Find expressions for the equivalent capacitance of (a) N identical capacitors C in parallel and (b) N identical capacitors C in series.

Verified step by step guidance

1. To solve part (a), recall that for capacitors in parallel, the equivalent capacitance is the sum of the individual capacitances. For N identical capacitors, each with capacitance \( C \), the equivalent capacitance \( C_{eq} \) is given by: \( C_{eq} = C + C + \dots + C \) (N terms). This simplifies to \( C_{eq} = N \cdot C \).
2. For part (b), recall that for capacitors in series, the reciprocal of the equivalent capacitance is the sum of the reciprocals of the individual capacitances. For N identical capacitors, each with capacitance \( C \), the equation becomes: \( \frac{1}{C_{eq}} = \frac{1}{C} + \frac{1}{C} + \dots + \frac{1}{C} \) (N terms).
3. Simplify the series equation: \( \frac{1}{C_{eq}} = \frac{N}{C} \).
4. Take the reciprocal to find the equivalent capacitance for capacitors in series: \( C_{eq} = \frac{C}{N} \).
5. Summarize the results: (a) For N capacitors in parallel, \( C_{eq} = N \cdot C \). (b) For N capacitors in series, \( C_{eq} = \frac{C}{N} \).

Key Concepts

Capacitance

Capacitance is the ability of a system to store electric charge per unit voltage. It is measured in farads (F) and is defined as the ratio of the electric charge (Q) stored on a conductor to the potential difference (V) across it. Understanding capacitance is essential for analyzing how capacitors behave in circuits.

Capacitors in Parallel

When capacitors are connected in parallel, the total or equivalent capacitance (C_eq) is the sum of the individual capacitances. This is because each capacitor experiences the same voltage across its terminals, allowing them to store charge independently. The formula for N identical capacitors in parallel is C_eq = N * C, where C is the capacitance of one capacitor.

Capacitors in Series

In a series configuration, the total capacitance is found using the reciprocal of the sum of the reciprocals of the individual capacitances. This occurs because the charge stored on each capacitor is the same, but the voltage across each capacitor can differ. The formula for N identical capacitors in series is 1/C_eq = N/C, leading to C_eq = C/N.