/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 50 Calculate the inductance of an \... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 32: Problem 50

Calculate the inductance of an \(L C\) circuit that oscillates at 120 \(\mathrm{Hz}\) when the capacitance is 8.00\(\mu \mathrm{F}\) .

Short Answer

Expert verified
The inductance of the LC circuit is approximately \(2.749 \mathrm{mH}\).

Step by step solution

01

Recall the formula for the resonant frequency of an LC circuit

The resonant frequency of an LC circuit is given by the formula \(f = \frac{1}{2\pi\sqrt{LC}}\), where \(f\) is the frequency in hertz, \(L\) is the inductance in henries, and \(C\) is the capacitance in farads.
02

Convert the given frequency and capacitance to SI units

The frequency is already given in hertz, which is the SI unit for frequency. Convert the capacitance from microfarads to farads by multiplying by \(1\times10^{-6}\), so \(8.00\mu\mathrm{F} = 8.00\times10^{-6}\mathrm{F}\).
03

Rearrange the formula to solve for inductance, L

Rearrange the formula from Step 1 to solve for \(L\): \(L = \frac{1}{(2\pi f)^2C}\). Insert the converted values of frequency and capacitance into the rearranged formula.
04

Calculate the inductance, L

Substitute \(f = 120\mathrm{Hz}\) and \(C = 8.00\times10^{-6}\mathrm{F}\) into the equation and solve for \(L\): \(L = \frac{1}{(2\pi\times120)^2\times 8.00\times10^{-6}} = \frac{1}{(753.98)^2\times 8.00\times10^{-6}} = \frac{1}{4.545\times10^{9}\times 8.00\times10^{-6}} = 2.749 \times 10^{-3}\mathrm{H}\), which can also be written as \(2.749 \mathrm{mH}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Inductance Calculation
Understanding inductance in an LC circuit is crucial for analyzing its behavior. Inductance, represented by the symbol 'L', is a property of an electrical conductor which opposes the change of current passing through it. It's measured in henries (H).

To calculate inductance within an LC circuit, one must understand the relationship between inductance (L), capacitance (C), and the resonant frequency (f). An LC circuit will naturally oscillate at a resonant frequency that depends on the values of L and C. The resonant frequency formula for an LC circuit is \(f = \frac{1}{2\pi\sqrt{LC}}\), where f is the resonant frequency in hertz (Hz), L is the inductance in henries (H), and C is the capacitance in farads (F).

When given resonant frequency and capacitance values, one can rearrange this formula to solve for L, which gives \(L = \frac{1}{(2\pi f)^2C}\). It's always important to use SI units when performing these calculations for correct results.
Capacitance Conversion
Capacitance is a measure of a component's ability to store an electrical charge. It is denoted by the letter 'C' and is measured in farads (F). In many practical instances, capacitance is provided in units like microfarads (\(\mu\)F), picofarads (pF), or nanofarads (nF), which are sub-multiples of the farad.

To convert these units to farads (the SI unit for capacitance), one can use the following conversions: \(1\mu\mathrm{F} = 1\times10^{-6}\mathrm{F}\), \(1n\mathrm{F} = 1\times10^{-9}\mathrm{F}\), and \(1p\mathrm{F} = 1\times10^{-12}\mathrm{F}\).

In the given exercise, the conversion from microfarads to farads is necessary for consistency and accuracy when using the resonant frequency formula. By multiplying by \(1\times10^{-6}\), one can effectively utilize the values within calculations that require the standard units of farads for capacitance.
Resonant Frequency Formula
The resonant frequency of an LC circuit is a fundamental concept in understanding how these circuits operate. This frequency denotes the rate at which the circuit naturally oscillates without an external energy source.

The formula to calculate the resonant frequency of an LC circuit is \(f = \frac{1}{2\pi\sqrt{LC}}\), where 'f' is the frequency in hertz (Hz), 'L' is the inductance in henries (H), and 'C' is the capacitance in farads (F).

While dealing with resonant frequency, it's imperative that students remember to use the correct units and accurately convert them to SI units if they are not already provided as such. This formula is derived from the fact that the energy in an LC circuit swings back and forth between the inductor and the capacitor, creating a harmonic oscillator. The resonant frequency marks the point at which this energy transfer is most efficient, causing the circuit to resonate at that particular frequency.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

A coil has an inductance of 3.00 \(\mathrm{mH}\) , and the current in it changes from 0.200 \(\mathrm{A}\) to 1.50 \(\mathrm{A}\) in a time of 0.200 \(\mathrm{s}\) . Find the magnitude of the average induced emf in the coil during this time.

Consider two ideal inductors \(L_{1}\) and \(L_{2}\) that have zero internal resistance and are far apart, so that their magnetic fields do not influence each other. (a) Assuming these inductors are connected in series, show that they are equivalent to a single ideal inductor having \(L_{\mathrm{eq}}=L_{1}+L_{2} .\) (b) Assuming these same two inductors are connected in parallel, show that they are equivalent to a single ideal inductor having \(1 / L_{\mathrm{eq}}=1 / L_{1}+1 / L_{2}\) . (c) What If? Now consider two inductors \(L_{1}\) and \(L_{2}\) that have nonzero internal resistances \(R_{1}\) and \(R_{2},\) respectively. Assume they are still far apart so that their mutual inductance is zero. Assuming these inductors are connected in series, show that they are equivalent to a single inductor having \(L_{\mathrm{eq}}=L_{1}+L_{2}\) and \(R_{\mathrm{eq}}=\) \(R_{1}+R_{2} .\) (d) If these same inductors are now connected in parallel, is it necessarily true that they are equivalent to a single ideal inductor having \(1 / L_{\mathrm{eq}}=1 / L_{1}+1 / L_{2}\) and \(1 / R_{\mathrm{eq}}=1 / R_{1}+1 / R_{2} ?\) Explain your answer.

An \(L C\) circuit consists of a \(20.0-\mathrm{mH}\) inductor and a \(0.500-\mu \mathrm{F}\) capacitor. If the maximum instantaneous current is 0.100 \(\mathrm{A}\) , what is the greatest potential difference across the capacitor?

Two solenoids \(A\) and \(B\) , spaced close to each other and sharing the same cylindrical axis, have 400 and 700 turns, respectively. A current of 3.50 A in coil A produces an average flux of 300\(\mu\) Whb through each turn of \(A\) and a flux of 90.0\(\mu \mathrm{Wb}\) through each turn of \(\mathrm{B}\) . (a) Calculate the mutual inductance of the two solenoids. (b) What is the self- inductance of \(\mathrm{A}\) ? (c) What emf is induced in \(\mathrm{B}\) when the current in \(\mathrm{A}\) increases at the rate of 0.500 \(\mathrm{A} / \mathrm{s}\) ?

Assume that the magnitude of the magnetic field outside a sphere of radius \(R\) is \(B=B_{0}(R / r)^{2},\) where \(B_{0}\) is a constant. Determine the total energy stored in the magnetic field outside the sphere and evaluate your result for \(B_{0}=\) \(5.00 \times 10^{-5} \mathrm{T}\) and \(R=6.00 \times 10^{6} \mathrm{m},\) values appropriate for the Earth's magnetic field.
See all solutions

Recommended explanations on Physics Textbooks

Electromagnetism

Read Explanation

Work Energy and Power

Read Explanation

Wave Optics

Read Explanation

Force

Read Explanation

Mechanics and Materials

Read Explanation

Linear Momentum

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.