Module 4 Electricity

7/30/2019 Module 4 Electricity

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MINISTRY OF EDUCATION AND TRAINING

NONG LAM UNIVERSITY

FACULTY OF FOOD SCIENCE AND TECHNOLOGY

Course: Physics 1

Module 1: Electricity and Magnetism

Instructor: Dr. Son Thanh Nguyen

Academic year: 2008-2009


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Physic 1 Module 4: Electricity and magnetism 2

Contents

Module 4: Electricity and magnetism

4.1. Electromagnetic concepts and law of conservation of electric charge

1. Electromagnetic concepts

2. Law of conservation of electric charge4.2. Electric current

1. Electric current

2. Current density

4.3. Magnetic interaction - Ampres law

1. Magnetic interaction

2. Ampres law for the magnetic field

4.4. Magnetic intensity

1. Magnetic intensity

2. Relationship between magnetic intensity and magnetic induction

4.5. Electromagnetic induction

1. Magnetic flux

2. Faradays law of induction4.6. Magnetic energy.

1. Energy stored in a magnetic field

2. Magnetic energy density


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4.1. Electromagnetic concepts and law of conservation of electric charge

1.Electromagnetic concepts

A magnetic fieldis a vector field which can exert a magnetic force on moving electric chargesand on magnetic dipoles (such as permanent magnets). When placed in a magnetic field,

magnetic dipoles tend to align their axes parallel to the magnetic field. Magnetic fields surround

and are created by electric currents, magnetic dipoles, and changing electric fields. Magneticfields also have their own energy, with an energy density proportional to the square of the field

magnitude.

The magnetic field forms one aspect of electromagnetism. A pure electric field in onereference frame will be viewed as a combination of both an electric field and a magnetic field in

a moving reference frame. Together, the electric and magnetic fields make up the

electromagnetic field, which is best known for underlying light and other electromagnetic

waves.

Electromagnetism describes the relationship between electricity and magnetism.Electromagnetism is essentially the foundation for all of electrical engineering. We use

electromagnets to generate electricity, store memory on our computers, generate pictures on a

television screen, diagnose illnesses, and in just about every other aspect of our lives that

depends on electricity.

Electromagnetism works on the principle that an electric current through a wire generates amagnetic field. We already know that a charge in motion creates a current. If the movement of

the charge is restricted in such a way that the resulting current is constant in time, the field thus

created is called a static magnetic field. Since the current is constant in time, the magnetic field

is also constant in time. The branch of science relating to constant magnetic fields is called

magnetostatics, or static magnetic fields. In this case, we are interested in the determination of

(a) magnetic field intensity, (b) magnetic flux density, (c) magnetic flux, and (d) the energy

stored in the magneticfi

eld.

Linking electricity and magnetism

There is a strong connection between electricity and magnetism. With electricity, there arepositive and negative charges. With magnetism, there are north and south poles. Similar to

charges, like magnetic poles repel each other, while unlike poles attract.

An important difference between electricity and magnetism is that in electricity it is possible tohave individual positive and negative charges. In magnetism, north and south poles are always

found in pairs. Single magnetic poles, known as magnetic monopoles, have been proposed

theoretically, but a magnetic monopole has never been observed.

In the same way that electric charges create electric fields around them, north and south poleswill set up magnetic fields around them. Again, there is a difference. While electric field lines

begin on positive charges and end on negative charges, magnetic field lines are closed loops,

extending from the south pole to the north pole and back again (or, equivalently, from the north

pole to the south pole and back again). With a typical bar magnet, for example, the field goes

from the north pole to the south pole outside the magnet, and back from south to north inside the

magnet.


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Electric fields come from charges. So do magnetic fields, but from moving charges, orcurrents, which are simply a whole bunch of moving charges. In a permanent magnet, the

magnetic field comes from the motion of the electrons inside the material, or, more precisely,

from something called the electron spin. The electron spin is a bit like the Earth spinning on its

axis.

The magnetic field is a vector; the same way the electric field is. The electric field at a

particular point is in the direction of the force a positive charge would experience if it wereplaced at that point. The magnetic field at a point is in the direction of the force a north pole of a

magnet would experience if it were placed there. In other words, the north pole of a compass

points in the direction of the magnetic field that exerts a force on the compass.

The symbol for magnetic field induction or magnetic flux density is the letter B. The SI unit isthe tesla (T).

One of various manifestations of the linking between electricity and magnetism iselectromagnetic induction (see section 4.5). This involves generating a voltage (an induced

electromotive force) by changing the magnetic field that passes through a coil of wire.

In other words, electromagnetism is a two-way link between electricity and magnetism. Anelectric current creates a magnetic field, and a magnetic field, when it changes, creates a voltage.

The discovery of this link led to the invention of transformer, electric motor, and generator. It

also explained what light is and led to the invention of radio.

2. Law of conservation of electric charge

Electric charge

There are two kinds of charge, positive and negative. Like charges repel; unlike charges attract.

Positive charge results from having more protons than electrons; negative charge resultsfrom having more electrons than protons.

Charge is quantized, meaning that charge comes in integer multiples of the elementarycharge e.

Charge is conserved.

Probably everyone is familiar with the first three concepts, but what does it mean for charge tobe quantized? Charge comes in multiples of an indivisible unit of charge, represented by the

letter e. In other words, charge comes in multiples of the charge on the electron or the proton.

These things have the same size charge, but the sign is different. A proton has a charge of +e,

while an electron has a charge of -e. The amount of electric charge is only available in discrete

units. These discrete units are exactly equal to the amount of electric charge that is found on the

electron or the proton.

Electrons and protons are not the only things that carry charge. Other particles (positrons, forexample) also carry charge in multiples of the electronic charge. Putting "charge is quantized" in

terms of an equation, we say:

q = ne (79)


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q is the symbol used to represent charge, while n is a positive or negative integer (n = 0, 1, 2,

3, ), and e is the electronic charge, 1.60 x 10-19

coulombs.

Table of elementary particle masses and charges:

The law of conservation of charge

The law of conservation of charge states thatthe net charge of an isolated system remainsconstant. This law is inherent to all processes known to physics.

In other words, charge conservation is the principle that electric charge can neither be creatednor destroyed. The quantity of electric charge of an isolated systemis always conserved.

If a system starts out with an equal number of positive and negative charges, there is nothingwe can do to create an excess of one kind of charge in that system unless we bring in chargefrom outside the system (or remove some charge from the system). Likewise, if something starts

out with a certain net charge, say +100 e, it will always have +100 e unless it is allowed to

interact with something external to it.

Electrostatic charging

Forces between two electrically-charged objects can be extremely large. Most things areelectrically neutral; they have equal amounts of positive and negative charge. If this was not the

case, the world we live in would be a much stranger place. We also have a lot of control over

how things get charged. This is because we can choose the appropriate material to use in a given

situation.

Metals are good conductors of electric charge, while plastics, wood, and rubber are not. Theyare called insulators. Charge does not flow nearly as easily through insulators as it does through

conductors; that is why wires you plug into a wall socket are covered with a protective rubber

coating. Charge flows along the wire, but not through the coating to you.

Materials are divided into three categories, depending on how easily they will allow charge(i.e., electrons) to flow along them. These are:

conductors - metals, for example,

semi-conductors, silicon is a good example, and insulators, rubber, wood, plastic for example.

Most materials are either conductors or insulators. The difference between them is that inconductors, the outermost electrons in the atoms are so loosely bound to their atoms that they are

free to travel around. In insulators, on the other hand, the electrons are much more tightly bound

to their atoms, and are not free to flow. Semi-conductors are a very useful intermediate class, not

as conductive as metals but considerably more conductive than insulators. By adding certain

impurities to semi-conductors in the appropriate concentrations, the conductivity can be well-

controlled.


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There are three ways that objects can be given a net charge. These are:

1. Charging by friction - this is useful for charging insulators. If you rub one material with

another (say, a plastic ruler with a piece of paper towel), electrons have a tendency to be

transferred from one material to the other. For example, rubbing glass with silk or saran

wrap generally leaves the glass with a positive charge; rubbing PVC rod with fur generally

gives the rod a negative charge.

2. Charging by conduction - useful for charging metals and other conductors. If a charged

object touches a conductor, some charge will be transferred between the object and the

conductor, charging the conductor with the same sign as the charge on the object.

3. Charging by induction - also useful for charging metals and other conductors. Again, a

charged object is used, but this time it is only brought close to the conductor, and does not

touch it. If the conductor is connected to ground (ground is basically anything neutral that

can give up electrons to, or take electrons from, an object), electrons will either flow on to it

or away from it. When the ground connection is removed, the conductor will have a charge

opposite in sign to that of the charged object.

Electric charge is a property of the particles that make up an atom. The electrons that surroundthe nucleus of the atom have a negative electric charge. The protons which partly make up the

nucleus have a positive electric charge. The neutrons which also make up the nucleus have no

electric charge. The negative charge of the electron is exactly equal and opposite to the positive

charge of the proton. For example, two electrons separated by a certain distance will repel one

another with the same force as two protons separated by the same distance, and, likewise, a

proton and an electron separated by the same distance will attract one another with a force of the

same magnitude.

In practice, charge conservation is a physical law that states that the net change in the amountof electric charge in a specific volume of space is exactly equal to the net amount of charge

flowing into the volume minus the amount of charge flowing out of the volume. In essence,charge conservation is an accounting relationship between the amount of charge in a region and

the flow of charge into and out of that same region.

Mathematically, we can state the law as

q(t2) = q(t1) + qin qout (80)

where q(t) is the quantity of electric charge in a specific volume at time t, qinis the amount of

charge flowing into the volume between time t1 and t2, and qout is the amount of charge flowing

out of the volume during the same time period.

The SI unit of electric charge is the coulomb (C).

4.2. Electric current

1. Electric current

Electric current is the flow of electric charge, as shown in Figure 51. The movingelectric charges may be either electrons or ions.


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Figure 51: Charges inmotion through an area A.The time rate at whichcharge flows through thearea is defined as thecurrent intensity I. Thedirection of the current isthe direction in which

positive charges flowwhen freeto do so.

Whenever there is a net flow of charge through some region, a electric current is said to exist.To define current more precisely, suppose that the charges are moving perpendicular to a surface

of area A, as shown in Figure 51. This area could be the cross-sectional area of a wire, for

example.

The electric current intensity I is the rate at which charge flowsthrough this surface. IfQ is the amount of charge that passesthrough this area in a time interval t, the average currentintensity IAV is equal to the charge that passes through A per unit

time:

IAV = Q/ t (81)

If the rate at which charge flows varies in time, then the currentvaries in time; we define the instantaneous current intensity I as

the differential limit of average current:

I =t 0

Q dQlim

t dt

=

(82)

The SI unit of electric current intensity is the ampre (A):1 A = 1 C/1 s. That is, 1 A of current is equivalent to 1 C of charge passing through the surface

area in 1 s.

If the ends of a conducting wire are connected to form a loop, all points on the loop are at thesame electric potential, and hence the electric field is zero within and at the surface of the

conductor. Because the electric field is zero, there is no net transport of charge through the wire,

and therefore there is no current.

If the ends of the conducting wire are connected to a battery, all points on the loop are not atthe same potential. The battery sets up a potential difference between the ends of the loop,

creating an electric field within the wire. The electric field exerts forces on the electrons in the

wire, causing them to move around the loop and thus creating a current. It is common to refer to

a moving charge (positive or negative) as a mobile charge carrier. For example, the mobile

charge carriers in a metal are electrons.

Current direction

The charges passing through the surface, as shown in Figure 51, can be positive or negative, orboth. It is conventional to assign the current direction the same direction as the flow of

positive charge. In electrical conductors, such as copper or aluminum, the current is due to the

motion of negatively charged electrons. Therefore, when we speak of current in an ordinaryconductor, the direction of the current is opposite to that of flow of electrons. However, if we are

considering a beam of positively charged protons in an accelerator, the current is in the direction

of motion of the protons. In some cases - such as those involving gases and electrolytes, for

instance - the current is the result of the flow of both positive and negative charges.

An electric current can be represented by an arrow. The sense of the current arrow is definedas follows:

If the current is due to the motion of positive charges, the current arrow is parallel to the

charge velocity.


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Figure 52: Depicting the

electric current density.

If the current is due to the motion of negative charges, the current arrow is antiparallel to

the charge velocity.

2. Electric current density

Electric current densityJG

isa vector quantity whose magnitude is the ratio of the magnitude of

electric current flowing in a conductor to the cross-sectional area perpendicular to the current

flow and whose direction points in the direction of the current.

In other words, JG

is a vector quantity, and the scalar product of which with the cross-sectional

area vectorAG

is equal to the electric current intensity. By magnitude it is the electric current

intensity divided by the cross-sectional area.

If the current density is constant then

I = JG

.AG

(83)

(scalar product ofJG

and AG

).

If the current density is not constant, then

I = .J dAGG

(84)

where the current is in fact the integral of the dot product of the

current density vectorJG

and the differential surface element dAG

of the conductors cross-sectional area.

The SI unit of J is the ampre per square meter (A/m2).

Electric current density is important to the design of electricaland electronic systems. For example, in the domain of electrical

wiring (isolated copper), maximum current density can vary

from 4 A/mm2

for a wire isolated from free air to 6 A/mm2

for a

wire at free air.

Example:During 4.0 minutes a 5.0-A current is set up in a wire,

find (a) charge quantity in coulombs and (b) number of electrons

passing through any cross section across the wires width.

(Ans. (a) 1.2 x 103

C; (b) 7.5 x 1021

)


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4.3. Magnetic interaction - Ampres law

1. Magnetic interaction

Between two permanent magnets

There are no individual magnetic

poles(or magnetic charges).

Electric charges can

be separated, but magnetic poles

always come in pairs

- one north and one south.

Opposite poles (N and S)

attract and like

poles (N and N,

or S and S) repel.

These bar magnets will remain

"permanent"

until something

happens to eliminate

the alignment of

atomic magnets

in the bar of

iron, nickel,

or cobalt.

Figure 53: Magnetic interaction between two bar magnets(permanent magnets).


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Between an electric currennt and a compass

The connection between electric current and

magnetic field was first observed when the

presence of a current in a wire near a magnetic

compass affected the direction of the compass

needle. We now know that current gives riseto magnetic fields, just as electric charge gave

rise to electric fields.

Figure 54: Compass near a current-carrying

wire.

Magnetic force acting on a moving charge

A charged particle q when moving with velocity vG

in a magnetic field BG

experiences a

magnetic force FG

.

Experiments on various charged particles moving in a magnetic field give the followingresults:

The magnitude Fof the magnetic force exerted on the particle is proportional to the

charge magnitude |q| and to the speed v of the particle.

The magnitude and direction of FG

depend on the velocity vG

of the particle and

on the magnitude and direction of the magnetic field B

G

.

When a charged particle moves parallel to the magnetic field vector, the magnetic

force acting on the particle is zero.

When the particles velocity vector vG

makes any angle 0 with the magneticfield B

G, the magnetic force F

Gacts in a direction perpendicular to both v

Gand B

G; that

is, FG

is perpendicular to the plane formed by vG

and BG

(see Figure 55).


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Mathematiclly the force FG

is given by

F qv x B=G GG

(85)

where the direction of FG

is in the direction ofGG

v x B ifq is positive, which by definition of the

cross product is perpendicular to both vG

and BG

.

We can regard equation (85) as an operational definition of the magnetic field at some point inspace.

The magnitude of the magnetic force is

F = |q|vB sin (86)

where is the smaller angle between vG

and BG

. From this expression, we see that Fis zero when

vG

is parallel or antiparallel to BG

( = 0 or 180) and maximum, Fmax = |q|vB, when vG

is

perpendicular to BG

( = 90).

The direction of the cross product can

be obtained by using a right-hand rule:

the index fingerof the right hand points

in the direction of the first vector ( vG

) in

the cross product, then adjust your wrist

so that you can bend the rest fingers

toward the direction of the second

vector (BG

); extend the thumb to get the

direction of the magnetic force.

Figure 55: Magnetic force acting on a moving charge.

MOTION OF A CHARGED PARTICLE IN A UNIFORM MAGNETIC FIELD

We previously found that the magnetic force acting on a charged particle moving in amagnetic field is perpendicular to the velocity of the particle, and consequently the work done

on the particle by the magnetic force is zero.


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Let us now consider the special case of a positively charged particle moving in a uniformmagnetic field with the initial velocity vector of the particle perpendicular to the field. Let us

assume that the direction of the magnetic field is into the page. Figure 56 shows that the particle

moves in a circle in a plane perpendicular to the magnetic field.

The particle moves in this way because the magnetic force FG

is at right angles to both vG

and

BG

and has a constant magnitude qvB (sin = 1). As the force deflects the particle, the directionsof vG and FG change continuously, as shown in Figure 56.

Because FG

always points toward the center of the circle, it changes only the direction of vG

and not its magnitude. As Figure 56 illustrates, the rotation is counterclockwise for a positive

charge. Ifq were negative, the rotation would be clockwise.

Consequently, a charged particlemoving in a plane perpendicular to a

magnetic field will

move in a circular orbit with the

magnetic force playing the role

of centripetal force. The direction of theforce is given by the right-hand rule.

Equating the centripetal force with themagnetic force and solving for R the

radius of the circular path, we get

mv2/R = |q|vB and

R = mv/|q|B (87)

Figure 56: Motion of a charged particle in a constant

magnetic field.

Example: (a) A proton is moving in a circular orbit of radius 14 cm in a uniform 0.35-T

magnetic field perpendicular to the velocity of the proton. Find the linear speed of the proton.(Ans. v = 4.7 x 10

6m/s)

(b) If an electron moves in a direction perpendicular to the same magnetic field with

this same linear speed, what is the radius of its circular orbit? (Ans. R = 7.6 x 10-5

m)

MAGNETIC FORCE ACTING ON A CURRENT-CARRYING CONDUCTOR

If a magnetic force is exerted on a single charged particle when the particle moves in amagnetic field, it follows that a current-carrying wire also experiences a force when placed in a


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magnetic field. This follows from the fact that the current is a collection of many charged

particles in motion; hence, the resultant force exerted by the field on the wire is the vector sum

of the individual forces exerted on all the charged particles making up the current.

Similar to the force on a movingcharge in a B

Gfield, we have

for a conductor of length l carrying a

current of intensity I in a BG

field the

force experienced by the conductor:

F I l x B=GG G

(88)

where I = JG

.AG

, according to

equation (83).

Figure 57: Magnetic force on a moving charge in a

current-carrying conductor.

Magnetic force between two parallel current carrying wires

Figure 58: Magnetic interaction between two parallel current

carrying wires.


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Consider two long, straight, parallel wires separated by a distance a and carrying currents I1and I2 in the same direction, as illustrated by Figure 58. We can determine the force exerted on

one wire due to the magnetic field set up by the other wire. Wire 1, which carries a current I1,

creates a magnetic field1

BG

at the location of wire 2. The direction of1

BG

is perpendicular to wire

2, as shown in Figure 58. According to equation (88), the magnetic force on a length l of wire 2

is21 2 1

F I l x B=GG G

. Because lG

is perpendicular to1

BG

in this situation, the magnitude of21

FG

is

F21 = I2 l B1. Since the magnitude of 1BG is given by B1 =

0 1

2Ia

, we have

F21 = I2 l (0 1

2

I

a

) = 0 1 2

2

I Il

a

(89)

The direction of 21FG

is toward wire 1 because lG

x1

BG

is in that direction. If the field set up at

wire 1 by wire 2 is calculated, the force12

FG

acting on wire 1 is found to be equal in magnitude

and opposite in direction to21

FG

. This is what we expect because Newtons third law must be

obeyed.

When the currents are in opposite directions (that is, when one of the currents is reversed inFig 56), the forces are reversed and the wires repel each other. Hence, we find thatparallel

straight conductors carrying currents in the same direction attract each other, and parallel

straight conductors carrying currents in opposite directions repel each other.

Because the magnitudes of the forces are the same on both wires, we denote the magnitude ofthe magnetic force between the wires as simply FB. We can rewrite this magnitude in terms of

the magnetic force per unit length:

0 1 2

2

BI IF

l a

= (90)

The SI unit ofFB is the newton (N), and that ofFB/l is the newton per meter (N/m).

2.Ampres law

The magnetic field in space around an electric current is proportional to the electric currentwhich serves as its source, just as the electric field in space is proportional to the charge whichserves as its source. Ampres law states that for any closed loop path, the sum of the lengthelements times the magnetic field in the direction of the length element is equal to the

permeability times the electric current enclosed in the loop (as expressed by equation 91).

(91)


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Oersteds 1819 discovery about deflected compass needles demonstrates that a current-carrying conductor produces a magnetic field. Figure 59a shows how this effect can bedemonstrated in the classroom. Several compass needles are placed in a horizontal plane near along vertical wire. When no current is present in the wire, all the needles point in the samedirection (that of the Earths magnetic field), as expected.

When the wire carries a strong, steady current, the needles all deflect in a direction tangent to

the circle, as shown in Figure 59b. These observations demonstrate that the direction of themagnetic field produced by the current in the wire is consistent with the right-hand ruledescribed in Figure 30.3 (see Hallidays book, page 941).

When the current is reversed, the needles in Figure 59b also reverse. Because the compassneedles point in the direction ofB

G, we conclude that the lines ofB

Gform circles around the

wire, as discussed in the preceding section. By symmetry, the magnitude ofBG

is the sameeverywhere on a circular path centered on the wire and lying in a plane perpendicular to thewire. By varying the current intensity and distance a from the wire, we find thatB isproportional to the current intensity and inversely proportional to the distance from the wire, asdescribed by the following equation

B = 02

I

a

(92)

Now let us evaluate the dot product BG

. d sG

for a small length element ds on the circular path

defined by the compass needles (see Figure 59b) and sum the products for all elements over the

closed circular path. Along this path, the vectors d sG

and BG

are parallel at each point (see Fig.

59b), soBG

. d sG

=B ds. Furthermore, the magnitudeB is constant on this circle and is given by

equation (92). Therefore, the sum of the products BG

. d sG

over the closed path, which is

equivalent to the line integral ofBG

. d sG

, is

00. (2 )

2

IB ds B ds a I

a

= = =

G G

v v (93)

where 2ds a=v is the circumference of the circular path. Although this result was calculatedfor the special case of a circular path surrounding a wire, it holds for a closed path ofany shape

surrounding a currentthat exists in an unbroken circuit.

As a result, the general case, known as Ampres law, can be also stated as follows:The line integral ofB

G. d s

Garound any closed path equals 0I, where I is the total

continuous current passing through any surface bounded by the closed path.

0.B ds I=

G G

v (94)

Ampres law describes the creation of magnetic fields by all continuous currentconfigurations, but at our mathematical level it is useful only for calculating the magnetic field

of current configurations having a high degree of symmetry.


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Applications of Ampres law

1.Magnetic field created by an infinitely long straight wire carrying an electric

current

The magnetic field lines around a long wire which carries an electric current form concentriccircles around the wire. The direction of the magnetic field is perpendicular to the wire and is in

the direction the fingers of your right hand would curl if you wrapped them around the wire withyour thumb in the direction of the current (see Figure 58).

The magnitude of the magnetic field vectorBG

produced by a current-carrying straight wire

depends on the intensity of the current. It is also inversely proportional to the distance from the

wire, as given by equation (92).

Figure 59: (a) When no current is present in the wire, all

compass needles point in the same direction (toward the

Earths north pole).

(b) When the wire carries a strong current, the compass

needles deflect in a direction tangent to the circle, which is the

direction of the magnetic field created by the current.


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Magnetic field created by an infinitely long straight wire carrying an

electric current

The magnetic field of an infinitely longstraight wire can be obtained by applying

Ampere's law. The expression for the

magnitude magnetic field vector is

where r is the distance from the point of

interest to the wire. and 0 the permeability offree space

Figure 60: Depicting the magnetic field created by an infinitely long straight

wire carrying an electric current.

2.Magnetic field created by a long straight coil of wire (solenoid)carrying an

electric current

A long straight coil of wire can be used to generate a nearly uniform magnetic field similar tothat of a bar magnet. Such coils, called solenoids, have an enormous number of practical

applications. The field can be greatly strengthened by the addition of an iron core. Such cores

are typical in electromagnets.


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In equation (95) for the magnetic field BG

inside a solenoid carrying an electric current, n is the

number of turns per unit length, sometimes called the "turns density". The expression is an

idealization to an infinite length solenoid, but provides a good approximation to the field of a

long solenoid.

Solenoid field from Ampres law

Taking a rectangular path about which to evaluate Ampere's law such thatthe length of the side parallel to the solenoid field is Lgives a contribution BL

inside the coil. The field is essentially perpendicular to the sides of the path,

giving negligible contribution. If the end is taken so far from the coil that the

field is negligible, then the length inside the coil is the dominant contribution.

This admittedlyidealized case for

Ampres lawgives

This turns out to be agood approximation for

the solenoid field,

particularly in the case

of an iron core solenoid.

Figure 61: Magnetic field created by a long straight coil of wire(solenoid) carrying an electric current.

(95)

3. Magnetic field created by a toroid carrying an electric current

A device called a toroid(see Figure 62) is often used to create a magnetic field with almostuniform magnitude in some enclosed area. The device consists of a conducting wire wrapped

around a ring (a torus) made of a nonconducting material. For a toroid havingNclosely spaced

turns of wire, we calculate the magnetic field in the region occupied by the torus, a distance r

from the center.

To calculate this field, we must evaluate .G G

vB ds over the circle of radius r, as shownin Figure

62. By symmetry, we see that the magnitude of the field is constant on this circle and tangent to

it, so BG

. d sG

= B ds. Furthermore, note that the circular closed path surroundsNloops of wire,

each of which carries a currentI. Therefore, the right side of equation (93) is 0NIin this case.

Ampres law applied to the circle gives


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B = 0

2

NI

r

(96)

This result shows thatB varies as 1/rand hence is nonuniform in the region occupied by thetorus. However, ifris very large compared with the cross-sectional radius of the torus, then the

field is approximately uniform inside the torus.

For an ideal toroid, in which the turns are closely spaced, the external magnetic field is zero.This can be seen by noting that the net current passing through any circular path lying outside

the toroid (including the region of the hole in the doughnut) is zero. Therefore, from Ampres

law we find thatB = 0 in the regions exterior to the torus.

Finding the magnetic field inside atoroid is a good example of the power

of Ampres law. The current enclosed

by the dashed line is just the number of

loops times the current in each loop.

Amperes law then gives the magnetic

field by

(96)

The toroid is a useful device used ineverything from tape heads totokamaks.

Figure 62: Magnetic field created by a toroid

carrying an electric current.

Magnetic field created by a toroid carrying an electric current = permeability x turn density x

current.

4.4. Magnetic field intensity or magnetic field strength

There are two vectors namelyBG

and HG

characterizing a magnetic field. The vector field BG

is

known among electrical engineers as magnetic flux density ormagnetic induction, or simply

magnetic field, as used by physicists. The vector field HG

is known among electrical engineers as

the magnetic field intensity ormagnetic field strength and is also known among physicists as

auxiliary magnetic fieldormagnetizing field.


20/26


Figure 63: Depicting of the

magnetic.

The magnetic field BG

has the SI unit of teslas (T), equivalent to webers per square meter

(Wb/m). The vector field HG

is measured in amperes per meter (A/m) in the SI units. An older

unit of magnetic field strength is the oersted: 1 A/m = 0.01257 oersted.

The magnetic fields generated by currents and calculated from Ampere's law are characterizedby the magnetic field B

Gmeasured in teslas. However, when the generated fields pass through

magnetic materials which themselves contribute internal magnetic fields, ambiguities can ariseabout what part of the field comes from the external currents and what comes from the material

itself. It has been common practice to define another magnetic field quantity, usually called the

"magnetic field strength" and designated by HG

.

The commonly used form for the relationship between B and H is

B = H (97)

where is the permeability of the medium and given by

= Km0 (98)

0 being the magnetic permeability of free space and Km the relative permeability of the

material. If the material does not respond to the external magnetic field by producing any

magnetization, then Km = 1.

For paramagnetic ( > 0) and diamagnetic ( < 0) materials, the relative permeability is veryclose to 1. For ferromagnetic materials, is much greater than 0.

4.5. Electromagnetic induction

1.Magnetic flux

The magnetic flux, B, through an element of area perpendicular to the direction of magneticfield is given by the product of the magnetic field and the area element. More generally,

magnetic flux is defined by a scalar product of the magnetic field vector and the area element

vector. The SI unit of magnetic flux is the weber (Wb).

The magnetic flux through a surface is proportional to the number of magnetic field lines thatpass through the surface. This is the netnumber, i.e., the

number passing through in one direction minus the number

passing through in the opposite direction.

As illustrated by Figure 63, we divide the surface that hasthe loop as its border into small elements of area dA.For

each element we calculate the differential magnetic flux of

the magnetic field BG

through it:

dB = .B dAGG

= B.dA.cos (99)


21/26


Figure 64: Faradays experiments of induction;

(Left) A permanent magnet approaching a loop.

(Right) Switching the current in one loop induces

a current in another loop.

where is the angle between the normal vector n ( dAG

= n dA) and the magnetic field vectorBG

at the position of the element.

We then integrate all the terms

B = B.dA.cos B.dA = GG

(100)

2. Faradays law of induction

Faraday's experiments

These experiments helped formulate what is known as "Faraday's law of induction."

The circuit shown in the left panel of Figure 64 consists of a wire loop connected to asensitive ammeter (known as a "galvanometer"). If we approach the loop with a permanent

magnet, we see a current being registered by the galvanometer. The results can be summarized

as follows:

i. A current appears only if there is relative motion between the magnet and the loop.

ii. Faster motion results in a larger current intensity.

iii. Ifwe reverse the direction of motion or the polarity of the magnet, the current

reverses sign and flows in the

opposite direction.

The current generated is knownas "induced current"; the

electromotive force (emf) that

appears is known as "inducedemf"; the whole effect is called

"induction."

In the right panel of Figure 64,we show a second type of

experiment in which current is

induced in loop 2 when the switch

S in loop 1 is either closed or

opened. When the current in loop

1 is constant, no induced current

is observed in loop 2.

We see that the magnetic field in an induction experiment can be generated either by apermanent magnet or by an electric current in a coil.

Faraday summarized the results of his experiments in what is known asFaraday's law ofinduction.

An emf is induced in a loop when the number of magnetic field lines (or magnetic

flux) that pass through the loop is changing.


22/26


Figure 65: Depicting Lenzs law.

We can also express Faraday's law of induction in the following form:

The magnitude of the emf induced in a conductive loop is equal to the rate at which

the magnetic fluxB through the loop changes with time.

The corresponding formula is

= -B

d

dt

(101)

where is the induced emf.

If the circuit is a coil consisting ofNloops of the same area and ifBis the flux throughone loop, an emf is induced in every loop; thus, the total induced emf in the coil is given by the

expression

= -N Bd

dt

(102)

The negative sign in equations (101) and (102) is of important physical significance, as

described later.

The SI unif of emf is the volt (V).

Methods for changing the magnetic flux B through a loop

We see that the magnetic flux Bcan be changed and an emf is then induced in a circuit inseveral ways:

The magnitude ofBG

can change with time.

The area enclosed by the loop can change with time.

The angle between the magnetic field vectorBG

and the normal vector n to the loop

can change with time. Any combination of the above can be used.

Lenzs law

Faradays law of induction (equation 101 or equation 102) indicates that the induced emf andthe change in flux have opposite algebraic signs. This has a very real physical interpretation that

has come to be known as Lenzs law:

The polarity of the induced emf is such that it tends to produce a current that

creates a magnetic flux to oppose the change in magnetic flux through the area

enclosed by the current loop.

That is, the induced current tends to keep the original magnetic flux through the circuit fromchanging. This law is actually a consequence of the law of

conservation of energy.

We now concentrate on the negative sign in the equation thatexpresses Faraday's law. The direction of the flow of induced

current in a loop is accurately predicted by what is known as

Lenz's law (or Lenz's rule).


23/26



motional electromotive force.


self-induction.

To understand Lenzs law, we consider an example as shown in Figure 65. In the figure weshow a bar magnet approaching a loop. The induced current flows in the direction indicated

because this current generates an induced magnetic field that has the field lines pointing from

left to right. The loop is then equivalent to a magnet whose north pole faces the corresponding

north pole of the bar magnet that is approaching the loop. The loop then repels the approaching

magnet and thus opposes the change in the original magnetic flux that generated the induced

current.

Example: A coil consists of 200 turns of wire having a total resistance of 2.0 . Eachturn is a square of side 18 cm, and a uniform magnetic field directed perpendicular to the plane

of the coil is turned on. If the field changes linearly from 0 to 0.50 T in 0.80 s,

(a) what is the magnitude of the induced emf in the coil while the field is changing? and

(b) what is the magnitude (intensity) of the induced current in the coil while the field is

changing? (Ans. (a) || = 4.1 V; (b) I = ||/R = 2.05 A)

MOTIONAL ELECTROMOTIVE FORCE

In examples illustrated by Figure 63, we considered cases in which an emf is induced in astationary circuit placed in a magnetic field when the field changes with time. In this section we

describe what is called motional electromotive force, which is the emf induced in a straight

conductor moving through a constant magnetic field.

Consider a loop of width l shown in Figure 66. Part of the loop is located in a region where auniform magnetic field exists. The loop is being pulled outside the magnetic field region with

constant speed v. The magnetic flux through the loop is

B = Blx. This flux decreases with time; according toFaradays law, there is an induced emf given by

= - Bddt

= -Bl

dx

dt= -Blv (103)

Because the resistance of the circuitisR, the intensity (magnitude) of the

induced current in the loop is

I = || = Blv/R (104)

Self induction

If we change the current i through an inductor whose inductance is L,this causes a change in the magnetic flux B= Li through the inductoritself. Using Faraday's law we can determine the resulting emf known as

self-induced emfL


24/26


Figure 68:A series RL circuit.

As the current intensity

increases toward its maximum

value, an emf that opposes the

increasing current is induced

in the inductor.

L = - Bd

dt

= -L

di

dt (105)

We have assumed that L is constant.

If the inductor is an ideal solenoid of cross-sectional area A with N turns, its inductance isgiven by

L = 0(N2/l)A = 0n

2Al (106)

where 0 is the permeability of free space, and n = N/l is the number of turns per unit length or

the turn density of the solenoid.

The permeability may be changed by putting a soft iron core into the solenoid, greatlyincreasing the inductance of the solenoid. In this case we must replace 0 by = Km0 where Km

is the relative permeability of the core; for iron Km is much greater than 1.

The SI unit of L isthe henry (H).

Example: (a) Calculate the inductance of an air-core solenoid containing 300 turns if the

length of the solenoid is 25.0 cm and its cross-sectional area is 4.0 cm2.

(b) Calculate the self-induced emf in the solenoid if the current through it is

decreasing at the rate of 50.0 A/s. (Ans. (a) 0.181 mH; (b) 9.05 mV)

4.6. Magnetic energy

1. Energy stored in a magnetic field

RL circuit

Consider a series RL circuit as shown in Figure 68. When theswitch S is closed, the current immediately starts to increase.

The induced emf (or back emf) in the inductor is large, as the

current is changing rapidly. As time goes on, the current

increases more slowly, and the potential difference across the

inductor decreases.

It takes energy to establish a current in an inductor; thisenergy is carried by the magnetic field inside the inductor.

Considering the emf needed to establish a particular currentand the power involved, we find:

As the current intensity through the coil increases, the magnetic field of the coil also

increases and electrical energy is stored in the coil as a magnetic field. The magnetic energy UB

stored in the coil is given by

UB =1

2LI

2(107)


25/26


In capacitors we found that energy is stored in the electric field between their plates. In

inductors, energy is similarly stored, only now in the magnetic field. Just as with capacitors,

where the electric field is created by a charge on the capacitor and electric energy is stored

inside the capacitors, we now have a magnetic field created when there is a current through the

inductor. Thus, just as with the capacitors, the magnetic energy is stored inside the inductor.

Again, although we introduce the magnetic field energy when talking about energy in

inductors, it is a generic concept whenever a magnetic field is created, it takes energy to do so,and that energy is stored in the field itself.

The SI unit of magnetic energy is the joule (J).

2. Magnetic energy density

For simplicity, consider an ideal solenoid whose inductance is given by

L = o(N2/l)A = on

2Al

The magnetic field inside a solenoid is given by B = onI. As a result I = B/on

Substituting the expressions for Land for I into equation (107) leads to

UB =2

02

B

Al (108)

Because Al = V is the volume of the solenoid, the energy stored per unit volume in themagnetic field orthe magnetic energy density, uB = UB/V,inside the inductor is

(109)

Although this expression was derived for the special case of a solenoid, it is valid for anyregion of space in which a magnetic field exists regardless of its source. From equation (109),

we see that magnetic energy density is proportional to the square of the square of the field

magnitude.

The SI unit of magnetic energy density is the joule per cubic meter (J/m3).

Example: The earths magnetic field in a certain region has the magnitude 6.0 x 10-5

T.

Find the magnetic energy density in this region. (Ans. 1.4 x 10-3 J/m3)


26/26

REFERENCES

1) Halliday, David; Resnick, Robert; Walker, Jearl. (1999) Fundamentals of Physics 7th ed.

John Wiley & Sons, Inc.

2) Feynman, Richard; Leighton, Robert; Sands, Matthew. (1989) Feynman Lectures on Physics.

Addison-Wesley Publishing Company.

3) Serway, Raymond; Faughn, Jerry. (2003) College Physics 7th ed. Thompson, Brooks/Cole.

4) Sears, Francis; Zemansky Mark; Young, Hugh. (1991) College Physics 7th ed. Addison-

Wesley Publishing Company.

5) Beiser, Arthur. (1992) Physics 5th ed. Addison-Wesley Publishing Company.

6) Jones, Edwin; Childers, Richard. (1992) Contemporary College Physics 7th ed. Addison-

Wesley Publishing Company.

7) Alonso, Marcelo; Finn, Edward. (1972) Physics 7th ed. Addison-Wesley Publishing

Company.

8) Michels, Walter; Correll, Malcom; Patterson, A. L. (1968) Foundations of Physics 7th ed.

Addison-Wesley Publishing Company.

9) Hecht, Eugene. (1987) Optics 2th ed. Addison-Wesley Publishing Company.

10) Eisberg, R. M. (1961) Modern Physics, John Wiley & Sons, Inc.

11) WEBSITES

http://ocw.mit.edu/OcwWeb/Physics/8-02TSpring-2005/LectureNotes/index.htm

http://physics.bu.edu/~duffy/PY106/Charge.html

http://science.jrank.org/pages/1729/Conservation-Laws-Conservation-electric-charge.html

http://web.pdx.edu/~bseipel/ch31.pdf

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcur.html#c1

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/magcon.html#c1

Module 4 Electricity

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