This image contains a physics problem involving a capacitor, resistors, and Kirchhoff’s loop rule. Here's the breakdown:

Problem Details:

The circuit in the image has:

• A power supply with constant emf (denoted as ε).
• Two switches, $S_1$ and $S_2$.
• Resistors $R_1$ and $R_2$.
• A capacitor with capacitance $C$.

Part (a)(i) Summary:

You are asked to use Kirchhoff's loop rule to derive, but not solve, the differential equation that could be used to find the charge $Q$ on the capacitor after switch $S_1$ is closed at time $t = 0$, while $S_2$ remains open.

The given differential equation is:

$\epsilon - \frac{dQ}{dt} + \frac{Q}{C} = 0$

This equation accounts for the emf, the rate of change of charge on the capacitor, and the relationship between charge and voltage across the capacitor.

Part (a)(ii) Summary:

You are then asked to solve this differential equation to find an expression for $Q(t)$. Using the steps shown in the image:

• The equation transforms to: $\epsilon - \frac{dQ}{dt} + \frac{Q}{C} = 0$
• Rewriting gives: $\frac{dQ}{dt} = \frac{Q}{RC} - \epsilon$
• After integration and applying boundary conditions (i.e., at $t = 0, Q = 0$), the final solution is:

$Q(t) = \epsilon C \left( 1 - e^{-\frac{t}{RC}} \right)$

This describes the charge on the capacitor as a function of time after the switch is closed.

Questions:

1. Would you like more detailed steps for the derivation of this equation?
2. Do you need clarification on Kirchhoff's loop rule or how it applies here?
3. Would you like help understanding how the capacitor charges over time?
4. How does this differential equation relate to the time constant $\tau = RC$?
5. Do you want to explore how the voltage across the capacitor changes over time?

Tip: Understanding the physical meaning of the solution $Q(t)$ helps in visualizing the capacitor’s charging process: initially, the charge grows exponentially and asymptotically approaches $\epsilon C$.