Ch 27: Current and Resistance

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 27: Current and Resistance

Problem 1

Chapter 27, Problem 1

The electron drift speed in a 1.0-mm-diameter gold wire is 5.0 x 10⁻⁵ m/s. How long does it take 1 mole of electrons to flow through a cross section of the wire?

Verified step by step guidance

Determine the cross-sectional area of the wire using the formula for the area of a circle:

A = π r^2

, where the radius

r

is half the diameter. Convert the diameter of 1.0 mm to meters before calculating.

Calculate the current density

J

using the drift velocity formula:

J = n e v

_d, where

n

is the number density of conduction electrons in gold,

e

is the elementary charge, and

v

_d is the drift velocity. Look up the number density of conduction electrons for gold.

Find the total charge carried by 1 mole of electrons using Avogadro's number

N

_A and the elementary charge

e

. The total charge is given by

Q = N

_Ae.

Relate the current

I

to the drift velocity using the formula

I = J A

, where

J

is the current density and

A

is the cross-sectional area of the wire.

Calculate the time it takes for 1 mole of electrons to flow through the cross section of the wire using the formula

t = Q / I

, where

Q

is the total charge and

I

is the current.

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Key Concepts

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

Drift Speed

Drift speed refers to the average velocity that a charge carrier, such as an electron, attains due to an electric field in a conductor. It is typically much slower than the speed of the electric signal in the wire, as it represents the net movement of electrons in the direction of the electric field. Understanding drift speed is crucial for calculating how quickly charge can flow through a conductor.

Current

Current is the rate of flow of electric charge in a circuit, measured in amperes (A). It can be calculated using the formula I = nAvq, where n is the charge carrier density, A is the cross-sectional area, and q is the charge of an electron. Knowing the current allows us to determine how long it takes for a specific amount of charge, such as 1 mole of electrons, to pass through a given cross-section of the wire.

Mole of Electrons

A mole of electrons refers to Avogadro's number (approximately 6.022 x 10²³), which quantifies the number of particles in one mole of a substance. In the context of electric charge, one mole of electrons carries a total charge of about -96,485 coulombs. This concept is essential for converting between the number of electrons and the total charge when calculating the time required for that charge to flow through a conductor.