Heddle / Robinson / Hiller Calculations in Fundamental Physics

Electricity and Magnetism
1. Auflage 2013
ISBN: 978-1-4831-3791-9
Verlag: Elsevier Science & Techn.
Format: EPUB
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 264 Seiten

Calculations in Fundamental Physics, Volume II: Electricity and Magnetism focuses on the processes, methodologies, and approaches involved in electricity and magnetism. The manuscript first takes a look at current and potential difference, including flow of charge, parallel conductors, ammeters, electromotive force and potential difference, and voltmeters. The book then discusses resistance, networks, power, resistivity and temperature, and electrolysis. Topics include shunts and multipliers, resistors in series, distribution circuits, balanced potentiometers, heating, resistance thermometry, and thermistors. The text explains electrolysis and thermoelectricity, including electroplating, Avogadro's number, and thermoelectric power. The manuscript describes magnetic fields and circuits and inductors. Concerns include straight conductors, series circuits, magnetic moments, stored energy, and mutual inductance. The book also takes a look at electric fields, transients, and direct current generators and motors. The manuscript is a dependable reference for readers wanting to be familiar with electricity and magnetism.

Autoren/Hrsg.

Heddle, T.

Robinson, Robert

Hiller, N.

Leseproben

CHAPTER 1

CURRENT AND POTENTIAL DIFFERENCE

Publisher Summary

Two types of charged particles are involved in the structure of any material—the positively charged nuclei of the atoms and the negatively charged electrons moving around the nuclei. When no current flows in a stationary conductor, the continual orbiting and jostling of these particles lead to no resultant large-scale movement of either positive or negative charge one way or another. If an electric cell is connected in a closed circuit of conductors, the cell tends to drive negative particles one way around the circuit and positive the other. The direction of flow of the positive charge is the direction of the current. In most solid conductors, electric currents consist only of those electrons that are free to drift through the solid, while the positive nuclei, with orbiting electrons, remain localized in the solid structure. Many effects of currents are the same whether because of positive particles drifting in one direction or to negative in the other. When an electric current flows in a wire, a magnetic field is produced in the surrounding space or medium. Evidence of this field is provided by the force exerted on a magnetic compass or on another conductor carrying a current. Such a force can be attributed to the existence of magnetic flux, which is imagined to pass around the current causing the field, and to have the same direction as a compass needle.

Flow of Charge

1.1 WORKED EXAMPLE

Two lamps are connected in parallel across a battery of cells. A current of 1.6 A flows through the battery when the current through one lamp is 7.5×1018 electrons per second. Calculate the current through the second lamp (a) in amperes, and (b) in electrons per second.

Introduction

Two types of charged particles are involved in the structure of any material: the positively charged nuclei of the atoms and the negatively charged electrons moving around the nuclei. When no current flows in a stationary conductor, the continual orbiting and jostling of these particles leads to no resultant large scale movement of either positive or negative charge one way or another. If, however, an electric cell is connected in a closed circuit of conductors, the cell tends to drive negative particles one way around the circuit and positive the other. The direction of flow of the positive charge is said to be the direction of the current. Yet in most solid conductors, electric currents consist only of those electrons which are free to drift through the solid, while the positive nuclei, with orbiting electrons, remain localized in the solid structure.

Many effects of currents are the same whether due to positive particles drifting in one direction or to negative in the other. Also the magnitudes of currents are more easily measured by their effects than by attempting to count the numbers or rates of flow of the particles. Thus the unit of current, the ampere (A), is determined by a convenient effect, namely the force exerted between parallel wires carrying currents (as inFig. 1.1). The ampere is that constant current which, in each of two parallel, infinitely long, and straight wires of negligible circular cross-section, spaced one metre apart in a vacuum, produces on each wire a force of 2×10-7 newton per metre length. The number 2×10-7 makes this definition agree with another referring to curved conductors, but now obsolete.

FIG. 1.1 Forces of attraction between current-carrying conductors.

The unit of charge corresponding to the ampere of current is the coulomb. Thus one coulomb (C) is the resultant charge passed in one second through any full cross-section of a constant current of one ampere. In terms of this unit the very small negative charge of each electron is found to be 1.6×10-19 C. Many charges are measured in microcoulombs (µC); 1 µC = 10-6 C. In general, the charge which passes when a constant current flows for a time is given ([A-z]+)

(1.1)

where is in amperes (A), is in coulombs (C), and is in seconds (s).

The question relates to the circuit shown inFig. 1.2, where the number of electrons arriving at any point in any time interval equals the number leaving in the same interval. No means of storing an accumulation of electrons is to be considered in this circuit. Thus at either terminal or ,

FIG. 1.2 Ex. 1.1.

(1.2)

where is the battery current and 1 and 2 are the lamp currents.

Solution

Using eqn. (1.1), the current through the first lamp

Then, from eqn. (1.2),

(a)

Now from eqn. (1.1) the charge passed by 2 in one second = 0.4×1 C.

But each electron is a charge of 1.6×10-19 C. Therefore the rate of flow of electrons in 2 is

b

1.2

Calculate the number of electrons which pass the terminals of an electric heater which draws a continuous current of 8.0 A for a period of 12 h. [2.2×1024.]

1.3

How long would it take to supply a charge of one microcoulomb at the rate of one million electrons per millisecond? What would be the current in amperes? [6.3×103 s; 1.6×10-10 A.]

Parallel Conductors

1.4 WORKED EXAMPLE

Two long, thin, straight, parallel conductors are spaced 6.0 cm apart in a vacuum. They carry currents of 25 A and 30 m A respectively. Calculate (a) the force per unit length exerted on each conductor, and (b) the flux density of the magnetic field across the axis of each conductor.

Introduction

When an electric current flows in a wire, a magnetic field is produced in the surrounding space or medium. Evidence of this field is provided by the force exerted on a magnetic compass or on another conductor carrying a current, as in ex. 1.1. Such a force may be attributed to the existence of magnetic flux, which is imagined to pass around the current causing the field, and to have the same direction as a compass needle.

With two conductors as in the question, each current sets up a field of magnetic flux around it, and thereby exerts a force on the other conductor. In accordance with Newton’s law of action and reaction, the forces on the two conductors are equal and opposite. Hence, in the following analysis, the suffixes 1 and 2 could be interchanged.

In Fig. 1.3 the loops show some of the directions of the magnetic flux produced by current 1 and exerting the force 2 on current 2. The force is perpendicular to both the flux and the current where they cross.

FIG. 1.3 Magnetic field of one conductor crossing another

The force per unit length, 2/2 or 1/1, is found to be proportional to both currents 1 and 2 and inversely proportional to the distance between the conductors. Thus

(1.3)

where is a constant which depends on the system of units and the medium in which the field is set up.

In M.K.S. units, the ampere (A) is defined as that value of both 1 and 2 which makes / = 2×10-7 newton per metre (N m-1), when = 1 metre (m) in a vacuum (see also ex. 1.1).

(1.4)

in vacuum.

The strength of the magnetic field in any small region is represented by the flux density , i.e. the ratio of the quantity of flux passing through the region to the cross-section area through which CFPVII 2

it passes perpendicularly (seeFig. 1.3). Since decreases with increase of , so also must the flux density be less at greater distances from the current which causes it. This is represented roughly by the spacing of the loops of flux inFig. 1.3.

An expression for appears on rearranging eqn. (1.3):

where the factor (1/) indicates the influence of the flux and is therefore the measure of the flux density 1 due to 1 at distance .

Then

(1.5)

and

(1.6)

where 2 is in newtons, 1 and 2 are in amperes, 2 and are in metres, and the unit of 1 is called either a tesla (T) or a weber per square metre (Wb/m2), the weber being the unit of flux.

Solution

(a) 

    Then eqns. (1.3) and (1.4) give the force per unit length:

(b) Across 2, the flux density due to 1 is given by eqn. (1.5):...