Module 4 electricity 2023 (1)

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

MINISTRY OF EDUCATION AND TRAINING

NONG LAM UNIVERSITY

FACULTY OF CHEMICAL ENGINEERING AND FOOD TECHNOLOGY

Course: Physics 1

Module 4: Electricity and magnetism

Instructor: Dr Nguyen Thanh Son

Academic year: 2022-2023

Contents

Module 4: Electricity and magnetism

4.1 Electromagnetic concepts and law of conservation of electric charge

4.1.1 Electromagnetic concepts

4.1.2 Law of conservation of electric charge

4.2 Electric current

4.2.1 Electric current

4.2.2 Electric current density

4.3 Magnetic interaction - Ampère’s law

4.6.1 Energy stored in a magnetic field

4.6.2 Magnetic energy density

4.1 Electromagnetic concepts and law of conservation of electric charge

• Magnetic field forms one aspect of electromagnetic field A pure electric field in one reference

frame will be viewed as a combination of both an electric field and a magnetic field in a moving

reference frame Together, electric and magnetic fields make up 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 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 a magnetic 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 field is called magnetostatics, or static magnetic field 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 magnetic field

♦ Linking electricity and magnetism

• There is a strong connection between electricity and magnetism With electricity, there are positive and negative charges With magnetism, there are north and south poles Similar to electric 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 to have 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 poles will 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

• Electric fields come from electric charges So do magnetic fields, but from moving charges, or currents, 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 that a positive charge would experience if it were placed at that point The magnetic field at a point is in the direction of the force that 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 of

B is the tesla (T)

• One of various manifestations of the linking between electricity and magnetism is

electromagnetic 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 An electric 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

4.1.2 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 results from having more electrons than protons

• Charge is quantized, meaning that charge comes in integer multiples of the elementary charge e

• Charge is conserved

• Probably everyone is familiar with the first three concepts, but what does it mean for charge to

be 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 of the electron or the proton A

proton has a charge of +e, while an electron has a charge of -e The amount of electric charge of any object 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, for example) also carry charge in multiples of the electronic charge Putting "charge is quantized" in terms of an equation, we say:

♦ The law of conservation of electric charge

• The law of conservation of charge states that the net charge of an isolated system remains

constant This law is inherent to all processes known to Physics

• In other words, electric charge conservation is the principle that electric charges can neither

be created nor destroyed The quantity of electric charge of an isolated system is always

conserved

• If a system starts out with an equal number of positive and negative charges, there is nothing

we can do to create an excess of one kind of charge in that system unless we bring in some charge from 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 are electrically 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 They are called insulators Charge does not flow nearly as easily through insulators as it does through conductors; that is the reason 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

• In fact, 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, plastics, for example

• Most materials are either conductors or insulators The difference between them is that in conductors, 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

• 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 surround the 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 amount

of electric charge in a specific volume of space is exactly equal to the net amount of charge

Figure 51 Charges in

motion through an area A The time rate at which charge flows through the area is defined as the current intensity I The direction of the current is the direction in which positive charges flow when free to do so

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 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, q in is the amount of

charge flowing into the volume between time t 1 and t 2 , and q out is the amount of charge flowing out of the volume during the same time period

• Another statement for this law is the net electric charge of an isolated system remains

constant

• The simple version of (80) is

q = constant for an isolated system (80’)

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

4.2 Electric current

4.2.1 Electric current

• Electric current is the flow of electric charge, as shown in Figure 51

The moving electric charges may be either electrons or ions or both

• Whenever there is a net flow of charge through some region, an

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 flows

through this surface If ∆Q is the amount of charge that passes through

this area in a time interval ∆t, the average current intensity Iave is equal to the charge that passes through A per unit time:

• The SI unit of electric current intensity is the ampère (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 the same 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 electric current

• If the ends of the conducting wire are connected to a battery, all points on the loop are not at the 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 an electric 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

♦ Electric current direction

• The charges passing through the surface, as shown in Figure 51, can be positive or negative, or

both It is conventional to assign the electric current direction the same direction as the flow of

positive charge In electrical conductors, such as copper or aluminum, the electric

current is due to the motion of negatively charged electrons Therefore, when we speak of electric current in an ordinary conductor, the direction of the current is opposite to that of the 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 electric 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 electric current arrow is

defined as follows:

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

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

to the charge velocity

Example: During 4 minutes a 5.0 A current is set up in a metal wire, how many (a)

coulombs and (b) electrons pass through any cross section across the wire’s width? ANS: (a) Q = It = 1.2x103 C (b) N = Q/e = 7.5 x 1021

Solution

(a) From (82’) we have Q = It; plugging numbers leads to Q = It = 1.2x103 C

(b) N = Q/|qe| = Q/e = 7.5 x 1021 (e = 1.60 x 10-19 C)

4.2.2 Electric current density

• Electric current density J is a 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, J is a vector quantity, and the scalar product of which with the cross-sectional area vector A is equal to the electric current intensity By magnitude it is the electric current intensity divided by the cross-sectional area

If the electric current density is constant then

(scalar product of J and A) If the current density is not constant, then

Figure 52 Depicting the

electric current density

where the current is in fact the integral of the dot product of the current density vector J and the differential surface element dA

of the conductor’s cross-sectional area

• The SI unit of J is the ampère per square meter (A/m2)

• Electric current density is important to the design of electrical and 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 to 6 A/mm2 for a wire in free air

If J is constant over the whole cross section of the wire and perpendicular to the cross sectional area, Equation 84 becomes

where A is the area of the wire’s cross section In this case, J is parallel to the arrow showing the direction of the electric current

Example: The electric current density in a cylindrical wire of radius R = 2 mm is

uniform across a cross section of the wire and is J = 4.0x105 A/m2 Find the intensity of the electric current through the wire

Solution

J is constant, using (84’) we have I = JA with A = πR2 I = J(πR2) Plugging numbers leads to I = 5.029 A

4.3 Magnetic interaction - Ampère’s law

4.3.1 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 Unlike 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)

♦ Between an electric current 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 electric current gives rise to magnetic fields, just as electric charge gives rise to electric fields

Figure 54 Compass near an electric

current-carrying wire

♦ Magnetic force acting on a moving charge

• A charged particle q when moving with velocity v in a magnetic field B experiences a

• The magnitude and direction of Fdepend on the velocity vof the particle and

on the magnitude and direction of the magnetic field B

• When a charged particle moves parallel to the magnetic field vector, the magnetic force acting on the particle is zero

• When the particle’s velocity vector v makes any angle φ ≠ 0 with the magnetic

field B, the magnetic force F acts in a direction perpendicular to both vand B; that

is, F is perpendicular to the plane formed by vand B (see Figure 55)

• Mathematically, the magnetic force F is given by

where φ is the smaller angle between vand B From this expression, we see that F is zero when

v is parallel or antiparallel to B (φ = 0 or 180°) and maximum, Fmax = |q|vB, when v is

perpendicular to B (φ = 90°)

The direction of the cross product of the two vectors can be obtained by using a right-hand rule:

The index finger of the right hand points in the direction of the first vector

( v ) in the cross product;

then adjust your wrist so that you can bend the rest fingers toward the direction of the second vector (B );

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 a

magnetic field is perpendicular to the velocity of the particle, and consequently the work done

on the particle by the magnetic force is zero

• Let us now consider the special case of a positively charged particle moving in a uniform

magnetic field with the initial velocity vector perpendicular to the field vector B 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 F is at right angles to both v and

B has a constant magnitude |q|vB (sinφ = 1) As the force deflects the particle, the directions of

v and F change continuously, as shown in Figure 56

• Because F always points toward the center of the circle, it changes only the direction of v

and not its magnitude As Figure 56 illustrates, the rotation is counterclockwise for a positive charge If q were negative, the rotation would be clockwise

• Consequently, a charged particle moving in a plane perpendicular to a constant magnetic field will move in a circular orbit with the magnetic force playing the role of centripetal force The direction of the force is given by the right-hand rule

• Equating the centripetal force with the magnetic 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

(uniform) 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 106 m/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)

Solution

(a) Because magnetic field is perpendicular to the velocity of the proton, we can use (87)

v = R|q|B/m; plugging numbers leads to v = 4.7 x 106 m/s (qp = 1.60 x 10-19 C and

mp = 1.67 x 10-27 kg)

(b) Again using (87), we have R = 7.6 x 10-5 m (qe = –1.60 x 10-19 C and me = 9.1 x 10-31

kg)

♦ MAGNETIC FORCE ACTING ON A CURRENT - CARRYING CONDUCTOR

• If a magnetic force is exerted on a single charged particle when the particle moves in a magnetic field, it follows that a current-carrying wire also experiences a force when placed in a magnetic field This follows from the fact that the electric 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 charged particles making up the current

• Similar to the force on a moving charge in a B field, a conductor of

length l carrying an electric current of

I in a B field experiences a magnetic

force given by:

Figure 57 Magnetic force on a current-carrying

conductor placed in a uniform magnetic field

♦ Magnetic force between two parallel electric current - carrying wires

• 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 field B1 at the location of wire 2 The direction of B1 is perpendicular to wire

2, as shown in Figure 58 According to Equation 88, the magnetic force on a length l of wire 2

isF21=I l x B2 1 Because l is perpendicular to B1 in this situation, the magnitude of F21 is

F21 = I2lB1 Since the magnitude of B1 is given by B1 = 0 1

2

I a

µ

π ) =

0 1 22

I I l a

• When the currents are in opposite directions (that is, when one of the currents is reversed in

Fig 58), the forces are reversed and the wires repel each other Hence, we find that parallel

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

Figure 58 Magnetic interaction between two parallel electric

current - carrying wires