Two equal capacitors are first connected in series and then in parallel. The ratio of the equivalent capacities in the two cases will be:

Options

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

Answer

To solve the problem of finding the ratio of the equivalent capacities of two equal capacitors connected in series and parallel, we can follow these steps:

### Step 1: Define the capacitance of the capacitors
Let the capacitance of each capacitor be \( C \).

### Step 2: Calculate the equivalent capacitance when capacitors are in series
When two capacitors \( C_1 \) and \( C_2 \) are connected in series, the formula for the equivalent capacitance \( C_{\text{series}} \) is given by:
\[ \frac{1}{C_{\text{series}}} = \frac{1}{C_1} + \frac{1}{C_2} \]
Since both capacitors are equal and have capacitance \( C \):
\[ \frac{1}{C_{\text{series}}} = \frac{1}{C} + \frac{1}{C} = \frac{2}{C} \]
Thus, the equivalent capacitance in series is:
\[ C_{\text{series}} = \frac{C}{2} \]

### Step 3: Calculate the equivalent capacitance when capacitors are in parallel
When two capacitors \( C_1 \) and \( C_2 \) are connected in parallel, the formula for the equivalent capacitance \( C_{\text{parallel}} \) is:
\[ C_{\text{parallel}} = C_1 + C_2 \]
Again, since both capacitors are equal and have capacitance \( C \):
\[ C_{\text{parallel}} = C + C = 2C \]

### Step 4: Find the ratio of the equivalent capacitances
Now we need to find the ratio of the equivalent capacitance in series to that in parallel:
\[ \text{Ratio} = \frac{C_{\text{series}}}{C_{\text{parallel}}} = \frac{\frac{C}{2}}{2C} \]
Simplifying this gives:
\[ \text{Ratio} = \frac{C}{2} \cdot \frac{1}{2C} = \frac{1}{4} \]

### Step 5: Express the ratio in a standard form
The ratio \( \frac{1}{4} \) can be expressed as:
\[ 1 : 4 \]

### Conclusion
The ratio of the equivalent capacities in the two cases is \( 1 : 4 \).

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