Five equal capacitors connected in series have a resultant capacitance 4uF. What is the ratio of energy stored when the capacitors are connected in series and then parallel and connected to the same source of emf in both the cases is

Options

A. `1 :5`
B. `5 :1`
C. `1 :25`
D. `25 :1`

Answer

To solve the problem, we need to find the ratio of energy stored in capacitors when they are connected in series and parallel. Let's go through the steps systematically.

### Step 1: Determine the value of each capacitor
We know that five equal capacitors are connected in series and the resultant capacitance is given as 4 µF.

Using the formula for capacitors in series:
\[ \frac{1}{C_s} = \frac{1}{C} + \frac{1}{C} + \frac{1}{C} + \frac{1}{C} + \frac{1}{C} = \frac{5}{C} \]
Thus, we have:
\[ C_s = \frac{C}{5} \]
Given \(C_s = 4 \, \mu F\), we can set up the equation:
\[ \frac{C}{5} = 4 \implies C = 20 \, \mu F \]
So, each capacitor has a capacitance of \(20 \, \mu F\).

### Step 2: Calculate energy stored in series
The energy stored in a capacitor is given by the formula:
\[ U = \frac{1}{2} C V^2 \]
For the series combination, the energy stored \(U_S\) is:
\[ U_S = \frac{1}{2} C_s V^2 = \frac{1}{2} \times 4 \, \mu F \times V^2 = 2 V^2 \]

### Step 3: Calculate equivalent capacitance when connected in parallel
When the same capacitors are connected in parallel, the equivalent capacitance \(C_p\) is given by:
\[ C_p = C + C + C + C + C = 5C \]
Substituting the value of \(C\):
\[ C_p = 5 \times 20 \, \mu F = 100 \, \mu F \]

### Step 4: Calculate energy stored in parallel
Now, we calculate the energy stored \(U_P\) when the capacitors are connected in parallel:
\[ U_P = \frac{1}{2} C_p V^2 = \frac{1}{2} \times 100 \, \mu F \times V^2 = 50 V^2 \]

### Step 5: Find the ratio of energies
Now we find the ratio of energy stored in series to that stored in parallel:
\[ \text{Ratio} = \frac{U_S}{U_P} = \frac{2 V^2}{50 V^2} = \frac{2}{50} = \frac{1}{25} \]

### Final Answer
Thus, the ratio of energy stored when the capacitors are connected in series to that when they are connected in parallel is:
\[ \text{Ratio} = 1 : 25 \]