Chapter 4. Topics to cover: 1) Magnetic Flux and Flux Density, 2) Biot-Savart Law 3) Magnetic Field Strength 4) Magnetic Scalar Potential 5) Ampere's Circuital Law

Magnetostatics 6) 7) 8) 9)

Self and Mutual Inductances Magnetic Energy Magnetic Force Magnetic Properties of Materials

Introduction In 1785 Coulomb developed a law for the force between magnetic poles which was precisely similar to the law he had stated for the force between electric charges. Based on this law the modern western theory of magnetostatics developed. In 1820 Oersted showed that magnetic effects could be produced by an electric current flowing in a circuit. The nature of fields generated this way was further explored by Ampere. As earlier development was based on permanent magnets, Ampere expressed his results in terms of a distribution of magnetic poles that was equivalent to the current carrying circuit producing the magnetic effect. A fundamental difference between electrostatics and magnetostatics is that while positive and negative electric charges exist separately, north and south magnetic poles always occur in pairs. (Some researchers are still searching for a isolated magnetic pole.) It has been concluded that if a fundamental magnetic particle exists then it takes the form of a dipole; a pair of equal and opposite poles separated by a very small difference. Ampere thought that these dipoles might arise from small (atomic scale) loops of current. It turns out that all observed magnetic effects, whether resulting from magnetised bodies or current carrying circuits in free space can be described in terms of the distribution of dipoles or in terms of the distribution of current. In this course we will describe magnetic effects in terms of the distribution of current. Biot-Savart Law In electrostatics we saw how the properties of an electrostatic field could be specified based on a knowledge of the electric field strength vector, E, at each point in the field. In the magnetic field, a corresponding vector quantity, the magnetic flux density, designated B with unit of tesla (T) or weber per square metre (Wm-2), will be used. The properties of this vector are based on experiment. B can be calculated for any circuit carrying a known current. In the following diagram, let dl be a small element of the circuit C, which carries current I. The distance of point P from dl is r and the angle between dl and the radius vector is θ. The magnitude of the part of the flux density B due to the current element (in Idl) is denoted dB. An experimental law asserts that dB is found as follows dB = (µI dl sin θ ) 4πr 2

48531 EMS – Chapter 4. Magnetostatics

where µ=µrµo is the magnetic permeability of the medium, µr the relative permeability, and µo=4π×10−7 the permeability of free space. The unit for permeability is henry per metre (Hm−1).

C

dl

θ The direction of dB is along a line perpendicular to the plane containing both dl and r. Furthermore, it points in a direction that I r a right handed screw would move when a screwdriver moves through θ from dl to r. For the case shown, dB is directed into the paper. P The law of superposition can be applied to determine B due to I in all of C or, indeed, B due to current flowing in several circuits.

To provide a convenient vector notation for the above equation the vector or cross product is defined. The vector product or cross product of two vectors A and B is a vector C with magnitude C given by C = AB sin θ

where θ is the angle between the two vectors. Its direction is determined by the righthand screw rule. The vector or cross product is denoted as C= A×B

Note that

A × B = −B × A

With this new notation we can write dB =

µI dl × rˆ 4πr 2

where rˆ is the unit vector in the direction from dl to P. For a whole circuit we write B=

µI 4π

dl × rˆ ∫l r 2

The integral sign simply instructs us to add vectorially the components of B arising from the various dl and the circle on the integral sign denotes that the summation must be carried out around the complete circuit. This law (the above equation) is usually referred to as the Biot-Savart law. Example: Calculate the axial flux density of a circular current loop in air using the Biot-Savart law.

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48531 EMS – Chapter 4. Magnetostatics

dBn γ

dB

dBz

γ

R

z α

Ro

dl

I

dl'

Solution: Consider the above drawing. By the Biot-Savart law, the contribution of a small current carrying segment of the circuit (Idl) to the magnetic flux density on the axis is given by dB =

µ 0 Idl 4πR 2

as the current element is at right angles to the radius vector (θ=90o). The element Idl’ opposite Idl on the circuit contributes a component that cancels dBn. Thus the final contribution of each element Idl points up (z-directed) with magnitude dB z =

µ 0 Idl 4πR 2

cos α

Summing the contributions of all dl leads to B = B z = ∫ dBz =

µ 0 I cos α dl 4πR 2 ∫

The dl integration around the loop is 2 πR0 , and so B = Bz =

µ 0 IRo cos α 2R 2

Noting that cos α =

R0

R

and that the loop area, A is A = πR02 we can arrive at alternative forms such as B=

µ 0 IA 2πR 3

zˆ

where zˆ is upward directed unit vector.

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48531 EMS – Chapter 4. Magnetostatics

Properties of B B, the magnetic flux density, is a flux vector. The total magnetic flux, φ, through a surface, S, is found as φ = ∫ B • dA S

where dA is the elemental area vector. The unit of magnetic flux is the weber (Wb). φ depends only on the perimeter of the area S, which is the same for all surfaces having the same perimeter. As a consequence of this (or examination of the Biot-Savart law or because we know that pairs of magnetic poles cannot be separated) for any closed surface S we can write ∫ B • dA = 0 S

This is sometimes referred to as Gauss' Law for magnetics.

Magnetic field strength H If we are only concerned with magnetic effects in free space then a knowledge of the magnetic flux density, B, would be enough to describe the magnetic field. When magnetised bodies are considered, it will be necessary to use a second vector, the magnetic field strength, H. In a magnetic medium, H is defined by the equation B = µH where µ is the permeability of the medium. The Biot-Savart law could be also used to find H. H is sometimes called the magnetising force. Its units are amperes per meter (Am−1). Superficially there is an analogy between the above equation and the constitutive relationship for electric fields D = εE

but it should be noted that the physically measurable electrostatic quantity is E and, in magnetostatics, the physically observable effects depend directly on B. The other vectors are introduced for mathematical convenience.

Magnetic scalar potential and Ampere's circuital law Recall the electric scalar potential V and the relationship B

VBA = − ∫ E • dl A

A purely mathematical quantity, the magnetic scalar potential, U, is defined as

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48531 EMS – Chapter 4. Magnetostatics

B

U BA = − ∫ H • dl A

The unit for U is amperes (A). Ampere's law states that the sum of the magnetic potential around a contour, C, surrounding a current, I, will equal I. That is

∫ H • dl = I C

Example: Calculate the magnetic field strength R0 distance away from an infinitely long conductor carrying current I as in the diagram below dl=R0 dθ dθ R0

I

H

contour C l

I

dl θ R0

r

P

dB

(into page)

Solution: By Ampere's law

∑U = ∫ H • dl = I C

we can readily obtain H =

I 2 πR0

To verify this, consider the Biot-Savart law. The contribution to the field at P due to circuit element dl is given by dB =

µ 0 Idl sin θ 4π r2

and so the total field at P can be obtained as

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48531 EMS – Chapter 4. Magnetostatics

B= Since

l tan θ = R0 ,

where

R0

µ0 I 4π

∞

sin θ dl r2 −∞

∫

is constant, by differentiation, we can obtain cos θ sin θ dl = l dθ

As

r sin θ = R0

and r cos θ = l , we can write B=

µ0 I π µ I sin θ dθ = 0 ∫ 4πR0 0 2πR0

and therefore H =

I 2 πR0

The vector H forms concentric circles around the conductor in the direction shown. Now, looking at the Ampere's law again, substituting H from above and using the relationship dl = R0 dθ , we can write 2π

I

∑U = ∫ H ⋅ dl = ∫ 2πR C

R0dθ =

0

0

I [θ ]20π = I 2π

so confirming Ampere's law for this case.

Flux Linkage Consider a circuit C in a magnetic field B as shown below. The magnetic flux linking the circuit is φ = ∫ B • dA (webers) A

B

dA

area A circuit C (Note: if B||A and uniform, φ = BA .) If there are N circuits (N turns), the total flux linkage, λ, is λ = Nφ av where the average flux Φav is

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48531 EMS – Chapter 4. Magnetostatics

φav =

1 N

N

∑φ k =1

k

Inductances Consider two neighbouring coils, C1 (of N1 turns) and C2 (of N2 turns) bounding surfaces S1 and S2 as shown in the diagram below. If a current I1 flows in C1, a magnetic field B1 will be created. All flux links C1 and some of the flux will link C2, and the flux linkages can be calculated by λ11 = N 1φ11 = N 1 ∫ B1 • da S1

and λ 21 = N 2φ 21 = N 2 ∫ B1 • da S2

From the Biot-Savart law, we know that the flux density and hence the flux linkage is proportional to the current I1. We write λ 11 = L11 I1 and

λ 21 = L21 I1

where L11 and L12 are defined as the self inductance of coil 1 and the mutual inductance between the two coils, respectively. The self inductance of coil 2 can be obtained similarly by introducing a current in it. It can be shown that the mutual inductances calculated from both sides are equal or L21=L12. Neumann formula L12 =

µo 4π

dl1 • dl 2 R C1 C 2

∫∫

where N1 and N2 are absorbed in the contour integrals over C1 and C2, and R is the distance between dl1 and dl2. Circuital Symbols of Inductors The circuital symbol for an inductor of a single coil is

L In the case of magnetically coupled coils, a dot convention is commonly employed to mark the reference directions of the magnetic fields generated by the currents in those coils. As shown below, the terminals of two coils, A and B, are marked such that a positive current IA entering the dot marked terminal of circuit A produces in circuit B a

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48531 EMS – Chapter 4. Magnetostatics

flux in the same direction as would positive current IB entering the dot marked terminal of circuit B. A

B

IA

IB

The circuital symbol for a magnetic coupling between two circuits or coils is L 12

L 11

L 22

where L11 and L22 are the self inductances of coils 1 and 2, and L12 is the mutual inductance between two coils. Note that L12=L21.

Magnetic Energy In terms of field quantities the energy stored in a magnetic field can be determined by Wm =

1 H • Bdv' 2 V∫'

In a system consists of n coils, the magnetic energy can be expressed in terms of the indcutances as 1 n n Wm = ∑ ∑ L jk I j I k 2 j =1 k =1 Lorentz Force So far we have described the magnetic field in terms of the vectors B and H, and we have calculated these quantities. We will now consider the measurable physical effects calculated from a knowledge of the magnetic flux density B. We have already dealt with the force acting on a particle with charge Q in an electrostatic field E. If a particle is moving through a magnetic field (with velocity v) then an additional force Fm is experienced by the particle. The additional force obeys the following rules: (a) Fm has a magnitude of QvB sin θ , where θ is the angle between B and v; (b) It acts along a line which is normal to a plane containing B and v; and (c) It is in the direction moved by a right hand screw in twisting from v to B.

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48531 EMS – Chapter 4. Magnetostatics

Thus in vector notation we can write Fm = Qv × B

(N)

If an electrostatic field E as well as the magnetic field B is present the total force FTotal on the particle can be given as FTotal = Fe + Fm = Q( E + v × B ) where Fe is the force due to the electric field. The total force, FTotal, is known as the Lorentz force and the above relationship is known as the Lorentz Force Law or the Lorentz force equation.

Magnetic Force on a Current Carrying Conductor B

The force acting on a current carrying conductor C can be derived directly from the force acting on moving charges as Fm = I ∫ dl × B

Fm

l I

C

For a single conductor in a uniform magnetic field, we have Fm = Il × B

A current carrying conductor in a uniform magnetic field

Reading Material: Magnetic Properties of Materials Magnetization and Equivalent Magnetization Current Densities According to the elementary atomic model of matter, all materials are composed of atoms, each with a positively charged nucleus and a number of orbiting negatively charged electrons. The orbiting electrons cause circulating currents and form microscopic magnetic dipoles. In addition, both the electrons and the nucleus of an atom rotate (spin) on their own axes with certain magnetic dipole moments. The magnetic dipole moment of a spinning nucleus is usually negligible in comparison to that of an orbiting and spinning electron because of the much larger mass and lower angular velocity of the nucleus. The diagram below illustrates schematically the orbital motion and the spin of an electron. A complete understanding of the magnetic effects of materials requires a knowledge of quantum mechanics. (We give a qualitative description of the behaviour of different kinds of materials later in this section). In the absence of an external magnetic field the magnetic dipoles of the atoms of most materials (except permanent magnets) have random orientations, resulting in no net magnetic moment. The application of an external magnetic field cause both an alignment of magnetic moments of the spinning electrons and an induced magnetic moment due to a charge in orbital motion of electrons. To obtain a formula for determining the quantitative change in the magnetic flux density caused by the presence

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48531 EMS – Chapter 4. Magnetostatics

of a magnetic material, we let mk be the magnetic dipole moment of an atom. If there are n atoms per unit volume, we define a magnetization vector, M, as n∆v

M = lim

∑m k =1

∆v → 0

k

∆v

(A/m)

which is the volume density of magnetic dipole moment.

(a) Orbital motion and (b) spin of an electron

Since each spinning electron can be regarded as a small current loop, a volume density of magnetic dipole moment can be equivalent to a volume current density and a surface current density as qualitatively illustrated in the diagram on the right hand side. Analytically, such an equivalence can be expressed as

and

Jm = ∇ × M

(A/m2)

J ms = M × a n

(A/m)

where Jm and Jms are the equivalent magnetization volume and surface current densities, respectively. Magnetic Permeability

A cross section of a magnetized material

In a magnetized material, the magnetic flux density B has two components contributed respectively by the external magnetic field and the magnetization: B = µo ( H + M ) When the magnetic properties of the medium are linear and isotropic, the magnetization is directly proportional to the magnetic field strength: M = χm H where χm is a dimensionless quantity known as the magnetic susceptibility. Therefore,

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48531 EMS – Chapter 4. Magnetostatics

B = µo (1 + χm )H or

B = µo µr H = µH

where µr = 1 + χm is another dimensionless quantity known as the relative permeability, and µ = µo µr the absolute permeability (or sometimes just permeability). The SI unit for the absolute permeability is henry per meter or H/m. It is interesting to noticed that there is an analogy between the constitutive relation for magnetic fields and that for electric fields: D = εE Classification of Materials by Magnetic Properties In the last section, we described the macroscopic magnetic property of a linear, isotropic medium by defining the magnetic susceptibility χm, a dimensionless coefficient of proportionality between magnetization M and magnetic field strength H. The relative permeability µr is simply 1+χm. All materials can be roughly classified into three main groups in accordance with their µr values. A material is said to be Diamagnetic, if µr ≈ 1 and µr < 1 (χm is a very small negative number), or Paramagnetic, if µr ≈ 1 and µr > 1 (χm is a very small positive number), or Ferromagnetic, if µr >> 1 (χm is a large positive number). As mentioned before, a thorough understanding of microscopic magnetic phenomena requires a knowledge of quantum mechanics. In the following we give a qualitative description of the behaviour of the various types of magnetic materials based on the classical atomic model. In the atoms of a diamagnetic material, the electrons are arranged symmetrically, so that the magnetic moments due to the spin and orbital motion cancel out, leaving the atom with no net magnetic moment in the absence of an externally applied magnetic field. The application of an external magnetic field to this material produces a force on the orbiting electrons, causing a perturbation in the angular velocities. As a consequence, a net magnetic moment is created. This is a process of induced magnetization. According to Lenz's law of electromagnetic induction, the induced magnetic moment always opposes the applied field, thus reducing the magnetic flux density. The macroscopic effect of this process is equivalent to that of a negative magnetization that can be described by a negative magnetic susceptibility. This effect is usually very small, and χm for most known diamagnetic materials (bismuth, copper, lead, mercury, germanium, silver, gold, diamond) is of the order of −10−5. Diamagnetism arises mainly from the orbital motion of the electrons within an atom and is present in all materials. In most materials it is too weak to be of any practical importance. The diamagnetic effect is masked in paramagnetic and ferromagnetic materials. Diamagnetic materials exhibit no permanent magnetism, and the induced magnetic moment disappears when the applied field is withdrawn.

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48531 EMS – Chapter 4. Magnetostatics

In the atoms of more than one third of the known elements, the electrons are not arranged symmetrically, so that they do possess a net magnetic moment. An externally applied magnetic field, in addition to causing a very weak diamagnetic effect, tends to align the molecular magnetic moments in the direction of the applied field, thus increasing the magnetic flux density. The macroscopic effect is, then, equivalent to that of a positive magnetization that is described by a positive magnetic susceptibility. The alignment process is, however, impeded by the forces of random thermal vibrations. There is little coherent interaction, and the increase in magnetic flux density is quite small. Materials with this behaviour are said to be paramagnetic. Paramagnetic materials generally have very small positive values of magnetic susceptibility, of the order of 10−5 for aluminum, magnesium, titanium, and tungsten. Paramagnetism arises mainly from the magnetic dipole moments of the spinning electrons. The alignment forces, acting upon molecular dipoles by the applied field, are counteracted by the deranging effects of thermal agitation. Unlike diamagnetism, which is essentially independent of temperature, the paramagnetic effect is temperature dependent, being stronger at lower temperatures where there is less thermal collision. While the atoms of many elements have net magnetic moments, the arrangement of the atoms in most materials is such that the magnetic moment of one atom is canceled out by that of an oppositely directed (antiparallel) near neighbour. It is only five of the elements that the atoms are arranged with their magnetic moments in parallel so that they supplement, rather than cancel, one another. These five elements are known as ferromagnetic (to be further explained later in this section) elements. They are iron, nickel, cobalt, dysprosium, and gadolinium; the last two are metals of the rare earths and have limited industrial application. A number of alloys of these five elements, which include nonferromagnetic elements in their composition, also possess the property of ferromagnetism. The direction of alignment of the magnetic moments in a ferromagnetic material is normally along one of the crystal axes. It has been shown experimentally that a specimen of ferromagnetic material is divided into so-called magnetic domains, usually of microscopic size (their linear dimensions ranging from a few microns to about 1 mm) such that a single crystal may contain many domains, each aligned with an axis of the crystal, in each of which the atomic moments are aligned. These domains, each containing about 1015 or 1016 atoms, are fully magnetized in the sense that they contain aligned magnetic dipoles resulting from spinning electrons even in the absence of an applied magnetic field. Quantum theory asserts that strong coupling forces exist between the magnetic dipole moments of the atoms in a domain, holding the dipole moments in parallel. Between adjacent domains there is a transition region about 100 atoms thick called a domain wall. In an unmagnetized state the magnetic moments of the adjacent domains in a ferromagnetic material have different directions, as exemplified the diagram below by the polycrystalline specimen shown, where the arrows are intended to indicate the magnetic moment direction in each domain. However, it must be appreciated that the domain alignments may be randomly distributed in three dimensions, and hence viewed as a whole, the random nature of the orientations in the various domains results in no net magnetization. The magnetization of ferromagnetic materials can be many orders of magnitude larger than that of paramagnetic substances. Ferromagnetism can be explained in terms of

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48531 EMS – Chapter 4. Magnetostatics

magnetized domains. When a specimen of ferromagnetic material is placed in a magnetic field, the magnetic moments of its atoms tend to rotate into alignment with the direction of the applied field. Domains in the specimen in which the magnetic moments are more or less aligned with the applied Domain structure of a polycrystalline magnetic field increase in size at the expense of ferromagnetic specimen neighbouring domains that are more or less oppositely aligned to the applied field. The phenomenon is known as domain wall motion. The consequence of domain wall motion is that the specimen of material as a whole acquires a magnetic moment that may be considered as the resultant of all its atomic moments, and the magnetic flux density in the material is increased. For weak applied fields, say up to point P1, in the following diagram, domain wall movements are reversible. But when an applied field becomes stronger (past Pl), domain wall movements are no longer reversible, and domain rotation toward the direction of the applied field will also occur. For example, if an applied field is reduced to zero at point P2, the B-H relationship will not follow the solid curve P2P1O, but will go down from P2 to P'2, along the lines of the broken curve in the figure. This phenomenon of magnetization lagging behind the field producing it is called magnetic hysteresis, which is derived from a Greek word meaning "to lag". As the applied field becomes even much stronger (past P2 to P3), domain wall motion and domain rotation will cause essentially a total alignment of the microscopic magnetic moments with the applied field, at which point the magnetic material is said to have reached saturation. The curve OP1P2P3 on the B-H plane is called the normal magnetization curve. If the applied magnetic field is reduced to zero from the value at P3, the magnetic flux density does not go to zero but assumes the value at Br. This value is called the residual or remanent flux density (in Wb/m2 or T) and is dependent on the maximum applied field strength. The existence of a remanent flux density in a ferromagnetic material makes permanent magnets possible. To make the magnetic flux density of a specimen zero, it is necessary to apply a magnetic field strength Hc in the opposite direction. This required Hysteresis loops in the B-H plane for Hc is called coercive force, but a more appropriate ferromagnetic material name is coercive field strength (in A/m). Like Br, Hc also depends on the maximum value of the applied magnetic field strength. The hysteresis loops shown in the above diagram are known as the major loops. A minor loop (as depicted in the diagram below) would appear if a smaller higher harmonic field is superimposed upon the fundamental excitation field causing an extra reversal of magnetization. It is evident from the diagram above that the B-H relationship for a ferromagnetic material is nonlinear. Hence, if we write B = µH, the permeability µ itself is a function of the magnitude of H. Permeability µ also depends on the history of the material's

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48531 EMS – Chapter 4. Magnetostatics

magnetization, since − even for the same H − we must know the location of the operating point on a particular branch of a particular hysteresis loop in order to determine the value of µ exactly. In some applications a small alternating current may be superimposed on a large steady magnetizing current. The steady magnetizing field intensity locates the operating point, and the local slope of the hysteresis curve at the operating point determines the incremental permeability. Ferromagnetic materials for use in electric generators, Minor hysteresis loop motors, and transformers should have a large magnetization for a very small applied field; they should have tall, narrow hysteresis loops. As the applied magnetic field intensity varies periodically between ±Hmax, the hysteresis loop is traced once per cycle. The area of the hysteresis loop corresponds to energy loss (hysteresis loss) per unit volume per cycle. Hysteresis loss is the energy lost in the form of heat in overcoming the friction encountered during domain wall motion and domain rotation. Ferromagnetic materials, which have tall, narrow hysteresis loops with small loop areas, are referred to as "soft" materials since they are easy to magnetize and demagnetize; they are usually well-annealed materials with very few dislocations and impurities so that the domain walls can move easily. In general magnetic field analysis for engineering applications, the hysteresis effect on B-H relationship is often ignored and normal magnetization curves are used. The diagram below illustrates the normal magnetization curves of a few common soft magnetic materials. Good permanent magnets, on the other hand, should show a high resistance to demagnetization. This requires that they be made with materials that have large coercive field strengths Hc, and hence fat hysteresis loops. These materials are referred to as "hard" ferromagnetic materials for that they are hard to magnetize and demagnetize. The coercive field intensity of hard ferromagnetic materials (such as Alnico alloys) can be 105 (A/m) or more, whereas that for soft materials is usually 50 (A/m) or less. The diagram below shows the demagnetization curves (part of the hysteresis loop in the fourth quadrant). Normal magnetization curves of soft As indicated before, ferromagnetism is the result magnetic materials of strong coupling effects between the magnetic dipole moments of the atoms in a domain. Figure (a) in the diagram below depicts the atomic spin structure of a ferromagnetic material. When the temperature of a ferromagnetic material is raised to such an extent that the thermal energy exceeds the coupling energy, the magnetized domains become disorganized. Above this critical temperature, known as the curie temperature, a ferromagnetic material behaves like a paramagnetic substance. Hence, when a permanent magnet is heated above its curie temperature it loses its magnetization. The curie temperature of most ferromagnetic

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48531 EMS – Chapter 4. Magnetostatics

materials lies between a few hundred to a thousand degrees Celsius, that of iron being 770oC.

Demagnetization curves of permanent magnets

Some elements, such as chromium and manganese, which are close to ferromagnetic elements in atomic number and are neighbors of iron in the periodic table, also have strong coupling forces between the atomic magnetic dipole moments; but their coupling forces produce antiparallel alignments of electron spins, as illustrated in Figure (b) in the diagram below. The spins alternate in direction from atom to atom and result in no net magnetic moment. A material possessing this property is said to be antiferromagnetic. Antiferromagnetism is also temperature dependent. When an antiferromagnetic material is heated above its curie temperature, the spin directions suddenly become random, and the material becomes paramagnetic. There is another class of magnetic materials that exhibit a behavior between ferromagnetism and antiferromagnetism. Here quantum mechanical effects make the directions of the magnetic moments in the ordered spin structure alternate and the magnitudes unequal, resulting in a net nonzero magnetic moment, as depicted in Figure (c) in the diagram on the right hand side. These materials are said to be ferrimagnetic. Because of the partial cancellation, the maximum magnetic flux density attained in a ferrimagnetic substance is substantially lower than that in a ferromagnetic specimen. Typically, it is about 0.3 Wb/m2, approximately one-tenth that for ferromagnetic substances. Ferrites are a subgroup of ferrimagnetic material. One type of ferrites, called magnetic spinels, crystallize in a complicated spinel structure and have the formula XO-Fe2O3, where X

Schematic atomic spin structures for (a) ferromagnetic, (b) antiferromagnetic, and (c) ferrimagnetic materials.

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48531 EMS – Chapter 4. Magnetostatics

denotes a divalent metallic ion such as Fe, Co, Ni, Mn, Mg, Zn, Cd, etc. These are ceramiclike compounds with very low conductivities (for instance, 10-4 to 1 (S/m) compared with 107 (S/m) for iron). Low conductivity limits eddy-current losses at high frequencies. Hence ferrites find extensive uses in such high-frequency and microwave applications as cores for FM antennas, high-frequency transformers, and phase shifters. Ferrite material also has broad applications in computer magnetic-core and magneticdisk memory devices. Other ferrites include magnetic-oxide garnets, of which yttriumiron-garnet ("YIG," Y3Fe5O12) is typical. Garnets are used in microwave multiport junctions. Following diagrams show the hysteresis loops of materials commonly used as the magnetic cores of high frequency inductors/transformers and recording media, respectively. Ferrites are anisotropic in the presence of a magnetic field. This means that H and B vectors in ferrites generally have different directions, and permeability is a tensor. The relation between the components of H and B can be represented in a matrix form similar to that between the components of D and E in an anisotropic dielectric medium.

Hysteresis loops of a soft ferrite at different temperatures

Hysteresis loops of deltamax (50% Ni 50% Fe)

Exercises: 1.

Calculate the field intensity H and the flux density B at the centre of a current loop of radius 5cm, carrying current I=5A, when the coil is wound on: (a) air, (b) aluminium and, (c) iron ( µr = 10,000) cores. (Answer: (a) 6 . 28 × 10 − T , (b) 6 . 28 × 10 − T , (c) 0 . 628 T ) 5

5

2.

A plane square wire loop of side length l carries a clockwise current I. Show that at the centre H = 2 2 I πl and is directed into the paper.

3.

Determine the flux density at point P.

(Answer:

5 . 2 × 10 −4 T

)

I=10A P radius 10 mm

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48531 EMS – Chapter 4. Magnetostatics

4.

Consider the current sheet shown; Calculate the flux density at point P, 10cm from an edge on the same plane as the strip. Assume that the current is uniformly distributed and that there are no magnetic materials in the vicinity. (Answer: 1. 39 × 10 − T ) 2

long thin straight strip of copper I=10kA P 10cm

10cm

5.

Determine the mutual inductance between a conducting triangular loop and a very long straight wire as shown in the diagram. 3µ o  λ b  ( d + b) ln 1 +  − b H) (Answer: L12 = L21 = 21 =   I1 2π  d 

6.

Determine the force per unit length between two infinitely long parallel conducting wires carrying currents I1 and I2 in the same direction. The wires are separated by a distance d. µ I I (Answer: F12 = − F21 = − a y o 1 2 N/m) 2πd Z

I1 I2

d O

60o

Y

d+b X

(Problem 5)

d

(Problem 6)

7.

A 13.2 KV, 500A (rms), single phase transmission line has two conductors, each 2cm in diameter, 1.5m apart. The span between supporting poles is 200m. Determine the average force acting on the conductors, over one span, during short circuit conditions if the short circuit current is =12 × normal current. (ignore line sag). ( Answer: 960N )

8.

Two parallel circuits of an overhead power line consist of four conductors carried at the corners of a square. Find the flux, in webers/km of circuit, linking one of the circuits when a current of 1A flows in the other circuit. (Answer: 2 × 10 −4 log e 2 Wb/km)

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48531 EMS – Chapter 4. Magnetostatics

9.

Use the Lorentz force law to determine the force experienced by a 10 m long conductor carrying a current, I=100 A, due to the earth's magnetic field, B =5 ×10−5 T, angle of dip 60o. (Answer: 4.3 ×10−2 N) B

60 o conductor I

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