Page 277 - Schaum's Outline of Theory and Problems of Applied Physics

262 HEAT TRANSFER [CHAP. 22

SOLVED PROBLEM 22.6
If a cup of coffee cools from 85 to 75 C in 2 min in a room at 20 C, how long will it take to cool from
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45 to 35 C?
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The average temperature difference between the coffee and the air in the first period of cooling is T 1 =
1
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80 C − 20 C = 60 C, and in the second period it is T 2 = 40 C − 20 C = 20 C. Since T 2 = T 1 , according to the
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3
1
solution of Prob. 22.5 (Q/t) 2 = (Q/t) 1 , so the rate of cooling will be one-third as great. The coffee will therefore
3
take (3)(2 min) = 6 min to cool from 45 to 35 C.
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RADIATION
In radiation, energy is carried by the electromagnetic waves emitted by every object. Electromagnetic waves, of
8
which light, radio waves, and X-rays are examples, travel at the velocity of light (3 × 10 m/s = 186,000 mi/s)
and require no material medium for their passage. The better an object absorbs radiation, the better it emits
radiation. A perfect absorber of radiation is called a blackbody, and it is accordingly the best radiator.
The rate at which an object whose surface area is A and whose absolute temperature is T emits radiation is
given by the Stefan-Boltzmann law:
P 4
R = = eσ T
A
4
2
The constant σ (Greek letter sigma) has the value 5.67×10 −8 W/(m ·K ). The emissivity e has a value between 0
(for a perfect reflector, hence a nonradiator) and 1 (for a blackbody), depending on the nature of the radiating
surface.
With increasing temperature, the predominant wavelength of the radiation emitted by a body decreases.
Thus a hot body that glows red is cooler than one that glows bluish white since red light has a longer wavelength
than blue light (see Chapter 30). A body at room temperature emits radiation that is chiefly in the infrared part
of the spectrum, to which the eye is not sensitive.
SOLVED PROBLEM 22.7
If all objects radiate electromagnetic energy, why don’t the objects around us in everyday life grow colder
and colder?
Every object also absorbs electromagnetic energy from its surroundings, and if both object and surroundings are
at the same temperature, energy is emitted and absorbed at the same rate. When an object is at a higher temperature
than its surroundings and heat is not supplied to it, the object radiates more energy than it absorbs and cools down
to the temperature of its surroundings.

SOLVED PROBLEM 22.8
If a red star and a white star radiate energy at the same rate, can they be the same size? If not, which must
be the larger?

The red star must be cooler than the white one, so if it is to radiate energy at the same rate, it must have a larger
surface area and hence a larger size.

SOLVED PROBLEM 22.9
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A copper ball 2 cm in radius is heated in a furnace to 400 C. If its emissivity is 0.3, at what rate does it
radiate energy?
The surface area of the ball is
2
2
A = 4πr = (4π)(0.02 m) = 0.005 m 2
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and its absolute temperature is T = 400 C + 273 = 673 K. Hence
4
4
4
2
2
P = eσ AT = (0.3)[5.67 × 10 −8 W/(m ·K )](0.005 m )(673 K) = 1.74 W