A 8.0 uf (uf means microfarad) capacitor and a 6.0 uf capacitor are connected in series, and the combination is connected in parallel with a 4.0 uf capacitor. What is the equivalent capacitance (in microfarads) of this combination?
The Correct Answer and Explanation is :
To solve this problem, we need to first calculate the equivalent capacitance of the capacitors in the series combination and then find the total equivalent capacitance by considering the capacitors in parallel.
Step 1: Series Combination of 8.0 µF and 6.0 µF Capacitors
When capacitors are connected in series, the reciprocal of the total capacitance (C_{\text{total, series}}) is the sum of the reciprocals of the individual capacitances. The formula for two capacitors (C_1) and (C_2) in series is:
[
\frac{1}{C_{\text{total, series}}} = \frac{1}{C_1} + \frac{1}{C_2}
]
For the given values:
[
\frac{1}{C_{\text{total, series}}} = \frac{1}{8.0 \, \mu F} + \frac{1}{6.0 \, \mu F}
]
To solve this, we first find a common denominator:
[
\frac{1}{C_{\text{total, series}}} = \frac{3}{24} + \frac{4}{24} = \frac{7}{24}
]
Thus, the equivalent capacitance of the series combination is:
[
C_{\text{total, series}} = \frac{24}{7} \, \mu F \approx 3.43 \, \mu F
]
Step 2: Parallel Combination with 4.0 µF Capacitor
Now, the equivalent capacitance of the series combination ((C_{\text{total, series}} = 3.43 \, \mu F)) is connected in parallel with a 4.0 µF capacitor. The total capacitance (C_{\text{total, parallel}}) of capacitors in parallel is the sum of the individual capacitances:
[
C_{\text{total, parallel}} = C_{\text{total, series}} + C_{\text{parallel}}
]
Substituting the values:
[
C_{\text{total, parallel}} = 3.43 \, \mu F + 4.0 \, \mu F = 7.43 \, \mu F
]
Final Answer:
The equivalent capacitance of the combination is approximately 7.43 µF.
Explanation:
- Series Connection: The effective capacitance of capacitors in series is always less than any individual capacitor. The reciprocal of the total capacitance is the sum of the reciprocals of the individual capacitances.
- Parallel Connection: The effective capacitance of capacitors in parallel is the sum of their individual capacitances.
By following these steps, we obtained the total equivalent capacitance of the capacitor network.