For capacitors in series combination, total capacitance C is given by

Options

A. `(1)/(C)=(1)/(C_(1))+(1)/(C_(2))+ . . . .`
B. `C=C_(1)+C_(2)+ . . . .`
C. `C=C_(1)C_(2)+C_(2)C_(3)+` . . . .
D. `C+(1)/(C_(1)+C_(2)+ . .. . )`.

Answer

To find the total capacitance \( C \) for capacitors in series, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Configuration**:
When capacitors are connected in series, they are connected end to end, and the same charge \( q \) flows through each capacitor. This means that the charge on each capacitor is the same.

2. **Identifying the Potential Differences**:
Let’s denote the capacitances of the capacitors as \( C_1, C_2, \) and \( C_3 \). The potential differences across these capacitors will be \( V_1, V_2, \) and \( V_3 \) respectively. The total potential difference \( V \) across the series combination is given by:
\[ V = V_1 + V_2 + V_3 \]

3. **Using the Definition of Capacitance**:
The capacitance \( C \) of a capacitor is defined as:
\[ C = \frac{q}{V} \]
Therefore, for each capacitor, we can write:
\[ V_1 = \frac{q}{C_1}, \quad V_2 = \frac{q}{C_2}, \quad V_3 = \frac{q}{C_3} \]

4. **Substituting the Values**:
Substituting these values into the total potential difference equation gives:
\[ V = \frac{q}{C_1} + \frac{q}{C_2} + \frac{q}{C_3} \]

5. **Factoring Out the Charge**:
Since \( q \) is common in all terms, we can factor it out:
\[ V = q \left( \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} \right) \]

6. **Expressing Total Capacitance**:
From the definition of capacitance, we can express the total capacitance \( C_{\text{eq}} \) for the series combination as:
\[ C_{\text{eq}} = \frac{q}{V} \]
Substituting for \( V \) gives:
\[ C_{\text{eq}} = \frac{q}{q \left( \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} \right)} = \frac{1}{\left( \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} \right)} \]

7. **Final Expression**:
Therefore, the total capacitance \( C_{\text{eq}} \) for capacitors in series is given by:
\[ \frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} \]