Ch 32: AC Circuits

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 32: AC Circuits

Problem 19b

Chapter 32, Problem 19b

What are V_R and V_C if the emf frequency in FIGURE EX32.19 is 25 kHz?

Verified step by step guidance

Identify the components of the circuit in FIGURE EX32.19. Typically, this type of problem involves a resistor (R), a capacitor (C), and an inductor (L) in an AC circuit. The voltage across the resistor (Vᵣ) and the capacitor (Vc) need to be determined at a given frequency.

Recall the formulas for the voltage across the resistor and capacitor in an AC circuit. The voltage across the resistor is given by:

V_{r} = I R

, where I is the current and R is the resistance. The voltage across the capacitor is given by:

V_{c} = I X

_c, where

X_{c} = 1 / (2 π f C)

is the capacitive reactance.

Determine the capacitive reactance

X_{c}

using the formula:

X_{c} = 1 / (2 π f C)

. Substitute the given frequency (f = 25 kHz) and the capacitance (C) from the circuit diagram into the formula.

Calculate the current (I) in the circuit. Use Ohm's law for AC circuits:

I = \(\text{emf}\) / Z

, where Z is the total impedance of the circuit. The impedance Z can be calculated using the formula:

Z = \(\text{sqrt}\)(R^2 + (X

_L - X_C)^2), where

X_{L} = 2 π f L

is the inductive reactance.

Finally, substitute the values of I, R, and

X_{c}

into the formulas for

V_{r}

and

V_{c}

to determine the voltages across the resistor and capacitor, respectively.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Electromotive Force (emf)

Electromotive force (emf) is the voltage generated by a source such as a battery or generator when no current is flowing. It represents the energy per unit charge produced by the source and is crucial for understanding how electrical circuits operate. In the context of alternating current (AC) circuits, the frequency of the emf can affect the behavior of the circuit components.

Resonant Frequency

Resonant frequency is the frequency at which a system naturally oscillates when not subjected to a continuous external force. In electrical circuits, this concept is vital for understanding how inductors and capacitors interact, particularly in RLC circuits. At resonant frequency, the impedance is minimized, and the circuit can achieve maximum current flow, which is essential for analyzing the behavior of circuits at specific frequencies.

Voltage and Current Relationships in AC Circuits

In alternating current (AC) circuits, the relationship between voltage and current is influenced by the circuit's impedance, which includes resistance, inductance, and capacitance. The phase difference between voltage and current can lead to different effective values (RMS) for both quantities. Understanding these relationships is crucial for calculating the voltages (Vᵣ and Vc) across various components in response to a given emf frequency.

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07:14

RMS Current and Voltage

Related Practice

Textbook Question

FIGURE EX32.24 shows voltage and current graphs for an inductor. What is the value of the inductance L?

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Textbook Question

The peak current through an inductor is 10 mA. What is the peak current if the emf peak voltage is doubled (at the original frequency)?

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Textbook Question

A high-pass RC filter is connected to an AC source with a peak voltage of 10.0 V. The peak capacitor voltage is 6.0 V. What is the peak resistor voltage?

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Textbook Question

A low-pass filter consists of a 100 μF capacitor in series with a 159 Ω resistor. The circuit is driven by an AC source with a peak voltage of 5.00 V. What is the crossover frequency f_c?