To analyze this circuit, let’s consider it in the context of two key moments: immediately after the switch closes and after a very long time when the capacitors are fully charged. Given the initial information, this appears to be a setup with resistors rather than capacitors based on the image, though you mentioned capacitors initially.

a) Immediately after the switch closes

If there were capacitors, they would initially act as short circuits (since they have no initial charge), meaning current would flow through them as though they were wires. Since the image doesn't show any capacitors explicitly but shows resistors, we'll treat this as a resistive network unless clarified otherwise.

1. Assuming Capacitors in Parallel with Resistors (Hypothetical): If there are hypothetical capacitors parallel to resistors $R_1, R_2, R_3, R_4,$ and $R_5$, then immediately after the switch closes, each branch with a resistor and capacitor pair would carry current due to the capacitors acting as short circuits.

2. Pure Resistive Analysis (Based on Image): If this is just a resistive network, immediately after the switch closes, we would calculate the equivalent resistance of the network and then use Ohm’s Law with the 13 V power source to find the total current.

For resistors arranged in a bridge configuration:

• We would find the equivalent resistance across points $a$ and $b$ and then apply Ohm’s Law: $I = \frac{V}{R_{\text{eq}}}$, where $V = 13$ V.

b) After a Very Long Time (Fully Charged Capacitors Scenario)

If capacitors are indeed in the circuit:

• After a very long time, the capacitors would be fully charged, and no current would flow through the branches containing them, leaving only the resistive branches (if any). In that case, only resistors directly in the path of the battery would conduct current.

If this is a resistive-only network:

• The current would be the same as calculated in (a) for a pure resistive bridge.

Could you clarify whether capacitors are indeed involved in the circuit, or if it's purely resistive? This will help finalize the analysis for both $I_1$ and $I_2$ values.