Step 1: Equivalent Capacitance for Series Capacitors

Capacitors in series follow the rule:

First Series Group:

• C₁ = 6.52µF
• 6µF capacitor

1/Cₓ = 1/6.52 + 1/6
Cₓ = 3.125µF

Second Series Group:

• C₂ = 8.54µF
• 2µF capacitor

1/Cᵧ = 1/8.54 + 1/2
Cᵧ = 1.62µF

Step 2: Parallel Capacitor Combination

For capacitors in parallel:

C_eq = 3.125µF + 1.62µF = 4.745µF

Step 3: Calculate Total Charge

Using the formula:

V = 90V, C = 4.745µF
Q_T = 4.745 × 10⁻⁶ × 90 = 4.275 × 10⁻⁴ C

Step 4: Charge in Parallel Branches

Since Cₓ and Cᵧ are in parallel, they share the same voltage:

Vₓ = Vᵧ = 90V

Charge on Cₓ:

Charge on Cᵧ:

Step 5: Voltages in Series Capacitors

Group Cₓ: [6.52µF, 6µF]

Charge Q₁ = Q₆ = Qₓ = 2.8125 × 10⁻⁴ C

• V₁ = Q / C = 2.8125e-4 / 6.52e-6 = 43.14V
• V₆ = Q / C = 2.8125e-4 / 6e-6 = 46.87V

Group Cᵧ: [8.54µF, 2µF]

Charge Q₂ = Q₂µF = Qᵧ = 1.458 × 10⁻⁴ C

• V₂ = Q / C = 1.458e-4 / 8.54e-6 = 17.1V
• V₂µF = Q / C = 1.458e-4 / 2e-6 = 72.9V

Final Answers

(a) Equivalent Capacitance:

C_eq = 4.745µF

(b) Charges:

• Q₁ = Q₆ = 2.8125 × 10⁻⁴ C
• Q₂ = Q₂µF = 1.458 × 10⁻⁴ C

(c) Voltages:

• V₁ = 43.14V
• V₆ = 46.87V
• V₂ = 17.1V
• V₂µF = 72.9V

Concepts Used

• Series Capacitors: Reciprocal formula, same charge, divided voltage.
• Parallel Capacitors: Sum of capacitances, same voltage, different charges.
• Q = C × V and V = Q / C used for all calculations.

Use Cases

This analysis is widely applicable in:

• Power supply design
• Signal filtering and conditioning
• Timing and delay circuits
• Energy storage and release mechanisms