Chapter

5

Magnetostatics

Contents

5.1 Magnetic Forces and Torques . . . . . . . . . . . . . . 5-4

5.1.1 Force on Current Carring Conductors . . . . . . 5-5

5.1.2 Torque on a Current-Carrying Wire . . . . . . . 5-6

5.2 The Biot-Savart Law . . . . . . . . . . . . . . . . . . 5-11

5.2.1 Surface and Volume Current Distributions . . . . 5-12

5.2.2 Magnetic Dipole . . . . . . . . . . . . . . . . . 5-19

5.2.3 Force Between Conductors . . . . . . . . . . . . 5-20

5.3 Maxwell’s Magnetostatic Equations . . . . . . . . . . 5-21

5.3.1 Gauss’s Law for Magnetisim . . . . . . . . . . . 5-21

5.3.2 Ampère’s Law . . . . . . . . . . . . . . . . . . 5-22

5.4 Vector Magnetic Potential . . . . . . . . . . . . . . . . 5-28

5.5 Magnetic Properties of Materials . . . . . . . . . . . . 5-29

5.5.1 Magnetic Permeability . . . . . . . . . . . . . . 5-29

5.6 Magnetic Boundary Conditions . . . . . . . . . . . . . 5-33

5.7 Inductance . . . . . . . . . . . . . . . . . . . . . . . . 5-35

5.7.1 Self Inductance . . . . . . . . . . . . . . . . . . 5-38

5.7.2 Mutual Inductance . . . . . . . . . . . . . . . . 5-44

5-1 CHAPTER 5. MAGNETOSTATICS

5.8 Magnetic Energy . . . . . . . . . . . . . . . . . . . . . 5-45

5.9 Appendix: Solenoid Exact Magnetic Field Off Axis . . 5-46

5-2 ꢀ As we move from from electrostatics to magnetostatics we leave behind two of Maxwell’s equations and begin using the remaining two equations r ꢁ B D 0 r ꢂ H D J; where as a quick review B is the megnetic ﬂux density, H is the magnetic ﬁled intensity, and of course J is the current density

ꢀ Also recall that B D ꢀH, but in magnetics it is not always a linear relation; ferromagnetic materials are the exception we will see later in this chapter

ꢀ With electrostatics behind us it is good to look at what lies ahead in the context of the past

Table 5.1: Moving from electrostatics to magnetostatics.

Attribute Electrostatics Magnetostatics

Sources Stationary charges ρ Steady currents J

v

Fields and Fluxes E and D H and B

Constitutive parameter(s) ε and σ µ

Governing equations

• Differential form

∇·D = ρ ∇·B = 0 v

∇×E = 0 ∇×H = J

$$

♥D·ds = Q

♥E·dl = 0

♥B·ds = 0

♥H·dl = I

• Integral form

SS

$$

CC

Potential Scalar V, with Vector A, with

E = −∇V B = ∇×A

11

22

Energy density we = εE2 wm = µH2

Force on charge q

Fe = qE

Fm = qu×B

Circuit element(s) C and R L

5-3 CHAPTER 5. MAGNETOSTATICS

5.1 Magnetic Forces and Torques

ꢀ An electric ﬁeld acting on a charge produces a froce, so it is with magnetic ﬂux acting on a moving charge

Fm D qu ꢂ B .N/

ꢀ The right-hand rule will be used frequency in this chapter

+q

Fm = quB sin θ

θ

Bu

Figure 5.1: Right-hand rule for a moving charge with B present.

ꢀ With the dot product the projection of A ꢁ B always involves the cosine of the angle between, i.e., AB cos Â

ꢀ With the cross product the resultant now involves the sine of the angle between A and B, i.e., AB sin Â

ꢀ When both E and B are present the total force becomes

F D qE C qu ꢂ B D q.E C u ꢂ B/ .N/

„ƒ‚… „ ƒ‚ …

FFem

ꢀ It is somewhat peculiar that magnetic force does not exist unless the charged particle is moving, and even then no work is done as the force is perpendicular to the motion u

5-4

5.1. MAGNETIC FORCES AND TORQUES

5.1.1 Force on Current Carring Conductors

ꢀ A charged particle in motion receives a force from B, so it follows that B acting on current ﬂow in a wire imparts force to the charged particles in the wire and hence the wire itself

BB

I = 0

I

(a) (b)

Bzxy

I

(c)

Figure 5.2: Force on a wire carrying current under varous conditions.

ꢀ A detailed description of the force on the wire can be developed using differential charge traveling at velocity ue (electron

5-5

CHAPTER 5. MAGNETOSTATICS drift velocity), by writing dFm D dQ ue ꢂ B

ꢀ Work through the details of drift velocity, cross-sectional are, and the density of the charges particles, we can write dFm D I dl ꢂ B .N/ and then upon integrating over a closed contour (the circuit) arrive at

I

Fm D I dl ꢂ B .C/

C

ꢀ An important side note is that if B is uniform it can be factored outside the contour integral, leaving the integration over the closed contour all by it self, which is simply zero, i.e., in the end no net force

5.1.2 Torque on a Current-Carrying Wire

ꢀ In physics you learn that torque arises out of a rotational force, i.e.,

T D d ꢂ F .Nꢁ m/; where d is a moment arm to which a force is applied on one end and the other end is connected to a rotating shaft

– Don’t act too surprised about getting into some rotational mechanics

– Electrical engineers do deal with motors and machines

5-6

5.1. MAGNETIC FORCES AND TORQUES yx

Pivot axis d

θ

Tz

F

Figure 5.3: deﬁnition of torque T D d ꢂ F.

ꢀ Drilling down a bit, Figure 5.3 shows that

O

T D zr F sin Â .Nꢁ m/ where r D jdj and F D jFj

ꢀ For a CD positioner example see the Signals and Systems for

Dummies on-line example1

Magnetic Field Acting a Multi-Turn Loop

ꢀ Consider initially a single turn loop of wire in a uniform B ﬁeld

O

ꢀ We assume that B D xB0

1

5-7

CHAPTER 5. MAGNETOSTATICS y

2

I

Bx

Ob

3

1

B

4a

Pivot axis

(a) z

F1 za/2 yx

Od1 d3

F3

Loop arm 1

Loop arm 3

B

−z

(b)

Figure 5.4: A single loop of wire parallel to B and the force analysis that results in torque.

ꢀ Due to the right-hand rule only segments 1 and 3 receive a non-zero force

5-8 5.1. MAGNETIC FORCES AND TORQUES

ꢀ The moment arms of length a=2 receive the corresponding forces F1 and F2 respectively and produce a net torque of OO

T D d1 ꢂ F1 C d3 ꢂ F3 D yIabB0 D yIAB0 .Nꢁ m/; where I is the current in the loop, A is the area of the loop, and B0 D jBj and the current in the loop isI

ꢀ Assuming the shaft is free to turn, the nonzero T will change the position of the loop to the general scenario of Figure 5.5

ꢀ With the loop rotated to angle Â the torque is no longer at its maximum, but now

T D jTj D I A B0 sin Â .Nꢁ m/

Multiturn Loop

ꢀ If the single turn loop is modiﬁed to contain N turns of wire, the torque is increased to

T D jTj D N I A B0 sin Â .Nꢁ m/

ꢀ There is also a magnetic moment, m that lies along the normal

On to the loop

2

OOm D n N I A D n m .Aꢁ m / and the torque can be written as

T D m ꢂ B .Nꢁ m/

5-9 CHAPTER 5. MAGNETOSTATICS z

B

Pivot axis y

F1

F2

ˆn

2

1

θ

4

3

Ia

Bb

F3 x

F4

(a) m (magnetic moment)

F1

ˆn

Arm 1 a/2

θ

θ

B

(a/2) sin θ

O

Arm 3

F3

(b)

Figure 5.5: A single turn loop no longer parallel to B.

5-10

5.2. THE BIOT-SAVART LAW

5.2 The Biot-Savart Law

ꢀ Up to this point no means of generating a magnetic ﬂux or ﬁeld intensity has been considered

ꢀ Biot and Savart studied how a compass needed was deﬂected when current carrying conductors were brought nearby

ꢀ The result of their work is the Biot-Savart law, which begins with a description of the ﬁeld intensity, H, due to a differential current element

(dH out of the page)

PdH

R

ˆ

R

I

θdl

P' dH

(dH into the page)

Figure 5.6: Finding the magnetic ﬁeld dH intensity due to a current element.

ꢀ The Biot-Savart law states that

O

I dl ꢂ R dH D

.A/m/

4ꢁ R2 in spherical coordinates

5-11

CHAPTER 5. MAGNETOSTATICS

ꢀ Note the similarity to Coulomb’s law, except for the cross product

ꢀ Over a conductor path l we obtain H as the line integral

Z

O

Idl ꢂ R

H D

.A/m/

4ꢁ R2 l

5.2.1 Surface and Volume Current Distributions

ꢀ The Biot-Savart law also can be applied to surface and volume current distributions

Z

O

I

Js ꢂ R

H D

H D ds .A/m/ dV .A/m/

I

4ꢁ R2

S

Z

O

J ꢂ R

4ꢁ R2

V

Example 5.1: Field from a Length l Conductor

ꢀ This is a classic example of a length l conductor along the z

Oaxis carrying current I in the z direction

ꢀ We desire the H in the x ꢃ y plane at radial distance r

ꢀ We set the problem up using the ﬁgure below

5-12

5.2. THE BIOT-SAVART LAW z

Idl

ˆ

Rlr

P

θ

RdH into the page dl dθ

Figure 5.7: Finite length linear conductor carrying current I.

OO

ꢀ Writing out R, R, and dl ꢂ R we have

ꢀ ꢁ ꢀ ꢁ

OO

R D rr ꢃ zz

R D p

OOrr ꢃ zz

Or2 C z2

Â

Ã

O

ꢀ r dz

OOrr ꢃ zz

O

Odl ꢂ R D zdz ꢂ p

D p r2 C z2 r2 C z2

O

O O since z ꢂ r D ꢀ

ꢀ We are now ready to integrate over ꢃl=2 Ä z Ä l=2

ZZl=2

O

dl ꢂ R

IIdz

O

H D D ꢀ r

ꢀꢁ

3=2

4ꢁ R2 4ꢁ

r2 C z2 l

ꢃl=2

I l p

O

D ꢀ

.A/m/

2ꢁr 4r2 C l2

5-13 CHAPTER 5. MAGNETOSTATICS

ꢀ Converting to magnetic ﬂux

ꢀ0I l p

O

B D ꢀ0H D ꢀ

.T/

2ꢁr 4r2 C l2

ꢀ When l ꢄ r the wire can be treated as inﬁnite in length, thus giving the important result that

ꢀ0I

O

B D ꢀ

(T) (for inﬁnite length wire)

2ꢁr

ꢀ This result parallels the inﬁnite line charge by having the ﬁeld inversely proportional to distance

ꢀ The ﬁeld however forms concentric circles around the wire

I

Magnetic field

B

B

B

I

Figure 5.8: The ﬁeld of a long linear conductor encircles the conductor and falls off as 1=r.

ꢀ The ﬁnite length and inﬁnite cases are compared using Python as a function of r below:

5-14 5.2. THE BIOT-SAVART LAW

Figure 5.9: Python code used for the comparison.

Figure 5.10: Normalized Bꢂ comparisons versus r.

5-15 CHAPTER 5. MAGNETOSTATICS

Example 5.2: Circular Loop in the x ꢃ y Plane

ꢀ Another classical magnetostatics problem is a circular loop of radius a carrying current I, in the x ꢃ y plane

ꢀ The ﬁeld point is P D .0; 0; z/ zdH' dH'z dHz dH dH'r

θ

P = (0, 0, z) dHr

Rdl'

φ + π y

θ

φadl xI

Figure 5.11: The ﬁeld of a long linear conductor encircles the conductor and falls off as 1=r.

5-16

5.2. THE BIOT-SAVART LAW

OO

ꢀ Writing out R, R, dl ꢂ R, and dH we have

ꢀ ꢁ ꢀ ꢁ

OO

R D zz ꢃ ra

OO

ꢃra C zz

O

R D p r2 C z2

ÂÃ

OOO O ꢃra C zz za C rz

O

Odl ꢂ R D ꢀa dꢂ ꢂ p

D a dꢂp r2 C z2 dꢂ r2 C z2

OO

Ia za C rz dH D p

4ꢁR2 a2 C z2

OO

O O O O since ꢀ ꢂ ꢃr D z and ꢀ ꢂ z D r

ꢀ Finally, from Figure 5.11 we see that symmtery makes the radial component Hr D 0, and integrating over 0 Ä ꢂ Ä 2ꢁ gives

Z

2ꢁ

Ia2

O

H D z dꢂ

ꢀꢁ

3=2

4ꢁ a2 C z2

0

Ia2

O

D z

.A/m/

ꢀꢁ

3=2

2 a2 C z2

ꢀ Special Cases:

– When z D 0 we have

ˇ

I

ˇ

O

H

D z

.A/m/

.A/m/ zD0

2a

– When z ꢄ a we have

Ia2

ˇ

ˇ

H

O

D z zꢄa

3

2jzj

ꢀ Solution comparisons using Python are given below:

5-17 CHAPTER 5. MAGNETOSTATICS

Figure 5.12: Python code used for the comparison.

Figure 5.13: Normalized Hz comparisons versus z.

5-18 5.2. THE BIOT-SAVART LAW

5.2.2 Magnetic Dipole

For a single-turn loop we can deﬁne the magnetic dipole mo-

2

OOment mm D zI a , so

ˇm

ˇ

HO

D z

.A/m/ za

3

2 jzj

It can also be shown that in spherical coordinates with R a,

H.R; Â; / becomes ꢀꢁm

4 R3

O

O

H D

R2 cos Â C ꢀ sin Â where

ÂÃa

1

Â D cos pa2 C z2

The magnetic dipole and electric dipole have similarities

E

H

H

N

+

I

−

S

(a) Electric dipole (b) Magnetic dipole (c) Bar magnet

Figure 5.14: Dipole ﬁeld patterns plus bar magnet.

5-19

CHAPTER 5. MAGNETOSTATICS

5.2.3 Force Between Conductors

ꢀ Consider two parallel (inﬁnite length) conductors that pass through the y axis parallel to the z axis z

I1 I2

F' F'

12d/2 d/2 lyd

B1 x

Figure 5.15: Analyzing the force between parallel inﬁnite length conductors.

ꢀ In the case shown here the current in each wire ﬂows in the same direction so the wires attract

ꢀ Using the fact that Fm D I` ꢁ B and the azimuthal ﬁeld produced by an inﬁnite linear wire, it can be shown that the force

5-20

5.3. MAXWELL’S MAGNETOSTATIC EQUATIONS per unit length is

F1 ꢀ0I1I2

F10 D

F20 D

D y

.N/m/

Ol2ꢁd

F2 ꢀ0I1I2

O

D ꢂy

.N/m/ l2ꢁd

Note: The force magnitudes are equal

5.3 Maxwell’s Magnetostatic Equations

ꢀ Using Maxwell’s equation two more properties can be deﬁned

5.3.1 Gauss’s Law for Magnetisim

ꢀ For the case of the electric ﬁeld (ﬂux)

Ir ꢃ D , D ꢃ ds D Q

S

Also for a charge volume

ꢀ On the magnetic ﬁeld side,

Ir ꢃ B , B ꢃ ds D 0

S

– In magnetics there is no equivalent to a point charge (no magnetic monopole)

5-21

CHAPTER 5. MAGNETOSTATICS

5.3.2 Ampère’s Law

ꢀ In electrostatics the line integral around a closed contour is always zero, i.e.,

Ir ꢁ E D 0 ,

E ꢃ dl D 0

C

ꢀ Ampère’s law is the counterpart to this in magnetics

Ir ꢁ H , H ꢃ dl D I

C

ꢀ Just as in the electric ﬁeld case of Gauss’s law, Ampère’s law is best applied to scenarios where symmetry is present

– Here we seek symmetric current distributions so that convenient contours may be utilized

Example 5.3: A Long Wire

ꢀ Determine the magnetic ﬁeld H at a distance r Ä a and r ꢄ a from the axis of a wire having diameter a ꢀ For convenience choose the wire to lie along the z axis

ꢀ The Ampèrian contour is a circle of radius 0 Ä r1 Ä a or r1 ꢄ a O

ꢀ The symmetry lies in the fact that H D ꢀHꢂ, e.g. pure azimuthal

5-22

5.3. MAXWELL’S MAGNETOSTATIC EQUATIONS

Contour C2 for r2 ≥ a Ir2

C2 r1

C1 a

Contour C1 for r1 ≤ a z

(a) Cylindrical wire yr2

ˆ

φ

C2 r1

φ1 x

C1 a

(b) Wire cross section

Figure 5.16: Finding H for a long wire using Ampère’s law, with C1 for r Ä a and C2 for r ꢄ a.

ꢀ For 0 Ä r D r1 Ä a we have

IZ

2ꢁ

O O

H1 ꢃ dl1 D

H1 .ꢀ ꢃ ꢀ/r1 dꢂ D 2ꢁr1H1 D I1

1

C0

Rand on the right side the current from J ꢃ ds is the ratio of Sthe areas times I, i.e.,

ÂÃ

ꢀ Á

ꢁr12 r1

2

I1 D I D

I

ꢁa2 a5-23 CHAPTER 5. MAGNETOSTATICS

ꢀ Finally, we solve for H1 (H1 D I1=.2ꢁr1/)

I1 .r1=a/2I r1 OOO

H1 D ꢀ D ꢀ D ꢀ

I .A/m/

2ꢁr1 2ꢁa2 2ꢁr1

ꢀ For r D r2 ꢄ a I

H2 ꢃ dl2 D 2ꢁr2H2 D I2 D I;

C

2so

I

OO

H2 D ꢀH2 D ꢀ

.A/m/

2ꢁr2

H(r)

I

2πa

H(a) =

H1

H2 ra

O

Figure 5.17: Magnetic ﬁeld strength magnitude H.r/ D jꢀHꢂj versus r for a conductor of radius a.

Example 5.4: A Long Coax with Current I

ꢀ Consider a long coax with inner conductor current I

ꢀ What is the magnetic ﬁeld outside the coax?

5-24 5.3. MAXWELL’S MAGNETOSTATIC EQUATIONS

ꢀ Visualize an Ampèrian contour outside the cable and consider the superposition of the ﬁeld due to CI in the inner conductor and ꢂI in the return path of the outer conductor

ꢀ The net ﬁeld is zero!

Example 5.5: Magnetic Field of a Toroid Coil

ꢀ Yet another classical example, is the toroid coil with inner radius a and outer radius b

ꢀ We assume N turns are tightly spaced around the toroid

H

ˆ

φr

I

Iab

C

Ampèrian contour

Figure 5.18: An N turn toroid coil.

ꢀ In Figure 5.18 the windings are spread apart to add notes, but in reality the windings are tight so each loop is parallel to the cross section of the toroid

5-25

CHAPTER 5. MAGNETOSTATICS

ꢀ Find H for r a, a Ä r Ä b, and r b

ꢀ Based on the direction of the current ﬂow and using the righthand rule, the ﬂux of each loop is normal to the loop cross

Osection in the ꢂꢀ direction

ꢀ To be clear, there is no ﬁeld present for r a and r b, why?

ꢀ Applying Ampère’s law for a Ä r Ä b yields

IZ

2ꢁ

OO

H ꢃ dl D

.ꢂꢀH/ ꢃ ꢀr dꢂ D ꢂ2ꢁrH D ꢂNI

C0

ꢀ Finally,

H D ꢂꢀH D

(

NI

O

ꢂꢀ2ꢁr (A/m); for a Ä r Ä b

O

0; otherwise

Example 5.6: An Inﬁnite Current Sheet

ꢀ Consider an inﬁnite sheet of current in the x ꢂ y plane with

O

J2 D xJs

ꢀ Find H everywhere

ꢀ A convenient Ampèrian contour is a rectangle in the yꢂz plane w by l

5-26 5.3. MAXWELL’S MAGNETOSTATIC EQUATIONS z

H

Ampèrian contour lyw

H

Js (out of the page)

Figure 5.19: An inﬁnite current sheet in the x ꢂ y plane.

ꢀ Using the right-hand rule the direction of H above and below the sheet is a shown in Figure 5.19, i.e.,

(

O

ꢂyH; z 0

H D

OyH; z 0

ꢀ Now apply Ampère’s law around the contour

I

H ꢃ dl D Hl C 0 C Hl C 0 D 2Hl D Jsl

Cwhere the zero entries come from the length w paths

ꢀ Finally solving for H we have

(

Js

O

ꢂy 2 ; z 0

H D

Js

Oy 2 ; z 0

5-27

CHAPTER 5. MAGNETOSTATICS

5.4 Vector Magnetic Potential

ꢀ With electrostatics we found that introducing the scalar potential V led to a convenient way of calculating E via ꢂrV

ꢀ A similar idea is possible with magnetic ﬁelds via the vector magnetic potential A, where

B D r ꢁ A .T = Wb/m2/; and webers per square meter is an alternate unit for tesla

ꢀ Working with some vector identities it can be shown that

Z

ꢀ

J

A D

0 dV0 .W/m/

0

4ꢁ R

V

ꢀ Note: It is also possible to write the above using a surface or line current distribution

ꢀ The vector magnetic potential often yields easier integrations that the Biot-Savart law when calculating B or H

Magnetic Flux ˆ Linking Surface S

ꢀ B is the magnetic ﬂux density, but when we consider the ﬂux passing through a surface S, the term magnetic ﬂux ˆ linking a surface, is often used, where

Z

ˆ D B ꢃ ds .Wb/ (5.1)

S

5-28

5.5. MAGNETIC PROPERTIES OF MATERIALS

ꢀ Via Stoke’s theorem with the vector magnetic potential, we can also calculate ˆ using a contour integral

ZI

ꢁꢂ

ˆ D r ꢁ A ꢃ ds D

A ꢃ dl .Wb/;

SCwhere the contour C bounds the surface S

ꢀ In practice choose the integral form which is easiest to evaluate

5.5 Magnetic Properties of Materials

ꢀ Earlier we saw that a loop of area A and current I has magnetic moment of magnitude m D IA

ꢀ A permanent magnet has similar properties

ꢀ In the permanent magnet it is due to current loops at the atomic level

1. Orbital motion of electrons and protons with respect to the nucleus

2. Electron spin

ꢀ In both of the above it is electron motion that dominates

ꢀ Classiﬁcations: diamagnetic, paramagnetic, and ferromagnetic

5.5.1 Magnetic Permeability

ꢀ For the electric we saw that in a dielectric material D D ꢃ0EC

P, where P is the polarization vector

5-29

CHAPTER 5. MAGNETOSTATICS

ꢀ In a magnetic material we similarly have

ꢁꢂ

B D ꢀ0H C ꢀ0M D ꢀ0 H C M ; where M is the magnetization vector

ꢀ In linear materials (most, but not all materials of interest)

M D ꢄmH; where ꢄm is the magnetic susceptability

ꢀ Since B D ꢀH we have that

ꢁꢂ

ꢀ D ꢀ0 1 C ꢄm

ꢀ

ꢀr D

D 1 C ꢄm

ꢀr

Table 5.2: Summary of Magnetic Material Properties.

Diamagnetism Paramagnetism Ferromagnetism

dipole moment

Permanent magnetic No Yes, but weak Yes, and strong

Primary magnetization Electron orbital Electron spin Magnetized mechanism magnetic moment magnetic moment domains

Direction of induced Opposite Same Hysteresis

magnetic ﬁeld [see Fig. 5-22]

(relative to external ﬁeld)

Common substances Bismuth, copper, diamond, Aluminum, calcium, Iron,

gold, lead, mercury, silver, chromium, magnesium, nickel, silicon cobalt niobium, platinum, tungsten

Typical value of χ

Typical value of µ ≈ 1 ≈ 1

≈ −10−5

≈ 10−5

|χ | ≫ 1 and hysteretic mm

|µ | ≫ 1 and hysteretic rr

Ferromagnetic Materials and Hysteresis

ꢀ Materials such as iron, nickel, and cobalt have magnetic moments that easily align with an applied magnetic ﬁeld

5-30 5.5. MAGNETIC PROPERTIES OF MATERIALS

ꢀ Permanent magnets are made of these materials and are key in the many applications of magnetics in engineering

ꢀ Ferromagnetic materials contain magnetized domains which contain magnetic moments that remain permanantly aligned with one another once magnetized

ꢀ At a high level the key result is the magnetization curve which relates B D jBj to H D jHj

B

A1

A2

Br

A3

H

O

A4

Figure 5.20: Ferromagnetic B ꢂ H hysteresis curve.

ꢀ An unmagnetized material starts at O and B increases as the applied H increases (A1); the curve ﬂattens as saturation is reached

ꢀ When H is decreased back to zero the point A2 is the residual ﬂux, or Br, that remains in the material and represents the magnetization

5-31 CHAPTER 5. MAGNETOSTATICS

ꢀ Reversing the direction of the applied H (e.g., chaning the current ﬂow in the wire) hysteresis requires H to reach A3 before

B again equals zero (demagnetized)

– The term hysteresis means lag behind

– The hysteresis loop shows that memory or material history is involved

ꢀ Saturation in the opposite direction is reached at A4 and the process repeats as H is decreased and then increased in the opposite direction

ꢀ Hard materials have wide loops and soft materials have narrow loops

BB

HH

(a) Hard material (b) Soft material

Figure 5.21: Hard vs soft ferromagnetic material hysteresis curves.

ꢀ A wide loop has large Br useful for magnets in motors, etc.

ꢀ A narrow loop is easy to magnetize/demagnetize

5-32 5.6. MAGNETIC BOUNDARY CONDITIONS

5.6 Magnetic Boundary Conditions

ꢀ The boundary conditions of the B and H ﬁelds at a material boundary ꢀ1=ꢀ2 parallels the electric ﬁeld case of Chapter 4

ˆl

1a

H1

Medium 1

μ1 b

ˆn2

H1n

∆h

2

∆h

2

ˆn

}

H1t

H2t

Js

ˆl

}

2dc

H2n

∆l

Medium 2

μ2

H2

Figure 5.22: Establishing magnetic boundary conditions.

ꢀ Gauss’s law for magnetics says that

I

B ꢃ ds D 0;

Cso from Figure 5.22 it must be that

B1n D B2n or ꢀ1H1n D ꢀ2H2n

ꢀ In words, B is continuous across a boundary

ꢀ Note: The corresponding D ﬁeld is not continuous unless ꢅs D

0

ꢀ Using Ampère’s law a closed contour at the boundary (see Figure 5.22)) establishes that

ꢁꢂ

On2 ꢁ H1 ꢂ H2 D Js;

Owhere n2 is the normal at the material interface

ꢀ In words, the tangential components of H are:

5-33

CHAPTER 5. MAGNETOSTATICS

1. Continuous if they are parallel to Js

2. Discontinuous by Js if orthogonal to Js

ꢀ With that said, understand that surface current Js only exists on perfect conductor, e.g., superconductors

ꢀ When ﬁnite conductivity is involved, Js D 0 then it is always true that

H1t D H2t

O

Example 5.7: Finding the Angle Between H1 and n2

ꢁꢂ

OOOO

ꢀ Suppose that H2 D x3 C z2 (A/m), n2 D z, ꢀr D 2, ꢀr D

12

8, and Js D 0

O

ꢀ Find the angle Â1 between H1 and n2

ꢀ

Medium 1 Above

Hꢇ

ꢁ − ꢂ plane

ꢆꢇ

ꢃꢄ

1n�2

ꢁ

H2

2

Medium 2 Below

ꢆ2

ꢁ − ꢂ plane

ꢂ

ꢃꢄ

ꢅ

3

Figure 5.23: Description of the interface and the arrival angle of H2

5-34 5.7. INDUCTANCE

ꢀ From the interface boundary conditions we know that

H1x D H1t D H2t D H2x and so

ꢀr H1z D ꢀr H1n D ꢀr H2n D ꢀr H2z

1122

ÂÃÂ Ã

H2x 3

Â2 D tanꢂ1

Â1 D tanꢂ1

D tanꢂ1

D 56:31ı

H2z 2

ÂÃÂÃ

ꢀr H2x 2 3

1

D tanꢂ1

ꢃ D 20:56ı

ꢀr H2z 8 2

2

5.7 Inductance

ꢀ The inductor is to magnetic ﬁelds as the capacitor is to electric

ﬁelds

ꢀ Recall that a capacitor can store the energy of and electric ﬁeld, so too and inductor can store the energy of a magnetic ﬁeld

ꢀ Inductors often take the form of a solenoid, which is multiples turns of wire wound around a cylindrical core as shown in

Figure 5.24

5-35

CHAPTER 5. MAGNETOSTATICS

B

N

N

B

S

S

(a) Loosely wound (b) Tightly wound solenoid solenoid

Figure 5.24: Loosely and tightly wound solenoids.

ꢀ The tightly wound solenoid makes a nice case study, since it has a near uniform B ﬁeld at its interior (in the limit a cylindrical current sheet), which simpliﬁes the inductance calculation

ꢀ The ﬁrst step is to establish the ﬁeld inside the solenoid starting from the axial ﬁeld of a single circular loop

ꢀ Recall that

I0a2

O

H D z

;

ꢁꢂ

3=2

2 a2 C z2 where here I0 is the current in a single loop and a is the loop radius

5-36

5.7. INDUCTANCE

ꢀ If we stack N such loops a tightly wound solenoid of n D N=l turns per unit length, as shown in Figure 5.25, is obtained zadz

Bdθ z

θ2

θ

Pxl

θ1

I (out) I (in)

Figure 5.25: Cross section of a tightly wound solenoid of length l on the z axis, straddling the x ꢂ y plane.

ꢀ A dz length of the solenoid contains n dz turns and we can let

I0 D In dz, so that

ꢀnIa2

OdB D ꢀ dH D z

3=2 dz

ꢁꢂ

2 a2 C z2

5-37

CHAPTER 5. MAGNETOSTATICS

ꢀ Integrating over the length of the solenoid yields the total ﬁeld

Zl=2

ꢀnIa2

O

B D z

3=2 dz

ꢁꢂ

22

ꢂl=2

ꢀnIa2

2 a C z

Zl=2 dz

O

D z

ꢁꢂ

3=2

2

22

ꢂl=2

2 a C z

2

ꢀnIa2 2l

O

D z

ꢃ p a2 4a2 C l2

1

D z ꢀNI ꢃ p

.Wb/m2/;

O

4a2 C l2 where the last line comes by replacing n with N=l

ꢀ As a special case of interest, consider l ꢅ a ꢀNI

.Wb/m2/

O

B ' z l

ꢀ It is also true that under l ꢅ a the ﬁeld inside is constant independent of the radial distance r from the center, as long as z is not near the ends

5.7.1 Self Inductance

ꢀ The inductance in Henries (H) is deﬁned as the ratio of total magnetic ﬂux, ƒ linking the circuit or conducting structure divided by the current I ﬂowing through the structure

ƒ

L D

.H/;

Iwhere total ﬂux linkage ƒ is related to ﬂux linkage ˆ deﬁned earlier in (5.1), the vector magnetic potential section

5-38

5.7. INDUCTANCE

– Note: For an N turn solenoid we simply have ƒ D Nˆ, where ˆ is the ﬂux linkage in just one turn

ꢀ For a structure composed of two conductors, e.g., a coax or two-wire line, the ﬂux linkage is computed over the area between the conductors (see examples coming up shortly)

Example 5.8: N-Turn Tightly Wound Solenoid

ꢀ Using the fact that the interior ﬁeld of the solenoid is approximately uniform, we have for single loop

Z

ˆ D B ꢃ ds

S

ÂÃ

Z

NN

OO

D z ꢀ I ꢃ z ds D ꢀ I S .Wb/ ll

S

ꢀ It is tempting to say S ꢆ 2ꢁa2, but in reality the ﬂux linkage for a real solenoid is more involved

ꢀ The total ﬂux is N times this value since N loops are involved, i.e.,

N2

ƒ D Nˆ D ꢀ I S .Wb/ l

ꢀ Finally the inductance is found to be

N2 N2

L D ꢀ S ꢆ ꢀ

ꢃ ꢁa2 .H/ ll

ꢀ In reality winding factors and pitch factors come into play in

ﬁnd the inductance

5-39 CHAPTER 5. MAGNETOSTATICS

ꢀ Empirical formulas and inductance calculators can be found on the Web2

Table 5.3: Two of many inductance formulas from Wikipedia.

ꢀ From Table 5.3 we see that the ﬁrst entry is the same as the the tightly wound solenoid with the inclusion of the scale factor K

ꢀ Use Python to compare the tightly wound solenoid with K D 1 to the short air-core formula

Figure 5.26: Python code for inductance calculation.

2

5-40

5.7. INDUCTANCE

Figure 5.27: Comparison plot of the two inductance formulas.

ꢀ For longer lengths the two formulas are close, but at short lengths, where the inductance is larger, the difference is over one order of magnitude

ꢀ The short air-core result is likely the more accurate expression of the two

Example 5.9: Coaxial Transmission Line

ꢀ Consider the coax geometry in Figure 5.28, which also includes the appropriate ﬂux linking surface S

5-41 CHAPTER 5. MAGNETOSTATICS

I

I

Slcba

Figure 5.28: Coax transmission line including the linking surface S.

ꢀ From Ampère’s law the ﬂux generated by the inner conductor

(assumed inﬁnite length) is

ꢀI

O

B D ꢀ

2ꢁr

– No ﬂux inside due to the outer conductor

ꢀ The ﬂux per unit length linking the area S inside the coax is

Z Z lb

ꢀI

ˆ D dr dz

2ꢁr

0a