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Chapter 23: Problem 85

A 20.0 kHz, 16.0 V source connected to an inductor produces a 2.00 A current. What is the inductance?

Short Answer

Expert verified
The inductance is approximately 63.7 碌H.

Step by step solution

01

Understand the Relationship between Voltage, Current, and Inductance

The voltage across an inductor can be expressed using the formula for an inductor's inductive reactance, which is given by the equation: \( V = I \times 2\text{蟺}fL \), where \( V \) is the voltage across the inductor, \( I \) is the current through the inductor, \( f \) is the frequency, and \( L \) is the inductance. Solving for \( L \), the inductance, we rearrange the equation to: \( L = \frac{V}{I \times 2\text{蟺}f} \).
02

Insert Given Values into the Formula

Substitute the given values into the rearranged formula: \( L = \frac{16.0\text{ V}}{2.00\text{ A} \times 2\text{蟺} \times 20,000\text{ Hz}} \).
03

Calculate the Inductance

Carry out the calculation to find the inductance \( L \): \( L = \frac{16.0}{2 \times 2\text{蟺} \times 20,000} \) henries. Doing the math yields: \( L = \frac{16.0}{4\text{蟺} \times 20,000} \approx \frac{16.0}{251,328} \approx 6.37 \times 10^{-5} \) henries or 63.7 碌H.

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Key Concepts

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

Inductive Reactance
Understanding inductive reactance is crucial when learning about AC circuits and their components. Inductive reactance (\( X_L \) ) represents the opposition that an inductor presents to alternating current (AC) due to the electromagnetic induction it creates. This reactance is proportional to the frequency (\( f \) ) of the AC signal and the inductance (\( L \) ) of the coil.

  • \( X_L = 2\text{蟺}fL \) where \text{蟺} represents the constant Pi (approximately 3.14159), \text{f} is the frequency in hertz (Hz), and \text{L} is the inductance in henries (H).
  • The higher the frequency or the greater the inductance, the greater the reactance will be, which results in a lower current for a given voltage.
  • This concept is particularly important in designing and analyzing AC circuits with inductive components such as transformers, motors, and inductors.

By integrating this knowledge into the exercise solution, we observed that the voltage and current in an AC circuit were used to determine the inductor's inductance, illustrating the role of inductive reactance in limiting the current through the inductor.
Electromagnetic Induction
Electromagnetic induction is the process by which a change in magnetic field generates an electromotive force (EMF) or voltage in a conductor. Discovered by Michael Faraday, it is one of the fundamental principles underlying much of our modern-day electrical technology.

  • Induction occurs when a conductor is exposed to a varying magnetic field, leading to the production of a current called induced current.
  • This process is governed by Faraday's law of electromagnetic induction, which states that the induced electromotive force in any closed circuit is equal to the rate of change of the magnetic flux through the circuit.
  • In the context of our exercise, the inductor coil generates an EMF opposing the change in current due to its inductance, a property that quantifies the magnetic field produced by the current.

The phenomenon of electromagnetic induction is essential for understanding how inductors work in AC circuits, as the inductance directly depends on how effectively the coil converts electrical energy into a magnetic field, and vice versa.
AC Circuits
AC circuits are composed of components that operate with alternating current, which changes direction periodically. Unlike direct current (DC), AC can be easily transformed to different voltages, making it ideal for large-scale power distribution.

  • In an AC circuit, the voltage and current vary sinusoidally with time, often leading to phase differences between them.
  • Key components in an AC circuit include resistors, inductors, and capacitors. Each component reacts differently to AC, presenting resistance, inductive reactance, and capacitive reactance, respectively.
  • Ohm's Law for AC circuits takes these reactances into account, altering the simple DC equation to consider both resistance and reactance, which affects how current flows.

In relation to the exercise, solving for the inductance within an AC circuit required us to understand how the inductive reactance affects the relationship between voltage, current, and frequency. The alternating nature of the circuit dictates how the inductive component influences the overall behavior, reaffirming the importance of considering each component's role when analyzing AC circuits.

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Most popular questions from this chapter

A 75-turn, 10.0 cm diameter coil rotates at an angular velocity of 8.00 rad/s in a 1.25 T field, starting with the plane of the coil parallel to the field. (a) What is the peak emf? (b) At what time is the peak emf first reached? (c) At what time is the emf first at its most negative? (d) What is the period of the AC voltage output?

(a) Estimate the mass of the luminous matter in the known universe, given there are \(10^{11}\) galaxies, each containing \(10^{11}\) stars of average mass \(1.5\) times that of our Sun. (b) How many protons (the most abundant nuclide) are there in this mass? (c) Estimate the total number of particles in the observable universe by multiplying the answer to (b) by two, since there is an electron for each proton, and then by \(10^{9}\), since there are far more particles (such as photons and neutrinos) in space than in luminous matter.

A very large, superconducting solenoid such as one used in MRI scans, stores \(1.00 \mathrm{M} \mathrm{J}\) of energy in its magnetic field when 100 A flows. (a) Find its self-inductance. (b) If the coils "go normal," they gain resistance and start to dissipate thermal energy. What temperature increase is produced if all the stored energy goes into heating the 1000 kg magnet, given its average specific heat is \(200 \mathrm{J} / \mathrm{kg} \cdot^{\circ} \mathrm{C} ?\)

Suppose you have a supply of inductors ranging from \(1.00 \mathrm{nH}\) to \(10.0 \mathrm{H},\) and capacitors ranging from \(1.00 \mathrm{pF}\) to 0.100 F. What is the range of resonant frequencies that can be achieved from combinations of a single inductor and a single capacitor?

What capacitance do you need to produce a resonant frequency of \(1.00 \mathrm{GHz},\) when using an \(8.00 \mathrm{nH}\) inductor?
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