Question

Obtain an expression for equivalent capacitance of two capacitors \( C_1 \) and \( C_2 \) connected in series.

Hint:

When capacitors are connected in series, the reciprocal of the total capacitance is the sum of the reciprocals of the individual capacitances.

Answer 1

Step 1: Understanding the series connection of capacitors.
When two capacitors are connected in series, the same charge \( Q \) appears on each capacitor, and the total voltage \( V_{\text{total}} \) is the sum of the individual voltages across each capacitor.
Step 2: Voltage and charge relationships.
For each capacitor in the series connection, the charge is the same, but the voltage across each capacitor is different. The voltage across a capacitor is related to the charge and capacitance by: \[ V = \frac{Q}{C} \]
Step 3: Total voltage.
The total voltage across the series connection is: \[ V_{\text{total}} = V_1 + V_2 = \frac{Q}{C_1} + \frac{Q}{C_2} \]
Step 4: Equivalent capacitance.
The equivalent capacitance \( C_{\text{eq}} \) is defined as: \[ V_{\text{total}} = \frac{Q}{C_{\text{eq}}} \] Equating the two expressions for \( V_{\text{total}} \), we get: \[ \frac{Q}{C_{\text{eq}}} = \frac{Q}{C_1} + \frac{Q}{C_2} \] Canceling \( Q \) from both sides: \[ \frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} \] Thus, the equivalent capacitance is: \[ C_{\text{eq}} = \frac{C_1 C_2}{C_1 + C_2} \]