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Chapter 23: Problem 47

At what frequency (in \(\mathrm{Hz}\) ) are the reactances of a \(52-\mathrm{mH}\) inductor and a \(76-\mu\) F capacitor equal?

Short Answer

Expert verified
The frequency at which the reactances are equal is approximately 80.08 Hz.

Step by step solution

01

Understand Reactance Concepts

The reactance of an inductor and a capacitor depends on frequency. The inductive reactance \(X_L\) is given by \(X_L = 2\pi f L\), and the capacitive reactance \(X_C\) is \(X_C = \frac{1}{2\pi f C}\). We need to find the frequency \(f\) where \(X_L = X_C\).
02

Set Up the Equation

To find the frequency at which the reactances are equal, set \(X_L\) equal to \(X_C\): \[ 2\pi f L = \frac{1}{2\pi f C} \].
03

Solve For Frequency

Rearrange the equation from Step 2 to solve for frequency \(f\):\[ 2\pi f L = \frac{1}{2\pi f C} \] \[ (2\pi f)^2 = \frac{1}{LC} \] \[ f = \frac{1}{2\pi \sqrt{LC}} \].
04

Substitute Values

Now substitute the given values into the formula. Let \(L = 52\, \mathrm{mH} = 52 \times 10^{-3}\, \mathrm{H}\) and \(C = 76\, \mu \mathrm{F} = 76 \times 10^{-6}\, \mathrm{F}\):\[ f = \frac{1}{2\pi \sqrt{52 \times 10^{-3} \times 76 \times 10^{-6}}} \].
05

Calculate

Calculate the frequency:1. Compute \( LC \): \[ 52 \times 10^{-3} \times 76 \times 10^{-6} = 3.952 \times 10^{-6} \]2. Find the square root: \[ \sqrt{3.952 \times 10^{-6}} = 1.9879 \times 10^{-3} \]3. Use the formula for \( f \): \[ f = \frac{1}{2\pi \times 1.9879 \times 10^{-3}} \approx 80.08 \mathrm{Hz} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reactance
Reactance is a measure of opposition that an inductor or capacitor provides to the change in current. It is an essential concept in alternating current (AC) circuits. Reactance is different from resistance, as it depends on the frequency of the applied AC signal.
There are two types of reactances:
  • Inductive Reactance ( X_L )
  • Capacitive Reactance ( X_C )
Reactance is measured in ohms (鈩�), just like resistance, but it varies with frequency. The overall reactance in a circuit determines how much the current leads or lags the voltage.
In general, reactance plays a crucial role in creating resonant circuits where alternating current can be managed efficiently.
Inductive Reactance
Inductive reactance occurs in a coil or inductor when it opposes changes in current. It works similarly to inertia in physics, resisting alterations.
The formula for inductive reactance is given by: \[ X_L = 2\pi f L \]where:
  • \(X_L\) is inductive reactance (in ohms)
  • \(f\) is the frequency of the AC signal (in Hz)
  • \(L\) is the inductance of the coil (in henries)
As the frequency increases, the inductive reactance increases. This causes current to lag behind voltage by 90 degrees in a purely inductive circuit.
Inductive reactance is used to filter signals and manage how circuits behave over various frequencies.
Capacitive Reactance
Capacitive reactance is seen in capacitors and opposes changes in voltage. Unlike inductive reactance, it allows current to flow more readily as frequency increases.
The formula for capacitive reactance is: \[ X_C = \frac{1}{2\pi f C} \]where:
  • \(X_C\) is capacitive reactance (in ohms)
  • \(f\) is the frequency of the AC signal (in Hz)
  • \(C\) is the capacitance (in farads)
With rising frequency, capacitive reactance decreases, allowing the current to lead the voltage by 90 degrees in a purely capacitive circuit.
Capacitive reactance is commonly used in tuning circuits and signal processing, providing a way to block DC while allowing AC to pass.

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Most popular questions from this chapter

A series circuit contains only a resistor and an inductor. The voltage \(V\) of the generator is fixed. If \(R=16 \Omega\) and \(L=4.0 \mathrm{mH},\) find the frequency at which the current is one-half its value at zero frequency.

A \(40.0-\mu \mathrm{F}\) capacitor is connected across a \(60.0-\mathrm{Hz}\) generator. An inductor is then connected in parallel with the capacitor. What is the value of the inductance if the rms currents in the inductor and capacitor are equal?

A capacitor is connected across an ac generator whose frequency is \(750 \mathrm{Hz}\) and whose peak output voltage is \(140 \mathrm{V} .\) The rms current in the circuit is \(3.0 \mathrm{A}\). (a) What is the capacitance of the capacitor? (b) What is the magnitude of the maximum charge on one plate of the capacitor?

A capacitor (capacitance \(C_{1}\) ) is connected across the terminals of an ac generator. Without changing the voltage or frequency of the generator, a second capacitor (capacitance \(C_{2}\) ) is added in series with the first one. As a result, the current delivered by the generator decreases by a factor of three. Suppose that the second capacitor had been added in parallel with the first one, instead of in series. By what factor would the current delivered by the generator have increased?

A circuit consists of a resistor in series with an inductor and an ac generator that supplies a voltage of 115 V. The inductive reactance is \(52.0 \Omega\), and the current in the circuit is 1.75 A. Find the average power delivered to the circuit.
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