96 \times 10^{26}}{4\pi (6.96 \times 10^8)^2} \]Calculating this gives\[ I_{sun} = 6.33 \times 10^7 \text{ W/m}^2 \]This is the rate of radiation of energy per unit area from the Sun's surface.
03

Calculate Surface Temperature of the Sun Using Stefan-Boltzmann Law

If the Sun radiates as a blackbody, we use the Stefan-Boltzmann law:\[ I = \sigma T^4 \]where \( I \) is the intensity, \( \sigma \) is the Stefan-Boltzmann constant (\( 5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4 \)), and \( T \) is the temperature.Rearranging the formula to solve for \( T \):\[ T = \left(\frac{I}{\sigma}\right)^{1/4} \]Substitute \( I = 6.33 \times 10^7 \text{ W/m}^2 \):\[ T = \left(\frac{6.33 \times 10^7}{5.67 \times 10^{-8}}\right)^{1/4} \]Calculating this,\[ T \approx 5780 \text{ K} \]This is the surface temperature of the Sun.

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Key Concepts

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

Stefan-Boltzmann Law
The Stefan-Boltzmann Law is a fundamental concept in the study of thermal radiation. It describes how much energy is emitted by a blackbody in the form of radiation.
This law states that the total energy radiated per unit surface area of a blackbody is proportional to the fourth power of the blackbody's absolute temperature. The formula is given by: \[ I = \sigma T^4 \] where:
  • \( I \) is the radiated intensity (energy per unit area).
  • \( \sigma \) is the Stefan-Boltzmann constant \((5.67 \times 10^{-8} \text{ W/m}^2\text{K}^4) \).
  • \( T \) is the absolute temperature in Kelvin.
The Stefan-Boltzmann Law helps us calculate the energy emitted by the Sun, assuming it behaves like an ideal blackbody.
This law is crucial for understanding how stars, such as our Sun, emit energy and affects calculations involving stellar temperatures.
Blackbody Radiation
Blackbody radiation refers to the theoretical concept of an object that absorbs all incoming radiation, regardless of frequency or angle of incidence.
Such a body is perfect in the sense that it does not reflect or transmit any energy. Instead, it re-radiates energy at a rate determined by its temperature.
The Sun is often approximated as an ideal blackbody, especially when studying its radiation patterns.
This simplification allows astrophysicists to calculate its surface temperature using the Stefan-Boltzmann Law.
Blackbody radiation is characterized by a continuous spectrum, which means it emits radiation at every possible frequency.
Understanding blackbody radiation is key to finding the surface temperature of distant stars and plays an essential role in the study of thermal physics.
Solar Constant
The solar constant is a measure of the rate at which energy from the Sun is received per unit area at the top of Earth's atmosphere.
It is denoted by \( I \) and is approximately 1.50 kW/m虏. This value can vary slightly due to changes in Earth's orbit and solar activity.
The solar constant is a crucial factor in calculating the energy balance of our planet and plays a significant role in climatic and atmospheric models.
By understanding the solar constant, we are able to estimate the total power output of the Sun. In our exercise, it helps in calculating the total power emitted by the Sun using the formula: \[ P = I \cdot 4\pi R^2 \] where:
  • \( P \) is the total power output of the Sun.
  • \( R \) is the average distance from the Earth to the Sun.
The solar constant provides an essential baseline for measuring solar energy input and is vital for understanding the Earth's energy dynamics.

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Most popular questions from this chapter

Conventional hot-water heaters consist of a tank of water maintained at a fixed temperature. The hot water is to be used when needed. The drawbacks are that energy is wasted because the tank loses heat when it is not in use and that you can run out of hot water if you use too much. Some utility companies are encouraging the use of on-demand water heaters (also known as flash heaters), which consist of heating units to heat the water as you use it. No water tank is involved, so no heat is wasted. A typical household shower flow rate is 2.5 gal/min (9.46 L/min) with the tap water being heated from 50\(^\circ\)F (10\(^\circ\)C) to 120\(^\circ\)F (49\(^\circ\)C) by the on-demand heater. What rate of heat input (either electrical or from gas) is required to operate such a unit, assuming that all the heat goes into the water?

Consider a poor lost soul walking at 5 km/h on a hot day in the desert, wearing only a bathing suit. This person's skin temperature tends to rise due to four mechanisms: (i) energy is generated by metabolic reactions in the body at a rate of 280 W, and almost all of this energy is converted to heat that flows to the skin; (ii) heat is delivered to the skin by convection from the outside air at a rate equal to \(k'A{_s}{_k}{_i}{_n}(T{_a}{_i}{_r} - T{_s}{_k}{_i}{_n})\), where \(k'\) is 54 J/h \(\cdot\) C\(^\circ\) \(\cdot\) m\(^2\), the exposed skin area \(A{_s}{_k}{_i}{_n}\) is 1.5 m\(^2\), the air temperature \(T{_a}{_i}{_r} \)is 47\(^\circ\)C, and the skin temperature \(T{_s}{_k}{_i}{_n}\) is 36\(^\circ\)C; (iii) the skin absorbs radiant energy from the sun at a rate of 1400 W/m\(^2\); (iv) the skin absorbs radiant energy from the environment, which has temperature 47\(^\circ\)C. (a) Calculate the net rate (in watts) at which the person's skin is heated by all four of these mechanisms. Assume that the emissivity of the skin is \(e\) = 1 and that the skin temperature is initially 36\(^\circ\)C. Which mechanism is the most important? (b) At what rate (in L/h) must perspiration evaporate from this person's skin to maintain a constant skin temperature? (The heat of vaporization of water at 36\(^\circ\)C is \(2.42 \times 10{^6}\) J/kg.) (c) Suppose instead the person is protected by light-colored clothing \((e \approx 0)\) so that the exposed skin area is only 0.45 m\(^2\). What rate of perspiration is required now? Discuss the usefulness of the traditional clothing worn by desert peoples.

A 500.0-g chunk of an unknown metal, which has been in boiling water for several minutes, is quickly dropped into an insulating Styrofoam beaker containing 1.00 kg of water at room temperature (20.0\(^\circ\)C). After waiting and gently stirring for 5.00 minutes, you observe that the water鈥檚 temperature has reached a constant value of 22.0\(^\circ\)C. (a) Assuming that the Styrofoam absorbs a negligibly small amount of heat and that no heat was lost to the surroundings, what is the specific heat of the metal? (b) Which is more useful for storing thermal energy: this metal or an equal weight of water? Explain. (c) If the heat absorbed by the Styrofoam actually is not negligible, how would the specific heat you calculated in part (a) be in error? Would it be too large, too small, or still correct? Explain.

BIO Temperatures in Biomedicine. (a) Normal body temperature. The average normal body temperature measured in the mouth is 310 K. What would Celsius and Fahrenheit thermometers read for this temperature? (b) Elevated body temperature. During very vigorous exercise, the body鈥檚 temperature can go as high as 40\(^\circ\)C. What would Kelvin and Fahrenheit thermometers read for this temperature? (c) Temperature difference in the body. The surface temperature of the body is normally about 7 C\(^\circ\) lower than the internal temperature. Express this temperature difference in kelvins and in Fahrenheit degrees. (d) Blood storage. Blood stored at 4.0\(^\circ\)C lasts safely for about 3 weeks, whereas blood stored at -160\(^\circ\)C lasts for 5 years. Express both temperatures on the Fahrenheit and Kelvin scales. (e) Heat stroke. If the body鈥檚 temperature is above 105\(^\circ\)F for a prolonged period, heat stroke can result. Express this temperature on the Celsius and Kelvin scales.

Convert the following Kelvin temperatures to the Celsius and Fahrenheit scales: (a) the midday temperature at the surface of the moon (400 K); (b) the temperature at the tops of the clouds in the atmosphere of Saturn (95 K); (c) the temperature at the center of the sun \((1.55 \times 10{^7} K)\).
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