a circuit is constructed with five capacitors and a battery as shown. the values for the capacitors are: c1 = c5 = 3.9 μf, c2 = 2.9 μf, c3 = 6.7 μf, and c4 = 3.2 μf. the battery voltage is v = 12 v.
The Correct Answer and Explanation is :
In this circuit, capacitors are arranged either in series or parallel. To find the total capacitance and answer questions about the circuit, we first need to identify the configuration of the capacitors.
1. Identifying the Configuration:
Capacitors can either be:
- In series, where the total capacitance is lower than any of the individual capacitors, or
- In parallel, where the total capacitance is the sum of the capacitances.
Let’s assume that the capacitors are connected in a mixture of series and parallel based on a typical arrangement.
2. Capacitances in Series:
When capacitors are in series, the formula for the total capacitance is:
[
\frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots
]
3. Capacitances in Parallel:
For parallel capacitors, the total capacitance is the sum:
[
C_{\text{total}} = C_1 + C_2 + \dots
]
4. Solution for Specific Configuration:
Let’s assume that capacitors (C_1) and (C_2) are in series and capacitors (C_3), (C_4), and (C_5) are in parallel.
- For (C_1) and (C_2) in series:
[
\frac{1}{C_{\text{series}}} = \frac{1}{3.9 \, \mu F} + \frac{1}{2.9 \, \mu F}
]
Solving for the series combination gives:
[
C_{\text{series}} = 1.646 \, \mu F
] - For (C_3), (C_4), and (C_5) in parallel:
[
C_{\text{parallel}} = 6.7 \, \mu F + 3.2 \, \mu F + 3.9 \, \mu F = 13.8 \, \mu F
]
5. Total Capacitance of the System:
Now, combine (C_{\text{series}}) with (C_{\text{parallel}}) in series:
[
\frac{1}{C_{\text{total}}} = \frac{1}{1.646 \, \mu F} + \frac{1}{13.8 \, \mu F}
]
Solving for (C_{\text{total}}):
[
C_{\text{total}} = 1.473 \, \mu F
]
6. Charge Stored by the Capacitors:
The total charge stored in the system is given by:
[
Q = C_{\text{total}} \times V
]
With (V = 12 \, V), we get:
[
Q = 1.473 \, \mu F \times 12 \, V = 17.68 \, \mu C
]
Thus, the total charge stored is (17.68 \, \mu C).