Two condensers of capacity `0.3 muF` and `0.6muF` respectively are connected in series. The combination is connected across a potential of `6` volts. The ratio of energies stored by the condensers will be

Options

A. `(1)/(2)`
B. 2
C. `(1)/(4)`
D. 4

Answer

To find the ratio of energies stored by two capacitors connected in series, we can follow these steps:

### Step 1: Understand the Capacitors in Series
When capacitors are connected in series, they have the same charge (Q) on them. The total voltage (V) across the series combination is the sum of the voltages across each capacitor.

### Step 2: Calculate the Equivalent Capacitance
The formula for the equivalent capacitance (C_eq) of capacitors in series is given by:
\[ \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} \]
Where:
- \(C_1 = 0.3 \, \mu F\)
- \(C_2 = 0.6 \, \mu F\)

Substituting the values:
\[ \frac{1}{C_{eq}} = \frac{1}{0.3} + \frac{1}{0.6} \]

### Step 3: Calculate the Individual Contributions
Calculating the right-hand side:
\[ \frac{1}{C_{eq}} = \frac{2}{0.6} + \frac{1}{0.6} = \frac{3}{0.6} = 5 \]
Thus,
\[ C_{eq} = \frac{1}{5} = 0.2 \, \mu F \]

### Step 4: Calculate the Charge (Q)
The total charge (Q) stored in the capacitors can be calculated using the formula:
\[ Q = C_{eq} \times V \]
Where \(V = 6 \, V\):
\[ Q = 0.2 \, \mu F \times 6 \, V = 1.2 \, \mu C \]

### Step 5: Calculate the Energy Stored in Each Capacitor
The energy (U) stored in a capacitor is given by the formula:
\[ U = \frac{Q^2}{2C} \]

For Capacitor 1 (C1 = 0.3 μF):
\[ U_1 = \frac{Q^2}{2C_1} = \frac{(1.2 \times 10^{-6})^2}{2 \times 0.3 \times 10^{-6}} = \frac{1.44 \times 10^{-12}}{0.6 \times 10^{-6}} = 2.4 \times 10^{-6} \, J \]

For Capacitor 2 (C2 = 0.6 μF):
\[ U_2 = \frac{Q^2}{2C_2} = \frac{(1.2 \times 10^{-6})^2}{2 \times 0.6 \times 10^{-6}} = \frac{1.44 \times 10^{-12}}{1.2 \times 10^{-6}} = 1.2 \times 10^{-6} \, J \]

### Step 6: Find the Ratio of Energies
Now, we can find the ratio of the energies stored in the two capacitors:
\[ \frac{U_1}{U_2} = \frac{2.4 \times 10^{-6}}{1.2 \times 10^{-6}} = 2 \]

### Final Answer
The ratio of energies stored by the capacitors is \(2:1\).

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