The three condensers of capacitances 10 , 20 and `30muF` are first connected in series and then connected in parallel. The ratio of the resultant capacitance in the two cases respectively is -

Options

A. `1:11`
B. `11:1`
C. `1:6`
D. `6:1`

Answer

To solve the problem of finding the ratio of the resultant capacitance when three capacitors (10 µF, 20 µF, and 30 µF) are connected in series and then in parallel, we will follow these steps:

### Step 1: Calculate the Equivalent Capacitance in Series
The formula for the equivalent capacitance \( C_S \) of capacitors in series is given by:

\[ \frac{1}{C_S} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} \]

Substituting the values of the capacitors:

\[ \frac{1}{C_S} = \frac{1}{10} + \frac{1}{20} + \frac{1}{30} \]

### Step 2: Find a Common Denominator
To add these fractions, we need a common denominator. The least common multiple of 10, 20, and 30 is 60.

\[ \frac{1}{C_S} = \frac{6}{60} + \frac{3}{60} + \frac{2}{60} = \frac{6 + 3 + 2}{60} = \frac{11}{60} \]

### Step 3: Calculate \( C_S \)
Now, we can find \( C_S \):

\[ C_S = \frac{60}{11} \, \mu F \]

### Step 4: Calculate the Equivalent Capacitance in Parallel
The formula for the equivalent capacitance \( C_P \) of capacitors in parallel is simply the sum of their capacitances:

\[ C_P = C_1 + C_2 + C_3 \]

Substituting the values:

\[ C_P = 10 + 20 + 30 = 60 \, \mu F \]

### Step 5: Find the Ratio of \( C_S \) to \( C_P \)
Now, we need to find the ratio of the capacitance in series to the capacitance in parallel:

\[ \text{Ratio} = \frac{C_S}{C_P} = \frac{\frac{60}{11}}{60} \]

This simplifies to:

\[ \text{Ratio} = \frac{60}{11 \times 60} = \frac{1}{11} \]

### Final Answer
The ratio of the resultant capacitance in series to that in parallel is:

\[ \text{Ratio} = \frac{1}{11} \]

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