Derive an expression for the equivalent capacity when capacitors are connected in series.

When capacitors are connected in series, the total or equivalent capacitance can be derived using a straightforward approach. Let's break it down step by step to understand how to arrive at the expression for the equivalent capacity.

Understanding Capacitors in Series

In a series connection, the capacitors are connected end-to-end, and the same charge flows through each capacitor. However, the voltage across each capacitor can vary. The key point to remember is that the total voltage across the series combination is the sum of the voltages across each individual capacitor.

Capacitance Basics

Capacitance (C) is defined as the ability of a capacitor to store charge (Q) per unit voltage (V). The relationship is given by the formula:

C = Q / V

Voltage and Charge in Series

For capacitors connected in series, if we have three capacitors, C1, C2, and C3, the total voltage (V_total) across the series is:

V_total = V1 + V2 + V3

Where V1, V2, and V3 are the voltages across each capacitor. Since the charge (Q) on each capacitor in series is the same, we can express the voltages in terms of charge and capacitance:

V1 = Q / C1

V2 = Q / C2

V3 = Q / C3

Combining the Voltages

Substituting these expressions into the total voltage equation gives us:

V_total = (Q / C1) + (Q / C2) + (Q / C3)

Factoring out the charge Q, we have:

V_total = Q * (1/C1 + 1/C2 + 1/C3)

Finding the Equivalent Capacitance

The equivalent capacitance (C_eq) for the series connection can be defined as:

C_eq = Q / V_total

Substituting our expression for V_total into this equation gives:

C_eq = Q / (Q * (1/C1 + 1/C2 + 1/C3))

After simplifying, we find:

C_eq = 1 / (1/C1 + 1/C2 + 1/C3)

Generalizing the Expression

This formula can be generalized for any number of capacitors connected in series. If you have n capacitors, the equivalent capacitance can be expressed as:

1/C_eq = 1/C1 + 1/C2 + 1/C3 + ... + 1/Cn

Example Calculation

To illustrate this, let’s consider three capacitors with values C1 = 2 µF, C2 = 3 µF, and C3 = 6 µF. Using our formula:

• 1/C_eq = 1/2 + 1/3 + 1/6
• 1/C_eq = 3/6 + 2/6 + 1/6 = 6/6
• C_eq = 1 µF

This means that the equivalent capacitance of these three capacitors in series is 1 µF.

Final Thoughts

Understanding how to derive the equivalent capacitance for capacitors in series is crucial for analyzing circuits. This knowledge helps in designing and troubleshooting electronic systems effectively. Remember, when capacitors are in series, the total capacitance is always less than the smallest individual capacitor in the series.