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Chapter 23: Problem 40

The current in an inductor is \(0.20 \mathrm{~A},\) and the frequency is \(750 \mathrm{~Hz}\). If the inductance is \(0.080 \mathrm{H},\) what is the voltage across the inductor?

Short Answer

Expert verified
The voltage across the inductor is 75.398 V.

Step by step solution

01

Identify Given Information

We are given the current \( I = 0.20 \mathrm{~A} \), frequency \( f = 750 \mathrm{~Hz} \), and inductance \( L = 0.080 \mathrm{~H} \). We need to find the voltage \( V \) across the inductor.
02

Use Formula for Inductive Reactance

The inductive reactance \( X_L \) can be calculated using the formula \( X_L = 2\pi f L \). Plug the values into the formula: \( X_L = 2\pi \times 750 \times 0.080 \).
03

Calculate Inductive Reactance

Calculate \( X_L \): \( X_L = 2\pi \times 750 \times 0.080 = 376.99 \mathrm{~惟} \).
04

Apply Ohm's Law for AC Circuits

In AC circuits, voltage \( V \) across the inductor is given by \( V = I \times X_L \). Use the current and the calculated inductive reactance in this formula: \( V = 0.20 \times 376.99 \).
05

Calculate Voltage Across Inductor

Complete the calculation: \( V = 0.20 \times 376.99 = 75.398 \mathrm{~V} \).

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Key Concepts

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

Ohm's Law for AC Circuits
In alternating current (AC) circuits, Ohm's Law adapts to accommodate the roles played by impedance, which is not just resistance but a combination of resistance, inductive reactance, and capacitive reactance. Ohm's Law for AC circuits is expressed as \( V = I \times Z \), where:
  • \( V \) is the voltage across the circuit element.
  • \( I \) is the current flowing through the circuit.
  • \( Z \) is the impedance, which can include resistance \( R \), inductive reactance \( X_L \), and capacitive reactance \( X_C \).
This formula plays a crucial role when dealing with inductive and capacitive components, such as inductors and capacitors in AC circuits. When focusing on an inductor, the impedance is primarily the inductive reactance \( X_L \). Thus, the formula modifies to \( V = I \times X_L \), emphasizing how frequency and inductance affect this relationship.
Inductance
Inductance is a fundamental property of electrical circuits that describes the ability of an inductor to store energy in a magnetic field when electrical current flows through it. It is denoted by \( L \) and measured in henrys (H). An inductor opposes changes in the current passing through it, with this opposition described by inductive reactance.
The inductance arises due to:
  • The physical makeup of the inductor: The number of turns in the coil and the core material.
  • The rate of change of current through the coil: Faster changes mean higher inductive reactance.
The formula for inductive reactance \( X_L \) is \( X_L = 2\pi f L \), where \( f \) is the frequency of the AC source. A higher frequency or a larger inductance will result in a higher inductive reactance, leading to a greater voltage drop across the inductor.
Voltage across Inductor
When alternating current flows through an inductor, a voltage is generated across it as a result of its inductive reactance, \( X_L \). This voltage can be calculated using the relationship derived from Ohm鈥檚 Law for AC circuits: \( V = I \times X_L \).
When the inductive reactance \( X_L \) is known, it can be computed from \( X_L = 2\pi f L \), where \( f \) is the frequency in hertz, and \( L \) is the inductance in henrys.
Key points about voltage across an inductor:
  • It is directly proportional to both the current \( I \) and the inductive reactance \( X_L \).
  • The higher the current or the inductive reactance, the higher the voltage.
  • This voltage leads the current in phase, meaning it reaches its maximum value before the current does in a cycle.
Understanding this helps in designing and analyzing circuits that include inductors, allowing for precise control of alternating voltage and current behavior.

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Most popular questions from this chapter

(a) An ac circuit contains only a resistor and a capacitor in series. Is the phase angle \(\phi\) between the current and the voltage of the generator positive or negative, and how is the impedance \(Z\) of the circuit related to the resistance \(R\) and the capacitive reactance \(X_{\mathrm{C}}\) ? (b) An ac circuit contains only a resistor and an inductor in series. Is the phase angle \(\phi\) positive or negative, and how is the impedance \(Z\) of the circuit related to the resistance \(R\) and the inductive reactance \(X_{\mathrm{L}}\) ? Account for your answers. Problem A series circuit has an impedance of \(192 \Omega\), and the phase angle is \(\phi=-75.0^{\circ}\). The circuit contains a resistor and either a capacitor or an inductor. Find the resistance \(R\) and the capacitive reactance \(X_{\mathrm{C}}\) or the inductive reactance \(X_{\mathrm{L}}\), whichever is appropriate.

An 8.2-mH inductor is connected to an ac generator \((10.0 \mathrm{~V} \mathrm{rms}, 620 \mathrm{~Hz})\). Determine the peak value of the current supplied by the generator.

A charged capacitor and an inductor are connected as shown in the drawing (this circuit is the same as that in Figure \(23-16 a\) ). There is no resistance in the circuit. As Section \(23.4\) discusses, the electrical energy initially present in the charged capacitor then oscillates back and forth between the inductor and the capacitor. (a) What is the amount of electrical energy initially stored in the capacitor? Express your answer in terms of its capacitance \(C\) and the magnitude \(q\) of the charge on each plate. (b) A little later, this energy is transferred completely to the inductor (see Figure \(23-16 b\) ). Write down an expression for the energy stored in the inductor. Give your answer in terms of its inductance \(L\) and the magnitude \(I_{\max }\) of the maximum current in the inductor. (c) If values for \(q, C\), and \(L\) are known, how could one obtain a value for the maximum current in the inductor? Remember that energy is conserved. Problem The initial charge on the capacitor has a magnitude of \(q=2.90 \mu \mathrm{C}\). The capacitance is \(C=3.60 \mu \mathrm{F}\), and the inductance is \(L=75.0 \mathrm{mH}\). (a) What is the electrical energy stored initially in the charged capacitor? (b) Find the maximum current in the inductor.

A circuit consists of a \(3.00-\mu F\) and a \(6.00-\mu F\) capacitor connected in series across the terminals of a 510 -Hz generator. The voltage of the generator is \(120 \mathrm{~V}\). (a) Determine the equivalent capacitance of the two capacitors. (b) Find the current in the circuit.

Multiple-Concept Example 3 reviews some of the concepts needed for this problem. An ac generator has a frequency of \(4.80 \mathrm{kHz}\) and produces a current of \(0.0400 \mathrm{~A}\) in a series circuit that contains only a \(232-\Omega\) resistor and a \(0.250-\mu F\) capacitor. Obtain (a) the voltage of the generator and (b) the phase angle between the current and the voltage across the resistor/capacitor combination.
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