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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

Physic 1 Module 4: Electricity and magnetism

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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 charge 4.2. Electric current 1. Electric current 2. Current density 4.3. Magnetic interaction - Ampère’s law 1. Magnetic interaction 2. Ampère’s 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. Faraday’s law of induction 4.6. 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 field is a vector field which can exert a magnetic force on moving electric charges and 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. Magnetic fields 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 one reference 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 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 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 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 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.


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• Electric fields come from 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 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 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 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.

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 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, for example) 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 that the net charge of an isolated system remains constant. This law is inherent to all processes known to physics.

• In other words, charge conservation is the principle that electric charge 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 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 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 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.


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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 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 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, qin is 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 moving

electric charges may be either electrons or ions.


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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.

• 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 flows through this surface. If ΔQ is the amount of charge that passes through this area in a time interval Δt, the average current intensity IAV is equal to the charge that passes through A per unit time:

IAV = ΔQ/ Δt (81) • If the rate at which charge flows varies in time, then the current varies in time; we define the instantaneous current intensity I as the differential limit of average current:

I =t 0

Q dQlimt dtΔ →

Δ=

Δ (82)

• 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 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 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, or both. 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 ordinary conductor, 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 defined as 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 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 current density is constant then

I = J . A (83) (scalar product of J and A ).

If the current density is not constant, then

I = .J dA∫ (84) 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 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 wire’s width.

(Ans. (a) 1.2 x 103 C; (b) 7.5 x 1021)


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4.3. Magnetic interaction - Ampère’s 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 rise

to 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 v in a magnetic field B experiences a magnetic force F . • Experiments on various charged particles moving in a magnetic field give the following results:

• The magnitude F of 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 F depend on the velocity v of 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 v and B ; that is, F is perpendicular to the plane formed by v and B (see Figure 55).


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• Mathematiclly the force F is given by

F qv x B= (85)

where the direction of F is in the direction of v x B if q is positive, which by definition of the cross product is perpendicular to both v and B . • We can regard equation (85) as an operational definition of the magnetic field at some point in space. • The magnitude of the magnetic force is F = |q|vB sinφ (86) where φ is the smaller angle between v and 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 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.


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• Let us now consider the special case of a positively charged particle moving in a uniform magnetic 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 F is at right angles to both v and B and has a constant magnitude qvB (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

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 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) ♦ 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


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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 moving charge in a B field, we have for a conductor of length l carrying a current of intensity I in a B field the force experienced by the conductor:

F I l x B= (88)

where I = J . A , 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 I1 and 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 1B at the location of wire 2. The direction of 1B is perpendicular to wire 2, as shown in Figure 58. According to equation (88), the magnetic force on a length l of wire 2 is 21 2 1F I l x B= . Because l is perpendicular to 1B in this situation, the magnitude of 21F is

F21 = I2 l B1. Since the magnitude of 1B is given by B1 = 0 1

2Ia

μπ

, we have

F21 = I2 l ( 0 1

2Ia

μπ

) = 0 1 2

2I I l

aμπ

(89)

• The direction of 21F is toward wire 1 because l x 1B is in that direction. If the field set up at

wire 1 by wire 2 is calculated, the force 12F acting on wire 1 is found to be equal in magnitude

and opposite in direction to 21F . This is what we expect because Newton’s third law must be obeyed. • When the currents are in opposite directions (that is, when one of the currents is reversed in Fig 56), 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. • Because the magnitudes of the forces are the same on both wires, we denote the magnitude of the 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

2B I IFl a

μπ

= (90)

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

2. Ampère’s law

• The magnetic field in space around an electric current is proportional to the electric current which serves as its source, just as the electric field in space is proportional to the charge which serves as its source. Ampère’s law states that for any closed loop path, the sum of the length elements 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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• Oersted’s 1819 discovery about deflected compass needles demonstrates that a current-carrying conductor produces a magnetic field. Figure 59a shows how this effect can be demonstrated in the classroom. Several compass needles are placed in a horizontal plane near a long vertical wire. When no current is present in the wire, all the needles point in the same direction (that of the Earth’s 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 the magnetic field produced by the current in the wire is consistent with the right-hand rule described in Figure 30.3 (see Halliday’s book, page 941). • When the current is reversed, the needles in Figure 59b also reverse. Because the compass needles point in the direction of B , we conclude that the lines of B form circles around the wire, as discussed in the preceding section. By symmetry, the magnitude of B is the same everywhere on a circular path centered on the wire and lying in a plane perpendicular to the wire. By varying the current intensity and distance a from the wire, we find that B is proportional to the current intensity and inversely proportional to the distance from the wire, as described by the following equation

B = 0

2Ia

μπ

(92)

• Now let us evaluate the dot product B . d s 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 s and B are parallel at each point (see Fig. 59b), so B . d s = B ds. Furthermore, the magnitude B is constant on this circle and is given by equation (92). Therefore, the sum of the products B . d s over the closed path, which is equivalent to the line integral of B . d s , is

00. (2 )

2IB ds B ds a Ia

μ π μπ

= = =∫ ∫ (93)

where 2ds aπ=∫ is the circumference of the circular path. Although this result was calculated for the special case of a circular path surrounding a wire, it holds for a closed path of any shape surrounding a current that exists in an unbroken circuit. • As a result, the general case, known as Ampère’s law, can be also stated as follows:

The line integral of B . d s around any closed path equals μ0I, where I is the total continuous current passing through any surface bounded by the closed path. 0.B ds Iμ=∫ (94) • Ampère’s law describes the creation of magnetic fields by all continuous current configurations, 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 Ampère’s 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 concentric circles 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 with your thumb in the direction of the current (see Figure 58).

• The magnitude of the magnetic field vector B 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 Earth’s 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 long straight 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 μ0 the permeability of free 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 to that 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 B 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 Ampère’s law

• Taking a rectangular path about which to evaluate Ampere's law such that the length of the side parallel to the solenoid field is L gives 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 admittedly idealized case for Ampère’s law gives

• This turns out to be a good 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 almost uniform 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 having N closely 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 .∫ B ds over the circle of radius r, as shown in Figure 62. By symmetry, we see that the magnitude of the field is constant on this circle and tangent to it, so B . d s = B ds. Furthermore, note that the circular closed path surrounds N loops of wire, each of which carries a current I. Therefore, the right side of equation (93) is μ0NI in this case. • Ampère’s law applied to the circle gives


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B = 0

2NI

rμπ

(96)

• This result shows that B varies as 1/r and hence is nonuniform in the region occupied by the torus. However, if r is 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 Ampère’s law we find that B = 0 in the regions exterior to the torus.

• Finding the magnetic field inside a toroid is a good example of the power of Ampère’s law. The current enclosed by the dashed line is just the number of loops times the current in each loop. Ampere’s law then gives the magnetic field by

(96)

• The toroid is a useful device used in everything from tape heads to tokamaks.

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 namely B and H characterizing a magnetic field. The vector field B is known among electrical engineers as magnetic flux density or magnetic induction, or simply magnetic field, as used by physicists. The vector field H is known among electrical engineers as the magnetic field intensity or magnetic field strength and is also known among physicists as auxiliary magnetic field or magnetizing field.


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Figure 63: Depicting of the magnetic.

• The magnetic field B has the SI unit of teslas (T), equivalent to webers per square meter (Wb/m²). The vector field H 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 characterized by the magnetic field B measured in teslas. However, when the generated fields pass through magnetic materials which themselves contribute internal magnetic fields, ambiguities can arise about 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 H .

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

B = μH (97)

where μ is the permeability of the medium and given by

μ = Kmμ0 (98)

μ0 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 (μ > μ0) and diamagnetic (μ < μ0) materials, the relative permeability is very close to 1. For ferromagnetic materials, μ is much greater than μ0. 4.5. Electromagnetic induction 1. Magnetic flux

• The magnetic flux, ΦB, through an element of area perpendicular to the direction of magnetic field 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 that pass through the surface. This is the net number, 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 has the loop as its border into small elements of area dA. For each element we calculate the differential magnetic flux of the magnetic field B through it:

dΦB = .B dA = B.dA.cosφ (99)


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Figure 64: Faraday’s experiments of induction; (Left) A permanent magnet approaching a loop. (Right) Switching the current in one loop induces a current in another loop.

where φ is the angle between the normal vector n̂ ( d A = n̂ dA) and the magnetic field vector B at the position of the element. • We then integrate all the terms

ΦB = B.dA.cos B.dA φ =∫ ∫ (100) 2. Faraday’s 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 a sensitive 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. If we 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 known as "induced current"; the electromotive force (emf) that appears is known as "induced emf"; 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 a permanent magnet or by an electric current in a coil. • Faraday summarized the results of his experiments in what is known as Faraday's law of induction.

An emf is induced in a loop when the number of magnetic field lines (or magnetic flux) that pass through the loop is changing.


22Figure 65: Depicting Lenz’s 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 flux ΦB through the loop changes with time. • The corresponding formula is

ε = - BddtΦ (101)

where ε is the induced emf.

If the circuit is a coil consisting of N loops of the same area and if ΦB is the flux through

one loop, an emf is induced in every loop; thus, the total induced emf in the coil is given by the expression

ε = -N BddtΦ (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 ΦB through a loop • We see that the magnetic flux ΦB can be changed and an emf is then induced in a circuit in several ways:

• The magnitude of B can change with time. • The area enclosed by the loop can change with time. • The angle φ between the magnetic field vector B and the normal vector n̂ to the loop

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

♦ Lenz’s law • Faraday’s law of induction (equation 101 or equation 102) indicates that the induced emf and the change in flux have opposite algebraic signs. This has a very real physical interpretation that has come to be known as Lenz’s 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 from changing. This law is actually a consequence of the law of conservation of energy. • We now concentrate on the negative sign in the equation that expresses 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

Figure 66: Depicting the motional electromotive force.

Figure 67: Depicting the self-induction.

• To understand Lenz’s law, we consider an example as shown in Figure 65. In the figure we show 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 Ω. Each turn 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) |ε| = 4.1 V; (b) I = |ε|/R = 2.05 A) ♦ MOTIONAL ELECTROMOTIVE FORCE • In examples illustrated by Figure 63, we considered cases in which an emf is induced in a stationary 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 a uniform magnetic field exists. The loop is being pulled outside the magnetic field region with

constant speed v. The magnetic flux through the loop is ΦB = Blx. This flux decreases with time; according to Faraday’s law, there is an induced emf given by

ε = - BddtΦ = -Bl dx

dt= -Blv (103)

• Because the resistance of the circuit is R, the intensity (magnitude) of the induced current in the loop is

I = |ε| = 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 ΦB = Li through the inductor itself. Using Faraday's law we can determine the resulting emf known as self-induced emf εL


24

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.

εL = - BddtΦ = -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 is given by

L = μ0(N2/l)A = μ0n2Al (106) where μ0 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, greatly increasing the inductance of the solenoid. In this case we must replace μ0 by μ = Kmμ0 where Km is the relative permeability of the core; for iron Km is much greater than 1. • The SI unit of L is the 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 the switch 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; this energy is carried by the magnetic field inside the inductor. • Considering the emf needed to establish a particular current and 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 = 12

LI2 (107)


25

• 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 = μo(N2/l)A = μon2Al • The magnetic field inside a solenoid is given by B = μonI. As a result I = B/μon • Substituting the expressions for L and for I into equation (107) leads to

UB = 2

02Bμ

Al (108)

• Because Al = V is the volume of the solenoid, the energy stored per unit volume in the magnetic field or the 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 any region 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 earth’s 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

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

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