**Educational Content: Analyzing an AC Circuit with a Resistor and Capacitor****Concept Overview:**In an AC (Alternating Current) circuit, a resistor with resistance $ R $ and a capacitor with capacitance $ C $ can be connected in series to an AC voltage source. The voltage across the capacitor is a time-dependent function, described by:\[ V_C(t) = V_0 \sin(\omega t) \]where:- $ V_0 $ is the peak voltage,- $ \omega $ is the angular frequency,- $ t $ represents time.**Problem Statement:***Part A: Determining the Amplitude of Current*The task is to find the amplitude $ I_0 $ of the total current $ I(t) $ in the circuit. The variables available for expressing the answer are $ R $, $ C $, $ V_0 $, and $ \omega $.**Proposed Solution Approach:**The current expression provided was:\[ I_0 = V_{C_0} \omega C \]Upon submission, it was noted that this answer is incorrect and does not rely on $ V_0 \omega C $. Students are encouraged to revisit the relationship and formula derivations considering both capacitive and resistive contributions to current.**Submission Feedback:**After entering the equation, feedback indicates "Incorrect; Try Again," with 3 attempts remaining. This suggests reevaluating the dependent variables and how they affect the amplitude of the current.

Answered: A resistor with resistance R and a…

College Physics

11th Edition

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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**Educational Content: Analyzing an AC Circuit with a Resistor and Capacitor**

**Concept Overview:**

In an AC (Alternating Current) circuit, a resistor with resistance $ R $ and a capacitor with capacitance $ C $ can be connected in series to an AC voltage source. The voltage across the capacitor is a time-dependent function, described by:

\[ V_C(t) = V_0 \sin(\omega t) \]

where:
- $ V_0 $ is the peak voltage,
- $ \omega $ is the angular frequency,
- $ t $ represents time.

**Problem Statement:**

*Part A: Determining the Amplitude of Current*

The task is to find the amplitude $ I_0 $ of the total current $ I(t) $ in the circuit. The variables available for expressing the answer are $ R $, $ C $, $ V_0 $, and $ \omega $.

**Proposed Solution Approach:**

The current expression provided was:

\[ I_0 = V_{C_0} \omega C \]

Upon submission, it was noted that this answer is incorrect and does not rely on $ V_0 \omega C $. Students are encouraged to revisit the relationship and formula derivations considering both capacitive and resistive contributions to current.

**Submission Feedback:**

After entering the equation, feedback indicates "Incorrect; Try Again," with 3 attempts remaining. This suggests reevaluating the dependent variables and how they affect the amplitude of the current.

Expert Solution

Step 1: Total current in RC circuit

In a series circuit with a resistor and a capacitor connected to an AC voltage source, the total current $I not stretchy left parenthesis t not stretchy right parenthesis$ can be expressed as:

$I left parenthesis t right parenthesis equals fraction numerator V subscript C subscript 0 end subscript over denominator square root of R squared plus left parenthesis fraction numerator 1 over denominator omega C end fraction right parenthesis squared end root end fraction space sin space left parenthesis omega t plus ϕ right parenthesis$

Where:
$V subscript C subscript 0 end subscript$ is the amplitude of the voltage across the capacitor,
$R$ is the resistance,
$C$ is the capacitance,
$omega$ is the angular frequency of the AC source,
$ϕ$ is the phase angle between the current and voltage.

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