Concept explainers
(i) Rank the following five capacitors from greatest to smallest capacitance, noting any cases of equality, (a) a 20-μF capacitor with a 4-V potential difference between its plates (b) a 30-μF capacitor with charges of magnitude 90 μC on each plate (c) a capacitor with charges of magnitude 80 μC on its plates, differing by 2 V in potential. (d) a 10-μF capacitor storing energy 125 μJ (e) a capacitor storing energy 250 μJ with a 10-V potential difference (ii) Rank the same capacitors in part (i) from largest to smallest according to the potential difference between the plates, (iii) Rank the capacitors in part (i) in the order of the magnitudes of the charges on their plates, (iv) Rank the capacitors in part (i) in the order of the energy they store.
(i)
The rank of the five capacitors from greatest to smallest capacitance.
Answer to Problem 26.12OQ
The rank of the five capacitors from greatest to smallest capacitance is
Explanation of Solution
Given info: The capacitances of the cases (a), (b) and (d) are
Formula to calculate the capacitance of the capacitor is,
Here,
For case (a):
The capacitance of the first capacitor is,
Here,
For case (b):
The capacitance of the second capacitor is,
Here,
For case (c):
The capacitance of the third capacitor is,
Here,
Substitute
Thus, the capacitance of the third capacitor is
For case (d):
The capacitance of the fourth capacitor is,
Here,
For case (e):
Formula to calculate the energy stored in a capacitor is,
Here,
Substitute
Thus, the capacitance of the fifth capacitor is
The rank of the capacitor is,
Conclusion:
Therefore, the rank of the five capacitors from greatest to smallest capacitance is
(ii)
The rank of the five capacitors from greatest to smallest capacitance according to the potential difference between plates.
Answer to Problem 26.12OQ
The rank of the five capacitors from greatest to smallest capacitance according to the potential difference between plates is
Explanation of Solution
Given info: The capacitances of the cases (a), (b) and (d) are
For case (a):
The electric potential across the first capacitor is,
For case (b):
The electric potential for across the second capacitor is,
Substitute
Thus, the electric potential for across the second capacitor is
For case (c):
The electric potential for across the third capacitor is,
Here,
For case (d):
Formula to calculate the energy stored in the fourth capacitor is,
Here,
Substitute
Thus, the energy stored in the fourth capacitor
For case (e):
The electric potential across the fifth capacitor is,
Here,
The rank of the electric potential from highest to lowest is,
From the above expression, the capacitance of the capacitor is inversely proportional to the square of voltage. Hence, the rank of the five capacitors from greatest to smallest capacitance according to the potential difference between plates is
Conclusion:
Therefore, the rank of the five capacitors from greatest to smallest capacitance according to the potential difference between plates is
(iii)
The rank of the five capacitors from greatest to smallest capacitance in the order of the magnitude of charge is
Answer to Problem 26.12OQ
The rank of the five capacitors from greatest to smallest capacitance in the order of the magnitude of charge is
Explanation of Solution
Given info: The capacitances of the cases (a), (b) and (d) are
For case (a):
The charge across the first capacitor is,
Here,
Substitute
Thus, the charge across the first capacitor is
For case (b):
The charge across the second capacitor is,
Here,
Thus, the charge across the second capacitor is
For case (c):
The charge across the third capacitor is,
Here,
Thus, the charge across the second capacitor is
For case (d):
Formula to calculate the energy stored in the fourth capacitor is,
Here,
Substitute
Thus, the charge across the fourth capacitor is
For case (e):
Formula to calculate the energy stored in the fifth capacitor is,
Substitute
Thus, the magnitude of the fifth capacitor is
The charge across the fifth capacitor is,
Here,
Substitute
Thus, the charge across the fifth capacitor is
The rank of the charge from highest to lowest is,
Since, the capacitance of the capacitor is proportional to the charge. Hence, the rank of the five capacitors from greatest to smallest capacitance in the order of the magnitude of charge is
Conclusion:
Therefore, the rank of the five capacitors from greatest to smallest capacitance in the order of the magnitude of charge is
(iv)
The rank of energy stored of the five capacitors from greatest to smallest capacitance.
Answer to Problem 26.12OQ
The rank of energy stored of the five capacitors from greatest to smallest capacitance is
Explanation of Solution
Given info: The capacitances of the cases (a), (b) and (d) are
For case (a):
The energy stored of the first capacitor is,
Here,
Thus, the energy stored of the first capacitor is
For case (b):
The energy stored of the first capacitor is,
Here,
Substitute
Thus, the energy stored of the second capacitor is
For case (c):
The energy stored of the third capacitor is,
Here,
Substitute
Thus, the energy stored of the third capacitor is
For case (d):
The energy stored of the fourth capacitor is,
Here,
Thus, the energy stored of the fourth capacitor is
For case (e):
The energy stored of the fifth capacitor is,
Here,
Thus, the energy stored of the fourth capacitor is
The rank of the charge from highest to lowest is,
Conclusion:
Therefore, the rank of energy stored of the five capacitors from greatest to smallest capacitance is
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Chapter 26 Solutions
Physics for Scientists and Engineers, Technology Update (No access codes included)
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