Two capacitors of equal capacitance when connected in series hae net capacitance `C_(1)` and when connected in parallel have net capacitance `C_(2)` what is the value of `C_(1)//C_(2)`?

Answer

To solve the problem, we need to find the net capacitance of two capacitors when connected in series and in parallel, and then compute the ratio \( \frac{C_1}{C_2} \).

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

### Step 2: Calculate the net capacitance when connected in series
For two capacitors in series, the formula for the net capacitance \( C_1 \) is given by:
\[ \frac{1}{C_1} = \frac{1}{C} + \frac{1}{C} \]
This simplifies to:
\[ \frac{1}{C_1} = \frac{2}{C} \]
Taking the reciprocal gives:
\[ C_1 = \frac{C}{2} \]

### Step 3: Calculate the net capacitance when connected in parallel
For two capacitors in parallel, the formula for the net capacitance \( C_2 \) is given by:
\[ C_2 = C + C = 2C \]

### Step 4: Calculate the ratio \( \frac{C_1}{C_2} \)
Now, we can find the ratio of \( C_1 \) to \( C_2 \):
\[ \frac{C_1}{C_2} = \frac{\frac{C}{2}}{2C} \]
This simplifies to:
\[ \frac{C_1}{C_2} = \frac{C}{2} \times \frac{1}{2C} = \frac{1}{4} \]

### Final Answer
Thus, the value of \( \frac{C_1}{C_2} \) is:
\[ \frac{C_1}{C_2} = \frac{1}{4} \]

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