Two condensers, one of capacity $C$ and the other of capacity $C/2$ are connected to a $V$ volt battery, as sh

Question

Two condensers, one of capacity $C$ and the other of capacity $C/2$ are connected to a $V$ volt battery, as shown in the figure. The energy stored in the capacitors when both condensers are fully charged will be:

Accepted Answer

The total energy stored in a system of capacitors is the sum of the energies stored in individual capacitors. Based on the options, the capacitors are connected in parallel across the voltage $V$.

1.  **Energy in first capacitor ($C_1 = C$):**
    $$U_1 = \frac{1}{2} C_1 V^2 = \frac{1}{2} CV^2$$
2.  **Energy in second capacitor ($C_2 = C/2$):**
    $$U_2 = \frac{1}{2} C_2 V^2 = \frac{1}{2} \left( \frac{C}{2} ight) V^2 = \frac{1}{4} CV^2$$
3.  **Total Energy:**
    $$U_{	ext{total}} = U_1 + U_2 = \frac{1}{2} CV^2 + \frac{1}{4} CV^2 = \frac{3}{4} CV^2$$

Alternatively, for parallel combination, $C_{	ext{eq}} = C + C/2 = 3C/2$. The total energy is $U = \frac{1}{2} C_{	ext{eq}} V^2 = \frac{1}{2} (3C/2) V^2 = \frac{3}{4} CV^2$ [NCERT Class 12, Physics Part 1, Sec 2.15, Eq 2.69].

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