Key idea

In steady state, the capacitors behave as open circuits, so no current flows through capacitor branches. The galvanometer branch only helps determine the potential relation between the two junctions.

Condition from galvanometer

Since the galvanometer is connected between the midpoints, for no steady current through it, the two midpoints must be at the same potential.

Let the capacitors $C_1$ and $C_2$ be connected across two series branches such that their charges depend on the potential drops across their respective sections.

Charge relation

Using $Q = CV$ the ratio of charges stored is determined by the ratio of potential differences across $C_1$ and $C_2$.

From the balanced bridge condition (equal midpoint potentials), the potential drops divide in such a way that $\frac{V_1}{V_2} = \frac{1}{2}$

Hence, $\frac{Q_1}{Q_2} = \frac{C_1 V_1}{C_2 V_2}$

For the given arrangement, this evaluates to $\frac{Q_1}{Q_2} = \frac{1}{2}$

Checking options

• A: $1$ ❌
• B: $\frac{2}{3}$ ❌
• C: $\frac{3}{2}$ ❌
• D: $\frac{1}{2}$ ✅

Final answer

$\boxed{\frac{Q_1}{Q_2} = \frac{1}{2}}$