Two identical capacitors are first connected in series and then in parallel. The ratio of equivalent capacitance is

Options

A. `1 : 1`
B. `1 : 2`
C. `1 : 3`
D. `1 : 4`

Answer

To solve the problem of finding the ratio of equivalent capacitance when two identical capacitors are connected in series and then in parallel, we can follow these steps:

### Step 1: Define the Capacitance of Each Capacitor
Let the capacitance of each identical capacitor be \( C \).

### Step 2: Calculate the Equivalent Capacitance for Series Connection
When two capacitors are connected in series, the formula for the equivalent capacitance \( C_{eq, series} \) is given by:

\[ \frac{1}{C_{eq, series}} = \frac{1}{C_1} + \frac{1}{C_2} \]

Since both capacitors have the same capacitance \( C \):

\[ \frac{1}{C_{eq, series}} = \frac{1}{C} + \frac{1}{C} = \frac{2}{C} \]

Thus, the equivalent capacitance for the series connection is:

\[ C_{eq, series} = \frac{C}{2} \]

### Step 3: Calculate the Equivalent Capacitance for Parallel Connection
When the same two capacitors are connected in parallel, the formula for the equivalent capacitance \( C_{eq, parallel} \) is:

\[ C_{eq, parallel} = C_1 + C_2 \]

Again, since both capacitors have the same capacitance \( C \):

\[ C_{eq, parallel} = C + C = 2C \]

### Step 4: Find the Ratio of Equivalent Capacitances
Now, we need to find the ratio of the equivalent capacitance in series to that in parallel:

\[ \text{Ratio} = \frac{C_{eq, series}}{C_{eq, parallel}} = \frac{\frac{C}{2}}{2C} \]

Simplifying this gives:

\[ \text{Ratio} = \frac{C}{2} \cdot \frac{1}{2C} = \frac{1}{4} \]

### Step 5: Express the Ratio
The ratio of the equivalent capacitance when the capacitors are in series to when they are in parallel is:

\[ \text{Ratio} = 1 : 4 \]

Thus, the final answer is that the ratio of equivalent capacitance of capacitors in series to that in parallel is \( 1 : 4 \).

---