Concept explainers
A spherical shell has inner radius 3.00 cm and outer radius 7.00 cm. It is made of material with thermal
(b) Next, prove that
where T is in degrees Celsius and r is in meters. (c) Find the rate of energy transfer through the shell. (d) Prove that
where T is in degrees Celsius and r is in meters. (e) Find the temperature within the shell as a function of radius. (f) Find the temperature at r = 5.00 cm, halfway through the shell.
(a)
Answer to Problem 46CP
Explanation of Solution
The law of thermal conduction is,
Here,
The expression for the surface area of the sphere ism,
Substitute
Since the value of the coefficient of thermal conductivity is constant and the radius of the spherical surface is also constant. The thermal gradient becomes constant.
The rate of energy transfer is directly proportional to the thermal gradient. Since the thermal gradient is constant so the rate of energy transfer through the spherical surface is same.
Conclusion:
Therefore, the rate of energy transfer through the spherical surface is same because the temperature gradient is constant
(b)
To show: The given relation,
Answer to Problem 46CP
Explanation of Solution
Let the temperature is
Rearrange the equation (1) to prove the relation,
Integrate at both sides for temperature from
Conclusion:
Therefore, the equation,
(c)
Answer to Problem 46CP
Explanation of Solution
From the equation (1),
Integrate at both sides for temperature from
Conclusion:
Therefore, the rate of energy transfer through the shell is
(d)
To show: The given equation,
Answer to Problem 46CP
Explanation of Solution
Let the temperature is
Rearrange the equation (1) to prove the relation,
Integrate at both sides for temperature from
Substitute
Put the value of the
Conclusion:
Therefore, the equation,
(e)
Answer to Problem 46CP
Explanation of Solution
From the equation (1),
Integrate the above equation,
Conclusion:
Therefore, the temperature within the cell as a function of radius is
(f)
Answer to Problem 46CP
Explanation of Solution
From the equation (5),
Substitute
Conclusion:
Therefore, the temperature in spherical shell at radius.
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Chapter 19 Solutions
Physics for Scientists and Engineers
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