10. What is the effective capacitance of this circuit?
a. 1.54 “F
b. 4.5 “F
c. 7.33 “F
d. 19 “F
The effective capacitance of the circuit is 7.33 μF, so option c. 7.33 μF is correct.
Understanding the circuit
In the diagram, the 10 μF capacitor (C1) and the 5 μF capacitor (C2) are connected in series between the same two nodes as the 4 μF capacitor (C3), which is directly connected across those nodes. This makes the series combination of C1 and C2 effectively in parallel with C3.
- Series pair: C1 = 10 μF, C2 = 5 μF
- Parallel branch: C3 = 4 μF
Step 1: Series combination of 10 μF and 5 μF
For capacitors in series, the reciprocal of the equivalent capacitance is the sum of the reciprocals:
1/Cseries = 1/C1 + 1/C2
1/Cseries = 1/10 + 1/5 = 0.1 + 0.2 = 0.3
Cseries = 1/0.3 ≈ 3.33 μF
This use of the standard series formula matches textbook treatments of capacitor networks.
Step 2: Parallel combination with 4 μF
The series equivalent Cseries = 3.33 μF is in parallel with C3 = 4 μF. For capacitors in parallel, capacitances simply add:
Ceq = Cseries + C3 = 3.33 + 4.00 = 7.33 μF
This matches standard parallel-capacitance rules where the total equals the sum of branch capacitances.
Evaluating each option
a. 1.54 μF: Much smaller than any single capacitor here; such a value would arise only if all three were in series, which they are not. Using the series formula for all three would not match the given diagram or node connections.
b. 4.5 μF: Slightly higher than 4 μF, suggesting an incorrect partial use of series/parallel rules (for example, adding 4 μF to an incorrect “0.5 μF” equivalent). This ignores the proper reciprocal calculation for the 10 μF and 5 μF series pair.
c. 7.33 μF (correct): Obtained by correctly combining 10 μF and 5 μF in series to get 3.33 μF and then adding 4 μF in parallel to reach 7.33 μF.
d. 19 μF: This is close to the result of adding all three as if they were in pure parallel (10+5+4=19 μF), which contradicts the series branch shown in the figure and the standard rules for series connection.
Thus, using the conventional series and parallel capacitor formulas, the effective capacitance of this circuit is 7.33 μF, corresponding to option c.
