Concept explainers
Two capacitors, C1 = 18.0 μF and C2 = 36.0 μF, are connected in series, and a 12.0-V battery is connected across the two capacitors. Find (a) the equivalent capacitance and (b) the energy stored in this equivalent capacitance. (c) Find the energy stored in each individual capacitor. (d) Show that the sum of these two energies is the same as the energy found in part (b). (e) Will this equality always be true, or docs it depend on the number of capacitors and their capacitances? (f) If the same capacitors were connected in parallel, what potential difference would be required across them so that the combination stores the same energy as in part (a)? (g) Which capacitor stores more energy in this situation, C1 or C2?
(a)

Answer to Problem 18P
Explanation of Solution
Given information: The value of capacitor 1 is
Explanation:
The capacitors
Formula to calculate the equivalent capacitance of the system when they are connected in series.
Here,
Substitute
Thus, the equivalent capacitance of the system is
Conclusion:
Therefore, the equivalent capacitance of the system is
(b)

Answer to Problem 18P
Explanation of Solution
Given information: The value of capacitor 1 is
Explanation:
Formula to calculate the energy stored in this equivalent capacitance.
Here,
Substitute
Thus, the energy stored in this equivalent capacitance is
Conclusion:
Therefore, the energy stored in this equivalent capacitance is
(c)

Answer to Problem 18P
Explanation of Solution
Given information: The value of capacitor 1 is
Explanation:
In series connection, the charge will be same in capactor 1 and capacitor 2,
It is given that the total voltage of the battery is
Write the expression to calculate the voltage across capacitor 1.
Substitute
Substitute
Thus, the voltage across capacitor 2 is
Substitute
Thus, the voltage across capacitor 1 is
Formula to calculate the energy stored in the capacitor 1.
Here,
Substitute
Thus, the energy stored in the capacitor 1 is
Formula to calculate the energy stored in the capacitor 2.
Here,
Substitute
Thus, the energy stored in the capacitor 2 is
Conclusion:
Therefore, the energy stored in the capacitor 1 is
(d)

To show: The sum of these two energies is the same as the energy found in part (b).
Answer to Problem 18P
Explanation of Solution
Given information: The value of capacitor 1 is
Explanation:
The energy stored in this equivalent capacitance is
The energy stored in the capacitor 1 is
The energy stored in the capacitor 2 is
Formula to calculate the sum of these two energies.
Here,
Substitute
Thus, the sum of these two energies is the same as the energy found in part (b).
Conclusion:
Therefore, the sum of these two energies is the same as the energy found in part (b) is
(e)

Answer to Problem 18P
Explanation of Solution
Given information: The value of capacitor 1 is
Explanation:
Formula to calculate the energy stored by the capacitor in series.
Here,
Formula to calculate the energy stored by the capacitor in parallel.
Here,
The value of the energy stored by the capacitor in series and the energy stored by the capacitor in parallel are equal so, this equality will always be true.
Thus, this equality will always be true because the energy stored in series and parallel for the capacitors is same.
Conclusion:
Therefore, this equality will always be true because the energy stored in series and parallel for the capacitors is same.
(f)

Answer to Problem 18P
Explanation of Solution
Given information: The value of capacitor 1 is
Explanation:
If the same capacitors are connected in parallel.
Formula to calculate the equivalent capacitance of the system when they are connected in parallel.
Here,
The energy stored in this equivalent capacitance is
Formula to calculate the required potential difference across them so that the combination stores the same energy as in part (b).
Substitute
Substitute
Thus, the required potential difference across them so that the combination stores the same energy as in part (b) is
Conclusion:
Therefore, the required potential difference across them so that the combination stores the same energy as in part (b) is
(g)

Answer to Problem 18P
Explanation of Solution
Given information: The value of capacitor 1 is
Explanation:
The capacitor
Thus, the capacitor
Conclusion:
Therefore, the capacitor
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