Ch 30: Electromagnetic Induction

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 30: Electromagnetic Induction

Problem 80

Chapter 30, Problem 80

In recent years it has been possible to buy a 1.0 F capacitor. This is an enormously large amount of capacitance. Suppose you want to build a 1.0 Hz oscillator with a 1.0 F capacitor. You have a spool of 0.25-mm-diameter wire and a 4.0-cm-diameter plastic cylinder. How long must your inductor be if you wrap it with 2 layers of closely spaced turns?

Verified step by step guidance

Determine the formula for the frequency of an LC oscillator. The frequency of an LC circuit is given by:

f = \frac{1}{2 π \sqrt{L C}}

, where

f

is the frequency,

L

is the inductance, and

C

is the capacitance.

Rearrange the formula to solve for the inductance

L

L = \frac{1}{4 π^{2} f^{2} C}

. Substitute the given values:

f = 1.0

Hz and

C = 1.0

Calculate the inductance

L

using the formula derived in the previous step. This will give you the required inductance for the oscillator circuit.

Determine the inductance of a solenoid. The inductance of a solenoid is given by:

L = \frac{μ^{0}}{n^{2}}

, where

μ 0

is the permeability of free space (

4π×10-7

T·m/A),

n

is the number of turns per unit length,

A

is the cross-sectional area of the solenoid, and

l

is the length of the solenoid.

Relate the geometry of the solenoid to the inductance. Use the given wire diameter (0.25 mm), cylinder diameter (4.0 cm), and the fact that there are 2 layers of closely spaced turns to calculate the number of turns per unit length

n

and the cross-sectional area

A

. Substitute these values into the solenoid inductance formula and solve for the required length

l

of the inductor.

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Key Concepts

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

Capacitance

Capacitance is the ability of a system to store an electric charge per unit voltage. It is measured in farads (F), where 1 farad is defined as the capacitance of a capacitor that stores one coulomb of charge at one volt. In this question, the 1.0 F capacitor indicates a significant capacity to store charge, which is crucial for the oscillator's operation.

Inductance

Inductance is the property of an electrical conductor that opposes a change in current. It is measured in henries (H) and is essential for the functioning of inductors in circuits. In this scenario, the inductor's design, including its length and the number of turns, will determine its inductance, which, along with the capacitance, will set the frequency of the oscillator.

Oscillator Frequency

The frequency of an oscillator is determined by the values of the inductor and capacitor in the circuit, following the formula f = 1/(2π√(LC)), where f is the frequency, L is the inductance, and C is the capacitance. For a 1.0 Hz oscillator, the relationship between the inductor and capacitor values must be carefully calculated to achieve the desired frequency, making it essential to understand how these components interact.

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