### Electrical Circuit Analysis#### Problem Statement:The switch in the circuit has been in position **a** for a long time, so the capacitor is fully charged. The switch is then changed to position **b** at $ t = 0 $.**Questions:**a. What is the current in the circuit immediately after the switch is changed to **b**?b. What is the current in the circuit 25 µs later?c. What is the charge $ Q $ on the capacitor 25 µs later?#### Circuit Diagram:- The circuit consists of a battery with a voltage of 9.0 V.- A capacitor with a capacitance of 1.0 µF.- A resistor with a resistance of 10 Ω.- A switch that can toggle between two positions: **a** and **b**.Initially:- The switch is in position **a**, allowing the capacitor to fully charge to the battery voltage.After switching:- At $ t = 0 $, the switch is flipped to position **b**. This creates a discharge path for the capacitor through the resistor.#### Detailed Analysis:**Immediate Current After Switching:**1. **At $ t = 0 $:** - Since the capacitor is initially fully charged to 9.0 V, right after changing the switch to position **b**, the initial current can be calculated using Ohm's Law: \[ I(0) = \frac{V}{R} = \frac{9.0 \, \text{V}}{10 \, \Omega} = 0.9 \, \text{A} \]**Current 25 µs Later:**2. **At $ t = 25 \, \mu s $:** - The current in an RC circuit discharges exponentially according to the formula: \[ I(t) = I(0) \cdot e^{-\frac{t}{RC}} \] - Here, $ I(0) = 0.9 \, \text{A} $, $ R = 10 \, \Omega $, $ C = 1.0 \, \mu F = 1.0 \times 10^{-6} \, F $ - The time constant $ \tau $ is given by: \[ \tau = R \cdot

R = 10 ohms C = 1 microfarade V = 9 volts

The switch in the circuit has been in position a for a long time, so the capacitor is fully charged. The switch is changed to position b at t = 0. a. What is the current in the circuit immediately after the switch is changed to b? b. What is the current in the circuit 25 µs later? C. What is the charge Q on the capacitor 25 µs later? 9.0 V 1.0 μF 10 Ω

The switch in the circuit has been in position a for a long time, so the capacitor is fully charged. The switch is changed to position b at t = 0. a. What is the current in the circuit immediately after the switch is changed to b? b. What is the current in the circuit 25 µs later? C. What is the charge Q on the capacitor 25 µs later? 9.0 V 1.0 μF 10 Ω

College Physics

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ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Question

### Electrical Circuit Analysis

#### Problem Statement:
The switch in the circuit has been in position **a** for a long time, so the capacitor is fully charged. The switch is then changed to position **b** at $ t = 0 $.

**Questions:**
a. What is the current in the circuit immediately after the switch is changed to **b**?

b. What is the current in the circuit 25 µs later?

c. What is the charge $ Q $ on the capacitor 25 µs later?

#### Circuit Diagram:
- The circuit consists of a battery with a voltage of 9.0 V.
- A capacitor with a capacitance of 1.0 µF.
- A resistor with a resistance of 10 Ω.
- A switch that can toggle between two positions: **a** and **b**.

Initially:
- The switch is in position **a**, allowing the capacitor to fully charge to the battery voltage.

After switching:
- At $ t = 0 $, the switch is flipped to position **b**. This creates a discharge path for the capacitor through the resistor.

#### Detailed Analysis:
**Immediate Current After Switching:**

1. **At $ t = 0 $:**
- Since the capacitor is initially fully charged to 9.0 V, right after changing the switch to position **b**, the initial current can be calculated using Ohm's Law:
\[
I(0) = \frac{V}{R} = \frac{9.0 \, \text{V}}{10 \, \Omega} = 0.9 \, \text{A}
\]

**Current 25 µs Later:**

2. **At $ t = 25 \, \mu s $:**
- The current in an RC circuit discharges exponentially according to the formula:
\[
I(t) = I(0) \cdot e^{-\frac{t}{RC}}
\]
- Here, $ I(0) = 0.9 \, \text{A} $, $ R = 10 \, \Omega $, $ C = 1.0 \, \mu F = 1.0 \times 10^{-6} \, F $
- The time constant $ \tau $ is given by:
\[
\tau = R \cdot

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