---
title: "A circuit is constructed with an ideal battery of emf \\( \\varepsilon \\), a switch S, two resistors of resistances \\( R_1 \\) and \\( R_2 \\), and an initially uncharged capacitor of capacitance \\( C \\), as shown in Figure 1. At time \\( t = 0 \\), the switch S is closed."
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url: "https://nerd-notes.com/ubq/117857/"
date_modified: "2026-08-04T07:59:41+00:00"
---

# A circuit is constructed with an ideal battery of emf \( \varepsilon \), a switch S, two resistors of resistances \( R_1 \) and \( R_2 \), and an initially uncharged capacitor of capacitance \( C \), as shown in Figure 1. At time \( t = 0 \), the switch S is closed.

A circuit is constructed with an ideal battery of emf \( \varepsilon \), a switch S, two resistors of resistances \( R_1 \) and \( R_2 \), and an initially uncharged capacitor of capacitance \( C \), as shown in Figure 1. At time \( t = 0 \), the switch S is closed.

![A circuit schematic showing a single rectangular loop that splits into two parallel branches on the right. The left vertical wire contains a battery with emf \( \varepsilon \), with the longer positive terminal on top. The top horizontal wire contains a switch labeled S, and to its right, a resistor labeled \( R_1 \). After \( R_1 \), the wire splits into two vertical branches. The inner vertical branch contains a resistor labeled \( R_2 \). The outer vertical branch, further to the right, contains a capacitor labeled \( C \). The bottom horizontal wires connect the bottoms of the two vertical branches and route back to the negative terminal of the battery. No other labels, lines, text, or axes appear.](https://nerd-notes.com/wp-content/uploads/ubq-frq-generated/stem-fig-1-1785830380-hj7rAu.jpg)

**Part a)** **Indicate** whether the current through resistor \( R_1 \) increases, decreases, or remains the same immediately after the switch is closed. - [ ] Increases - [ ] Decreases - [ ] Remains the same **Justify** your answer using physical principles. *(3 points)*

**Part b)** **Sketch** a graph of the potential difference \( V_C \) across the capacitor as a function of time \( t \) after the switch is closed. Explicitly label any maximum or asymptotic values on the vertical axis in terms of the given variables. *(2 points)*

**Part c)** **Derive** a differential equation that can be used to solve for the charge \( q \) on the capacitor as a function of time \( t \). Express your answer in terms of \( \varepsilon \), \( R_1 \), \( R_2 \), \( C \), \( q \), and physical constants, as appropriate. *(3 points)*

**Part d)** A student solves the differential equation and determines that the total current supplied by the battery as a function of time is given by the expression: \[ I_1(t) = \dfrac{\varepsilon}{R_1 + R_2} + \dfrac{\varepsilon R_2}{R_1(R_1 + R_2)} e^{-\dfrac{R_1 + R_2}{R_1 R_2 C} t} \] **Explain** how this expression is consistent with your qualitative claim in Part (a). *(2 points)*

**Part e)** Use the student's expression for \( I_1(t) \) from Part (d) to answer the following. *(3 points)*


*The answer key and step-by-step explanation are available to logged-in users at https://nerd-notes.com/ubq/117857/*