Two capacitors connected in parallel produce an equivalent capacitance of 32.9-μF, but when connected in series the equivalent capacitance is only 5.5 μF. What is the individual capacitance of each capacitor?

Verified step by step guidance

1. Step 1: Understand the problem. Two capacitors are connected in parallel and series, and their equivalent capacitances are given. The goal is to find the individual capacitances of each capacitor, denoted as \( C_1 \) and \( C_2 \).
2. Step 2: Recall the formulas for equivalent capacitance. For capacitors in parallel, the equivalent capacitance is \( C_{eq} = C_1 + C_2 \). For capacitors in series, the equivalent capacitance is given by \( \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} \).
3. Step 3: Use the parallel formula to set up the first equation. Substitute the given parallel equivalent capacitance: \( C_1 + C_2 = 32.9 \, \mu\text{F} \).
4. Step 4: Use the series formula to set up the second equation. Substitute the given series equivalent capacitance: \( \frac{1}{5.5} = \frac{1}{C_1} + \frac{1}{C_2} \). Rewrite this equation as \( \frac{1}{C_1} + \frac{1}{C_2} = 0.1818 \, \text{(approx)} \).
5. Step 5: Solve the system of equations. Use the two equations \( C_1 + C_2 = 32.9 \, \mu\text{F} \) and \( \frac{1}{C_1} + \frac{1}{C_2} = 0.1818 \) to find the values of \( C_1 \) and \( C_2 \). This involves algebraic manipulation or substitution to solve for the individual capacitances.

Key Concepts

Capacitance in Parallel

When capacitors are connected in parallel, the total or equivalent capacitance is the sum of the individual capacitances. This means that if two capacitors, C1 and C2, are connected in parallel, the equivalent capacitance (Ceq) can be calculated using the formula Ceq = C1 + C2. This configuration allows for increased capacitance, as the charge storage capability is enhanced.

Capacitance in Series

In a series connection, the equivalent capacitance is found using the formula 1/Ceq = 1/C1 + 1/C2. This means that the reciprocal of the total capacitance is equal to the sum of the reciprocals of the individual capacitances. Series connections result in a lower equivalent capacitance compared to the individual capacitors, as the charge must pass through each capacitor sequentially.

Solving for Individual Capacitances

To find the individual capacitances of capacitors connected in both parallel and series, one can set up a system of equations based on the known equivalent capacitances. By using the equations for capacitance in parallel and series, one can solve for the unknown values of C1 and C2. This often involves algebraic manipulation and substitution to isolate the variables.