full notes for electromagnetism course at ucl

PHAS3201: Electromagnetic Theory

Stan Zochowski

December 17, 2011

PHAS3201: Electromagnetic Theory

2011 2

PHAS3201: Electromagnetic Theory CONTENTS

Contents

1 Introduction 71.1 Mathematical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2 Overview of PHAS2201 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.2.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.2.2 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.3 Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.2.4 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Macroscopic Fields 132.1 Reminder of PHAS2201 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.1 Electrostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.1.2 Dielectrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Electric Field in Dielectric Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Magnetic Fields Revision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4 Magnetic Vector Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5 Magnetic Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.6 Interfaces and Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.7 Summary of Linear Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3 Atomic Mechanisms 293.1 Dipoles and Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Magnetic Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.3 Magnetic Dipoles and Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4 Diamagnetism and Paramagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Ferromagnetism 474.1 Atomic-level Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 B & H: Macroscopic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 504.3 Simple Examples of Electromagnetic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3.1 Solenoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.3.2 Bar Magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.3.3 Toroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3.4 Fluxmeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4 Energy Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.5 Summaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

5 Maxwells Equations and EM Waves 635.1 Displacement Current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635.2 Maxwells Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2.1 Differential Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2.2 Integral Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.2.3 Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.3 Plane Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.4 Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

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6 Reflection & Refraction 716.1 Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6.1.1 Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716.1.2 Phase velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.2 Reflection & Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.2.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 726.2.2 Two Laws: Reflection & Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 736.2.3 Changes of Amplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 746.2.4 Fresnel Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6.3 Special Angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.3.1 Brewster Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 766.3.2 Critical Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 776.3.3 Total Internal Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 786.3.4 Intensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

7 Waves in Conducting Media 817.1 Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

7.1.1 Origins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817.1.2 Dispersion Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817.1.3 Good Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817.1.4 Skin depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

7.2 Reflection At Metal Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 837.2.1 Refractive Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.3 Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847.3.1 Plasma Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 847.3.2 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

8 Energy Flow and the Poynting Vector 898.1 Poyntings Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

8.1.1 Energy Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 898.1.2 Energy Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908.1.3 Poyntings Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 908.1.4 Average flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

8.2 Pressure due to EM Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 918.2.1 Photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

9 Emission of Radiation 939.1 Retarded Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

9.1.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 939.1.2 Lorentz Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949.1.3 Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949.1.4 Retarded Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

9.2 Hertzian Dipole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969.2.1 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969.2.2 Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

10 Relativistic Transformations 10110.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

10.1.1 Basic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10110.1.2 Coordinate transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10110.1.3 Interval . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10210.1.4 Lorentz transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

10.2 Four-vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10410.2.1 Position-Time 4-vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10410.2.2 Other 4-vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

10.3 Transformations of Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10510.3.1 Current Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

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10.3.2 Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10610.3.3 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10610.3.4 Maxwells Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10710.3.5 Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

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2011 6

PHAS3201: Electromagnetic Theory CHAPTER 1. INTRODUCTION

Chapter 1

Introduction

Office hours: anytime you can find me, or email me ([email protected]) to set a time. Attendance sheets must befilled in. Theyll be given out at the start of a lecture and collated over the weeks.

Problem sheets will be given out through the term, roughly every two weeks. As detailed in the Preliminaries, the bestthree problem sheets will count for 10% of the final course mark. N.B. There will be four sheets during term.

Full sets of lecture notes will be made available a few days after the lecture. A complete PDF file will be available atthe end of the course.

1.1 Mathematical ToolsThe easy use of mathematical tools is vital to understanding electromagnetic theory.

The differential operators transform vectors and scalars:Grad : scalar to vector F(r) = (r) (1.1)

=

(

xi+

yj+

zk

)(1.2)

Div : vector to scalar q(r) = F(r) (1.3)=Fxx

+Fyy

+Fzz

(1.4)

Curl : vector to vector G(r) = F(r) (1.5)

=

i j kx

y

z

Fx Fy Fz

(1.6) These are also given in the Preliminaries handout for other coordinate systems; They should be reasonably familiar.

These should be reasonably familiar from the courses in the first and second years. Hopefully we wont have to spendmuch time on them, but they are important.

Integrals of vectors can produce scalars or vectors; There are 1-, 2- and 3-D integrals (line, surface and volume); These are all important in Electromagnetic theory! There are important theorems relating integrals of the differential operators: Divergence Theorem:

V

Fdv =S

F nda (1.7)

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Stokes Theorem: S

F nda =C

F dl (1.8)

Notice the importance of !Its useful to understand how a line integral works by considering the basic definition in terms of small steps: b

a C

F dl = limN

Ni=1

Fi dli, (1.9)

where C is the curve were integrating along. In other words, at each point along the curve we take the dot productbetween the function were integrating and the tangent to the curve, and then sum over all these points. It should be easyto see that, in general, the value of the line integral will depend on the curve chosen.

There are two standard ways of working out a line integral. If you are given F (x, y) along some line g(x, y), thenwe can replace every occurrence of y and dy in the integral below with some functions of x (found from g(x, y)) andintegrate:

C

F dl =C

Fx(x, y)dx+ Fy(x, y)dy (1.10)

=

C

Fx(x, y(x))dx+ Fy(x, y(x))dy

dxdx. (1.11)

The second way is using a parametric form. This is possible if the curve being used for the integral is given in termsof a parameter (e.g. angle around a circle). Then we have a curve l(t) which depends on a single parameter t. So wewrite:

C

F dl = ba

F(l(t)) dldtdt (1.12)

=

ba

Fx (x(t), y(t))dx

dtdt+ Fy (x(t), y(t))

dy

dtdt (1.13)

One important result is: ba

dl = (b) (a), (1.14)

so that the line integral of a gradient is independent of path.Surface integrals can be evaluated in a similar way, and we expect to have a surface defined; one way is like this:

r(u, v) = x(u, v)i+ y(u, v)j+ z(u, v)k. (1.15)

Then we know that ru rv is orthogonal to the surface. It is also useful to remember that for a plane passing through apoint r0, the normal is defined as:

n (r r0) = 0 (1.16) (a, b, c) (x x0, y y0, z z0) = 0. (1.17)

Its clear that a plane whose equation is ax+ by + cz = d has a normal vector given by (a, b, c).

1.2 Overview of PHAS2201

1.2.1 Fields In a vacuum, the basic fields are E and B; What are they? What are their units? When they interact with matter, there are changes;

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Why should this happen? We have the the fields D and H, with D = 0E+P and B = 0(H+M) where M is the magnetization; Be careful that you know what youre dealing with!

The electric and magnetic fields have SI units of newtons per coulomb (or volts per metre) and tesla (equivalent tokilograms per coulomb second!). Be careful with units: Gaussian units are quite different. Note that B is often called themagnetic induction field, or simply the magnetic induction.

Matter responds to fields: the atoms of molecules polarize in an electric field, and respond in varied ways to a magneticfield (both diminishing and amplifying it). The fields D and H reflect this.

Figure 1.1: Definition of r (point) and r (source)

1.2.2 Electrostatics For two charges, q1 and q2 at rest at points r1 and r2

Force Field

F(r2) =q1q2

4pi0|r2r1|2 r12 E(r2) =q1

4pi0|r2r1|2 r12

Energy Potential

U(r2) =q1q2

4pi0|r2r1| (r2) =q1

4pi0|r2r1|

(1.18)

How are the force and field directed? What are the values at r1? What is 0, and what are its units?

The force and field are directed along a line joining the two charges; the values would have equal magnitude but theopposite direction. 0 is the permittivity of free space (absolute permittivity), and is 8.854 1012Fm1 where the unitsare C2 N1 m2. This value has been chosen, not measured.

For a charge density (r)

First Maxwell Equation: Gauss Law

E(r) = (r)0

(1.19)

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This can also be written for a collection of charges:E nda = 1

0

i

qi (1.20)

The integral and differential forms are linked by the divergence theorem

Note thati qi =

Vdv

1.2.3 Magnetostatics For an element of a current loop, dl, carrying current I at r:

dB(r) =0I

4pi

dl (r r)|r r|3 (1.21)

We can perform a loop integral:B =

0I

4pi

C

dl (r r)|r r|3 (1.22)

Because F = 0, we can show that there is a magnetic vector potential A such that B =A, so

Second Maxwell Equation: No Magnetic Monopoles

B = 0 (1.23)

What is 0, and what are its units?

Remember that the Biot-Savart law is empirical: there is no underlying theory stating that there are no magnetic monopolesin the universe. But we havent found any yet! The result derived from the Biot-Savart law is the second Maxwell equa-tion.

0 is the permeability of free space, and is 4pi107T m A1 (which is equivalent to kilograms metres per coulomb2).

1.2.4 Electromagnetism For a surface S bounded by loop C,

c

B dl = 0I, (1.24)

where I is the current passing through the surface S We can write I as

SJ nda

Using Stokes Theorem, we find:c

B dl =S

B nda = 0I =S

0J nda (1.25)

Third Maxwell Equation: Ampres law

B = 0J (1.26)

This is incomplete.

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We will consider the detailed form of why Ampres law is incomplete later in the lectures, though you should alreadyhave seen this and understood it at some level. This will form our third Maxwell equation when complete.

If a conducting circuit, C, is intersected by a B field, then the flux is given by:

C =

S

B nda (1.27)

The EMF induced around the circuit isE = d

dt=

C

E dl (1.28)

As before, we can use Stokes Theorem to write:S

dB

dt nda =

C

E dl = S

E nda (1.29)

and derive:

Fourth Maxwell Equation: Faradays Law

E = dBdt

(1.30)

As you can see, the derivation is almost trivial: substitute Eq. (1.27) into Eq. (1.28), and then apply Stokes theoremto the loop integral of E. This is the final Maxwell equation.

Ampres law as described above is incomplete: it needs to account for time-varying electric fields When we do this, we can write (in a vacuum):

Maxwells Equations:

E = 0

(1.31)

B = 0 (1.32)B = 0J+ 00 dE

dt(1.33)

E = dBdt

(1.34)

To complete our set of equations, we have the force on a moving charge:

Lorentz Force

F = q (E+ v B) (1.35)

Once Maxwells equations and the Lorentz force law have been specified, classical electromagnetism is essentiallycomplete: the basic physics has not changed, though the details of the interaction of the fields with matter are still beingunderstood.

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2011 12

PHAS3201: Electromagnetic Theory CHAPTER 2. MACROSCOPIC FIELDS

Chapter 2

Macroscopic Fields

Maxwells equations have two major variants: the microscopic set of Maxwells equations uses total charge and total cur-rent including the difficult-to-calculate atomic level charges and currents in materials. The macroscopic set of Maxwellsequations defines two new auxiliary fields that can sidestep having to know these atomic sized charges and currents.Unlike the microscopic equations, "Maxwells macroscopic equations", also known as Maxwells equations in matter,factor out the bound charge and current to obtain equations that depend only on the free charges and currents. Theseequations are more similar to those that Maxwell himself introduced. The cost of this factorization is that additional fieldsneed to be defined: the displacement field D which is defined in terms of the electric field E and the polarization P ofthe material, and the magnetic-H field, which is defined in terms of the magnetic-B field and the magnetization M of thematerial. In this chapter, we will look at these macroscopic fields, D and H.

2.1 Reminder of PHAS2201We begin this section of the course by going over electrostatic concepts which should be very familiar from PHAS2201,including Gauss law and the effect of dielectrics on capacitance.

2.1.1 Electrostatics We start with a single charge, q, at r:

E(r) = q(r r)/(4pi0|r r|3) (2.1)

Taking a surface integral gives SE.nda = q/0

Increasing the number of charges, and using the principle of superposition, we get:S

E.nda =

dv/0 (2.2)

This leads directly to Gauss law: E = /0We will now go through a few worked examples on the use of Gauss Law.

Ex. 1 Find the electric field inside a sphere which carries a charge density proportional to the distance from the origin, = kr, for some constant k.

Ex. 2 A long coaxial cable carries a uniform volume charge density on the inner cylinder (radius a), and a uniformsurface charge density on the outer cylindrical shell (radius b). This surface charge is negative and of just the rightmagnitude so that the cable as a whole is electrically neutral. Find the electric field in each of the three regions: (i) insidethe inner cylinder (r < a), (ii) between the cylinders (a < r < b), (iii) outside the cable (r > b). Plot |E| as a function ofr.

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2.1.2 DielectricsA dielectric is an electrical insulator that can be polarized by an applied electric field. When a dielectric is placed in anelectric field, electric charges do not flow through the material, as in a conductor, but only slightly shift from their averageequilibrium positions causing dielectric polarization. Because of dielectric polarization, positive charges are displacedtoward the field and negative charges shift in the opposite direction. This creates an internal electric field which reducesthe overall field within the dielectric itself.

Recall that capacitance is defined by Q = CV Capacitance changes when a dielectric is added:

Cdielectric = Cvacuum (2.3)

A dielectric has no free charges: an insulator The polarization is P = 0eE (and is defined as dipole moment per unit volume) This gives the susceptibility, e The dielectric constant is = 1 + ePolarization reflects the fact that the atoms which make up the dielectric consist of separate positive (nucleus) and

negative (electrons) charges. These respond differently to the electric field, leading to a shift in the overall charge distri-bution of the dielectric, while keeping it neutral. We will consider the microscopic origin of polarization in detail in nextsection of the course.

Figure 2.1: Electronic polarization occurs due to displacement of the centre of the negatively charged electron cloudrelative to the positive nucleus of an atom by the electric field.

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2.2 Electric Field in Dielectric Media We want to develop a theory for electric fields in the presence of polarized media We will start by consider the field outside a piece of polarized dielectric This will introduce the ideas of polarization charge densities Then we will move onto the field inside a piece of polarized dielectric We will find a useful reformulation of Gauss Law

Figure 2.2: Electronic polarization occurs due to displacement of the centre of the negatively charged electron cloudrelative to the positive nucleus of an atom by the electric field.

We start by finding the potential at a point r due to a small volume of polarized material at a point r. We will thenintegrate this over the entire piece of dielectric material. First, note that the potential at r due to a dipole at r is:

(r) =1

4pi0

p (r r)| r r |3 (2.4)

Recall that p = qd and that P = p/v.Then we use the fact that the polarization is the dipole moment per unit volume to write:

(r) =vP (r) (r r)

4pi0 |r r|3(2.5)

When we take the limit v 0 and sum over the elements, we get an expression for the total potential:

(r) =

V

dvP (r) (r r)4pi0 |r r|3

(2.6)

We use the gradient of 1/ |r r|, derived as (worth remembering!):

(

1

|r r|)

=(r r)|r r|3 (2.7)

to transform this:

(r) =1

4pi0

V

P (r) (

1

|r r|)dv (2.8)

Using the formula for (F) from the Mathematical Identities, (F) = () F+ F (2.9)

and rearranging (we want F ) we can write, with F = P (r) and = 1|rr| ,

(r) =1

4pi0

V

{

(P (r)|r r|

) 1|r r| P (r

)}dv (2.10)

2011 15


Finally, we use the divergence theorem on the first term[V Fdv =

SF nda], to give the potential outside a

polarized dielectric object:

(r) =1

4pi0

S

P (r) n|r r| da

+1

4pi0

V

P (r)|r r| dv

(2.11)

The surface polarization charge density is defined:P = P n (2.12)

The volume polarization charge density is defined:P = P (2.13)

We can write the potential as:

(r) =1

4pi0

(S

P|r r|da

+V

P|r r|dv

)

(2.14)

=1

4pi0

dqP|r r| (2.15)

For uniform polarization, P = 0, so there is no bound charge within the material, but there will be bound chargeon the surface.

Bound charge: The charge within a material that is unable to move freely through the material. Small displacementsof bound charge are responsible for polarization of a material by an electric field.

Free charge: The charge in a conducting material associated with the conduction electrons that are free to movethroughout the material. These electrons can carry electric current.

Figure 2.3: Origin of surface charge density due to polarization.

We have considered the field due to a polarized dielectric, but only outside the dielectric. What is the field inside apolarized dielectric?

Consider three (small) charged conductors embedded in a dielectric They have charges q1, q2 and q3 (sum to Q) Now use Gauss Law:

S

E nda = 10

(Q+QP ) (2.16)

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We start by noting that:

QP =

S1+S2+S3

P nda+V

Pdv. (2.17)

It is important to realize that the arbitrary bounding surface S does not enter into this integral because there is no po-larization charge density on it (it is not a real surface). We use the divergence theorem

[V Fdv =

SF nda] to

transform the second integral into a surface integral. But we must take care: this time, we must include the surface Sbecause it bounds the volume V. It is also important to understand the directions of the surface normals. Explicitly, thisgives:

QP =

S1+S2+S3

P ndaS

P ndaS1+S2+S3

P nda (2.18)

= S

P nda (2.19)

Now we can use this is in Gauss law inside the dielectric, which was given as Eq. (2.16):S

E nda = 10Q 1

0

S

P nda. (2.20)

After a little manipulation, we can rewrite this in terms of the free or external charge, Q.

Using the divergence theorem yet again, V Fdv =

SF nda, we find that:

Q =

S

(0E+P) nda (2.21)D = 0E+P (2.22)

becomes V

(v)dv =

Ddv (2.23)

The electric displacement D is the field whose divergence is the free (or external) charge density So, if we consider a charge density, and use the divergence theorem, we get:

Divergence of D

D = (r) (2.24)

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External Charge

We have talked about free or external charge (as opposed to the bound charge) With a dielectric, the difference is clear Charge added from outside (external charge) is different to polarization charge But it is not free to move For a conductor, charge is free to move around It is important to be aware of the difference between charge added and charge already present

In general, the polarization P is a function of the material and the external field E We write P = 0eE in linear, isotropic, homogeneous media In these media, as e (the electric susceptibility) is constant:

D = 0E+ 0eE = E (2.25)

We call = 0(1 + e) the permittivity, and /0 the relative permittivity or dielectric constant Linear: P depends linearly on E Homogeneous: e does not vary with position Isotropic: P and E are parallel

[Non-examinable material] [It is important to realize that a sufficiently strong electric field can break apart the chargesin a material which form the microscopic dipoles. At this point, called dielectric breakdown, all approximations discussedto this point are invalid. For air, whose dielectric constant is 1.0006, the maximum field sustainable without breakdown isaround 3 106 V/m.

The reason that we refer to an isotropic dielectric for the relationP = 0e (E)E is that it implies that the polarizationhas the same direction as the external field. This is a good approximation for most media, but it is necessary in some mediato replace this with a tensor relationship, where the two vectors are not in the same direction. This type of behaviour ismore common in magnetic materials, which we will come to.]

Energy Density

What is the energy density of an electric field? We will consider this in two ways:

1. Charge flowing into a capacitor;

2. Adding a small charge to a field.

The final result is the same:

Energy Density of an Electric Field

U =1

2D E (2.26)

Considering a capacitor first, we assume that it is in the process of being charged. If we start with the expression forpower (which is rate of change of energy with time) for a current I(t) flowing at a voltage V (t) at time t, P (t) = V (t)I(t).Then the energy is:

W =

P (t)dt =

V (t)I(t)dt =

Q(t)

C

dQ

dtdt =

1

2

Q2

C(2.27)

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For a parallel plate capacitor with plates of area A separated by a distance d, we know that the capacitance is given byC = Ad . Using V = Q/C, we find that the electric field can be written:

E =V

d=

Q

Cd=

Qd

Ad=

Q

A. (2.28)

Of course, as D = E, we find that D = QA . So the energy density is given by:

U =W

Ad=

1

2

Q2

CAd

=1

2

Q2

A2

=1

2D E (2.29)

Another (more general) way to reach the same formula is to consider the work done bringing a charge from infinity tothe point where the energy density is required. We know that the energy of a point charge, q, in a potential is W = q.This can be generalized for a charge distribution given by the charge density (r):

W =

V

dv. (2.30)

Now, what would be change in electrostatic energy when adding a small amount of charge, ? We use our recentresult for Gauss theorem, D = :

W =

V

dv (2.31)

= D (2.32) ( D) = (D)D () (2.33)

W =

V

(D) D dv (2.34)

W =

S

D ndaV

D dv, (2.35)

where we have used the divergence theorem on the first part of the integral in the final line. But we know that E = ,and we can notice that the first term will fall off rapidly with distance (D with 1/r2 and with 1/r). This means that wecan write overall, as the volume being integrated tends to infinity:

W =

V

D Edv (2.36)

Now, if we assume a linear, dielectric medium, we know that D = E, and we can integrate over the field going from 0to D:

W =

D0

W =

D0

V

D Edv (2.37)

We can write:

W =1

2

E0

V

(E2)dv =

1

2

V

E2dv (2.38)

This of course gives us the result we derived above, namely U = E D/2.

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2.3 Magnetic Fields RevisionAn important point to note as we start the area of magnetic fields is that this is where the essential link between elec-tric fields and magnetic fields (leading to the unified area of electromagnetism) becomes apparent. Thus far we haveconsidered electrostatics only.

The magnetic field at r2 due to a circuit at r1, in both integral and differential forms:

Biot-Savart Field Law

B (r2) =04piI1

1

dl1 r12|r12|3

(2.39)

dB (r2) =04piI1dl1 r12|r12|3

(2.40)

Note that this is empiricially derived. For a current density, we find:

B (r2) =04pi

V

J (r1) r12|r12|3

dv1 (2.41)

This implies that2 B = 0, which indicates a lack of magnetic monopoles.We can show the last statement using the mathematical identity for (FG) = ( F) G (G) F:

2 B = 04pi

V

2 (J (r1) r12|r12|3

)dv1 (2.42)

=04pi

V

J (r1) (2 r12|r12|3

)dv1, (2.43)

where, since we are taking the divergence at point r2, the term involving2 J (r1) is zero. But now we can use twoidentities:

1. (1/r12) = r12/ |r12|3

2. () = 0This shows that the integral on the right-hand size of equation (2.43) is zero, and hence there are no magnetic monopoles(though note that we started from just this assumption: that the magnetic field arises from the line integral around acircuit!).

The original, integral form of Ampres Law is: C

B dl = 0I, (2.44)

where the current is that flowing through the area enclosed by the path.

The differential form comes from writing I = SJ ndaB = 0J (2.45)

But we have to account for time-varying E:

Ampre-Maxwell Law

B = 0J+ 00 Et

(2.46)

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We can understand why this is incomplete by considering a capacitor being charged with a constant current, I. UsingAmpres law (in original form) we see:

B dl = 0S

J nda (2.47)

Now consider a loop, C, around the wire leading to one plate of the capacitor, and two different surfaces, as shown inFig. 2.4:

1. A surface cutting the wire

2. A surface passing between the plates of the capacitor, and not cutting the wire

Figure 2.4: Amprian loops on a charging capacitor.

It is clear that these will give two different answers for the integral over the current density: in the first, the answer will beI, and in the second it will be zero. This is clearly wrong, as Ampres law insists that the choice of surface be arbitrary.

The resolution to the problem, using the continuity equation, will be considered later, in Chapter 5, on MaxwellsEquations.

Faradays Law

Electromotive force (emf) is equivalent to a potential difference Often encountered in terms of circuits, with inductance Around a circuit, the emf, E , is defined by:

E =C

E dl (2.48)

Faradays Law (integral form):E = d

dt(2.49)

We define the magnetic flux, , as:

=

S

B nda, (2.50)

in other words the magnetic field crossing a surface. Now, using the definition of emf we can related the electric field tothe derivative of the magnetic field:

C

E dl = ddt

S

B nda. (2.51)

Provided that the circuit being considered does not change with time, we can take the time derivative inside the integral.We can also use Stokes theorem

[CF dl =

S F nda] on the line integral of E to obtain the surface integral of

E: s

E nda = S

B

t nda. (2.52)

Since this must be true for all fixed surfaces S, we find:

2011 21


The differential form of Faradays Law:

E = Bt

(2.53)

When the magnetic field is static, this reduces to the conservative field E,E = 0. Notice the minus sign: Lenzs law states that any induced magnetic field opposes the change in flux that induced it

2.4 Magnetic Vector PotentialThe solution of many electrostatic problems is made easier by working in terms of the potential rather than the electricfield directly. The same idea can be applied to the magnetic field, though the eventual solution is rather more complex.

Since = 0 we know that we can write E = when B/t = 0 Similarly, we know that B = 0 The relevant identity here is (A) = 0 We can then write generally:

The Magnetic Vector Potential:

B =A, (2.54)

where A is the vector potentialWhen we consider the form of the vector potential, it should be immediately apparent (by analogy with the electric

field as gradient of the potential) that there is a freedom in choosing it:

A A+f (2.55)for any scalar function f results in the same B field since (f) = 0. This invariance under a transformation iscalled gauge invariance. It should not be surprising: the electrostatic potential, , is not defined up to an arbitrary additiveconstant (and all potentials are actually potential differences.

[Non-examinable] [This vector potential is not just something weve dreamt up: for instance, when considering theSchrdinger equation for a quantum particle in the presence of a magnetic field (even if it never passes through the regionwhere B > 0, the momentum operator needs to be altered: p p ecA).]

There are different ways of choosing the vector potential which help with different situations. Consider a situationwhere the electric field does not change with time. Then we write Ampres Law as:

B =A = 0J (2.56) ( A)2A = 0J. (2.57)

The Coulomb gauge is: A = 0 (2.58)

It leads to the following expression for the vector potential:2A = 0J (2.59)

By analogy with Poissons equation,2V = /0, we can write:

A (r1) =04pi

V

J (r2)

|r1 r2|dr2 (2.60)

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The current density determines the vector potential

There are other choices of gauge, for instance, the Lorentz gauge is A = o0 (V/t). Gauge invariance is a more general phenomenon Solving for vector potential is (generally) harder than solving for the electrostatic potential The electric field can no longer be expressed as the gradient of a scalar potential if there is a time-varying B field:

E(t) = At

(2.61)

This last change can be seen rather easily. Consider the Maxwell equation for the curl of the electric field:

E = Bt

(2.62)

and substitute in the form of B =A:

E+ tA = 0 (2.63)

The vector E+ A/t has zero curl. We know from identities that it can be written as a gradient of a scalar:

E+A

t= (2.64)

So, rearranging, we find that E = A/t

2.5 Magnetic IntensityAs we saw with the electric field, E, the introduction of a medium other than vacuum results in changes to Maxwellsequations. These changes can be handled by using an alternative field which includes the effects of the medium implicitly.We will now do the same for magnetic fields. A word of caution: non-linear magnetic media are much more commonthan non-linear electric media; we will deal with these rather interesting materials in Chapter 4 on Ferromagnetism.

Magnetization We introduced the polarization of a dielectric material, P E Similarly, we introduce a quantity, proportional to the magnetic induction B This is the magnetization, M It describes the response of a material to the magnetic induction Electrons can be modelled as moving in loops around atoms: we can use the magnetic dipole to model the response

Let us consider the vector potential at a point r1 due to a small volume of magnetised material at a point r2 (we willsee later that this is given by the expression below). This small volume will have magnetic moment m = M (r2) V2.Then we can write:

A (r1) =04pi

V

m r12|r12|3

(2.65)

=04pi

V

M (r2) r12|r12|3

dV2 (2.66)

=04pi

V

M (r2)2 1r12

dV2, (2.67)

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where weve used a standard result to get from Eq. (2.66) to Eq. (2.67). Now we use the expansion of (F), withF = M and = 1r12 to write:

F = F (F) (2.68)A (r1) =

04pi

V

{2 M (r2)r12

2 (M (r2)

r12

)}dV2 (2.69)

Now we use the theoremV FdV =

Sn Fda to write:

A (r1) =04pi

V

2 M (r2)r12

dV2 04pi

S

nM (r2)r12

da2 (2.70)

=04pi

V

2 M (r2)r12

dV2 +04pi

S

M n (r2)r12

da2 (2.71)

This then leads us to the magnetization current densities: We formally define:

JM = M (2.72)jM = M n (2.73)

JM is the volume magnetization current density jM is the surface magnetization current density

It is clear that there will be no bound current density where the magnetization is uniform. So within the bulk of the rodthere is a bound current density given by JM =M, and at the surface there is a bound surface current per unit lengthgiven by jM = M n is a unit vector in the direction of the outward normal to the surface. JM is a current per unitarea, where the area is perpendicular to the direction of flow, and jM is a current per unit length, where the length is in theplane of the surface and perpendicular to the direction of the surface current. These bound currents are the net effect ofthe microscopic currents associated with magnetic dipoles.

Figure 2.5: Origins of the magnetization surface current

We move on to considering how linear magnetic media behave . . . We know thatB = 0J We also have J = JM + Jf Here Jf is due to the motion of free charges, and JM =M

SoB = 0 (Jf +M) or(B0M

)= Jf

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We then define H, the magnetic intensity, as

Magnetic Intensity

H =B

0M (2.74)

This yieldsH = JfThe magnetic intensity serves a similar purpose to the electric displacement, in accounting for the response of the

medium as well as the magnetic induction. We can rewrite this, using Stokes theorem:S

H nda =S

Jf nda (2.75)C

H dl =S

Jf nda (= If ) (2.76)

This tells us that the integral of the intensity along a closed loop is equal to the current flowing across the surface definedby that loop. It also gives the units as amperes per metre (the same units as the magnetization).

It is important to note that the three quantities that we have defined so far (the magnetic induction, B, the magneti-zation, M and the magnetic intensity, H) are not necessarily parallel; this will be important when considering ferromag-netism in particular.

Magnetic Susceptibility

For a linear, isotropic material, we assert (based on experimental observations):M = mH (2.77)

where m is the magnetic susceptibility

We can write B = 0 (1 + m)H If m > 0 we have a paramagnetic material If m < 0 we have a diamagnetic material Note that m can depend on temperature, but is generally small for these materials (less than 105)

2.6 Interfaces and Boundary Conditions Understanding how the different field vectors change at interfaces is important We need to consider both medium/vacuum and medium/medium interfaces We will consider the electric and magnetic fields in two groups:

D and B together E and H together

We want to know what is conservedNormal components

First notice that we can write similar equations forD and B:

D = f B = 0

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Figure 2.6: Small cylinder at interface

Consider an interface with no free charges Consider the small cylinder of Fig. 2.6, height dh, area da.Gauss theorem tells us:

V

Ddv =v

fdv (2.78)

S

D nda =V

fdv (2.79)

For the magnetic field, we find: S

B nda = 0 (2.80)

What is the flux of D through the box? Take the limit dh 0, and for an interface with no free charge we find:S

D nda = D2 ndaD1 nda = 0 (2.81) D2 n = D1 n (2.82)

D1 = D2 (2.83)B1 = B2 (2.84)

where the opposite signs on the displacement vectors come from their opposing directions (compared to the surfacenormals). This implies that the normal components of D are continuous across an interface with no free charges, whilethe normal components of B are always continuous. This means that lines of D and B are conserved at an interface withno free charges.

Note that, in factD2 D1 = f (2.85)

Tangential components

Figure 2.7: Small loop at interface

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First notice that we can write similar equations for E and H:E = BtH = J

Consider an interface with no free current Consider the small loop of Fig. 2.7, height dh, length dl.

Stokes theorem tells us S

E nda = S

B

t nda (2.86)

C

E dl = S

B

t nda (2.87)

Taking the limit dh 0, da = dldh 0 we find:S

E nda =C

E dl = 0 (2.88)

But this can be written as E1 AB +E2 CD. As the vectors from A to B and from C to D have opposite directions,we write:

E1 dl = E2 dl (2.89)E1 = E2 (2.90)

And for an interface with no free surface current (surface magnetization currents are irrelevant) we have a similarresult for H:

H1 dl = H2 dl (2.91)H1 = H2 (2.92)

Note that, in factH2 H1 = Ifenc (2.93)

This implies that the tangential components of the E and H fields are conserved subject to the conditions explainedabove. This means that field lines are not conserved in general across the interface for these fields.

Normal components of B are continuous across an interface Normal components of D are continuous across an interface with no free charges Tangential components of E are continuous across an interface Tangential components of H are continuous across an interface with no free currents Field lines of E and H are not conserved across interfaces in general

2.7 Summary of Linear Media Linear: e is independent of E (or m of B) Isotropic: P is parallel to E (or M to H) Homogeneous: e is position independent D = 0E+P P = 0eE so D = E, with = 0 (1 + e)

2011 27


D = f H = B/0 M M = mH so B = 0rH with r = 1 + m H = Jf continuous across an interface:

B D when no free charges E H when no free currents

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PHAS3201: Electromagnetic Theory CHAPTER 3. ATOMIC MECHANISMS

Chapter 3

Atomic Mechanisms

3.1 Dipoles and Polarization

Dipoles

A dipole is a pair of equal and opposite charges separated by a small distance

The microscopic effect of an electric field on a dielectric can be modelled with dipoles

We can expand the field or potential of an arbitrary charge distribution in terms of multipoles

This will be discussed after we have looked at a dipole

Notice that if we have a pair of equal but opposite charges separated by a vector l, then the dipole moment is p = ql.

Figure 3.1: Geometry of a simple dipole.

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Dipole Geometry

E (r) =q

4pi0

(r+r2+ rr2

)(3.1)

V (r) =q

4pi0

(1

r+ 1r

)(3.2)

r+ = r l/2 (3.3)r = r+ l/2 (3.4)

V (r) is easier to work with than E(r)What is the electric field at a point r due to a dipole (length l) at the origin, oriented along the z axis? (Note that

we have decided to put the dipole at the origin, and chosen an easy orientation; these can be generalized without muchdifficulty). Define the vectors involved:

r+ = r l2

(3.5)

r = r+l

2(3.6)

We need to know the magnitude of r+ and r, so using Eqs. (3.5) and (3.6), we find

|r+| = r+ r+ (3.7)r+ r+ = r2 rl cos + l

2

4(3.8)

= r2(

1 lr

cos +l2

4r2

)(3.9)

|r+| = r

1 lr

cos +l2

4r2(3.10)

|r| = r

1 +l

rcos +

l2

4r2(3.11)

(3.12)

We can now write down 1/ |r+| and 1/ |r| and expand them to first order in l/r:1

|r+| =1

r

(1 l

rcos +

l2

4r2

) 12(3.13)

' 1r

(1 +

l

2rcos

)(3.14)

1

|r| '1

r

(1 l

2rcos

), (3.15)

where we have used (1 + )n ' 1 + n to first order in . Note that this is only valid when r l. Now,1

|r+| 1

|r| =l

r2cos , (3.16)

so the potential is given by:

Dipole Potential:

V (r) =ql cos

4pi0r2=

p r4pi0r2

, (3.17)

where p is the dipole moment, defined as:

p = ql (3.18)

p =

V

r (r) dv, (3.19)

2011 30


where the second form is for the dipole moment of a charge density in a small volume V . Now that we have the potential,we can calculate the field. Notice that we have naturally ended up working in spherical polar coordinates (with definedas the angle from the z-axis), and that there is no dependence on .

[The following brief discussion of multipole expansion follows Griffiths pp.149-150, and is not directly exam-inable; however, it is extremely useful to understand, and is well within the capability of students.]

In outline, we start by considering the potential at a point r due to an arbitrary charge distribution, (r). This can bewritten as:

V (r) =1

4pi0

(r)|R| dr

, (3.20)

where R = r r. In order to make this simpler, we need to rewrite |R| in terms of r and r. The magnitude of R can befound from a dot product:

R R = (r r) (r r) (3.21)= r2 + (r)2 2rr cos , (3.22)

where is the angle between r and r. By taking a factor of r2 outside, we see that we can write:

R2 = r2

(1 +

(r

r

)2 2

(r

r

)cos

)(3.23)

R = r1 + (3.24) =

(r

r

)((r

r

) 2 cos

)(3.25)

So we can expand 1/R using the binomial expansion; it is important to note that we will not make any approximation,and inherently carry the full expansion with us (though it will not be shown).

1

R=

1

r(1 + )1/2 =

1

r

[1 1

2+

3

82 5

163 + . . .

](3.26)

=1

r

[1 1

2

(r

r

)((r

r

) 2 cos

)+

3

8

(r

r

)2((r

r

) 2 cos

)2+ . . .

](3.27)

Now gathering terms in(rr

), this can be written:

1

R=

1

r

[1 +

(r

r

)cos +

(r

r

)23 cos2 1

2+

(r

r

)35 cos3 3 cos

2+ . . .

](3.28)

=1

r

0

(r

r

)nPn(cos ), (3.29)

where Pn(cos ) are the Legendre polynomials. Substituting this expression into Eq. (3.20) for the potential, we find:

V (r) =1

4pi0

[1

r

(r)dv +

1

r2

r cos (r)dv +

1

r3

(r)2(

3

2cos 1

2)(r)dv + . . .

](3.30)

=1

4pi0

0

1

rn+1

(r)nPn(cos )(r)dv (3.31)

This shows that for large distances, an arbitrary charge distribution behaves approximately like the total charge (thefirst term, which falls off with 1/r, is known as the monopole term). Other terms can be brought in to improve the

2011 31


approximation (and will be important at shorter distances): the dipole term scales with 1/r2, the quadrupole term scaleswith 1/r3, the octopole term with 1/r4 etc.

[End of multipole discussion]

Starting from the potential we just derived, and working in spherical polar coordinates, we can write:

E (r) = V = rVr 1

r

V

1

r sin

V

(3.32)

V

r= 2ql cos

4pi0r3(3.33)

V

= ql sin

4pi0r2(3.34)

V

= 0 (3.35)

Dipole Field and Potential

Er(r, , ) =ql cos

2pi0r3(3.36)

E(r, , ) =ql sin

4pi0r3(3.37)

The potential is zero along the centre line (x-y plane) The field decays as 1/r3, potential as 1/r2

As always, the electric field is always perpendicular to equipotentials

Figure 3.2: Electric field and equipotentials for a dipole.

Ultimately, we want to understand the response of matter to applied fields We looked at the potential of an arbitrary charge distribution Expanding 1/R we made the multipole expansion We then derived the potential & electric field for a dipole The potential falls off as 1/r2, the field as 1/r3

2011 32


Microscopic Dipoles

Dielectrics have no free charges Atoms consist of nuclei and electrons which respond to an applied field Positive charge moves with field, negative against it But the displacement is limited by a restoring force This results in a neutral material with a net dipole

Polarized Dielectric

Consider a small volume of a dielectric (Fig. 3.3)

Figure 3.3: A piece of unpolarised dielectric.

Apply a field: there is a net displacement of a (Fig. 3.4)

Figure 3.4: A piece of polarised dielectric.

Polarization per unit volume

Charge element dq = (r) dv In a dielectric, p (r) =

vrdq

Define macroscopic polarizationP (r) =

p (r)

v(3.38)

How can we relate it to induced charge densities? What is the field due to a polarized dielectric?

2011 33


There are two approaches: one simple (as found in Grant & Phillips), and one more rigorous. We consider the simpleone first, then expand to the rigorous one.

In both cases, we must be very careful about using the expression for potential outside a dielectric inside the dielectric;the problem is that we must average over some volume which is large enough that the effects of individual electrons andions are not considered but which is still smaller than the object considered. This then gives the macroscopic field; notethat the potential and field outside a polarized dielectric (which we derived earlier) also rely on this averaging, but weare far enough away that the details of the microscopic dipoles dont need to be considered. It turns out that this samepotential can be used, and we will sketch a justification (see Griffiths pp.173175 for a detailed discussion).

First, we need some results (I will quote them - they can be shown relatively easily); in both cases we need the averagepotential over a sphere or radius R.

Figure 3.5: Looking at a sphere inside a dielectric.

1. The potential, averaged over a sphere, due to a polarized dielectric outside the sphere is equal to the potential to thefield produced at the centre of the sphere (the same is true for the field). Thus we can use the formula for potentialfrom before:

Vout(r1) =1

4pi0

out

r12 P(r2)|r12|3 dr2 (3.39)

2. The field due to a collection of charges inside a sphere averaged over that sphere is:

Ein = 14pi0

p

R3, (3.40)

where p is the dipole moment of the charges relative to the centre of the sphere.

The potential inside a polarized dielectric due to the dielectric itself can then be constructed as follows. We considera sphere of radius R within the dielectric (which will be large enough to contain a few hundred or thousand atoms), andaverage the potential (or field) over that sphere. The potential due to the charges outside is given above in Eq. (3.39), andis the formula wed expect. The charges inside the sphere are a little harder: we need the total dipole moment, which issimply p = 43piR

3P. Then substituting into Eq. (3.40) we find:

Ein = 14pi0

43piR

3P

R3= 1

30P (3.41)

which is just the field for a uniformly polarized sphere (again, this result is easily proved). This means that, regardlessof the microscopic distribution of charges, the average field or potential is that of a uniformly polarized sphere. When weadd the two contributions, its clear that the macroscopic potential (and field) inside a polarized dielectric have the sameform as the potential (and field) outside a polarized dielectric.

Now onto a simple demonstration. Consider a small block of material of size xyz, located at (x, y, z). Since thepolarization, P, is defined as the dipole moment per unit volume, the amount of charge which has crossed the plane at xmust be Px(x)yz, and at x + x it is Px(x + x)yz. The net charge entering the small cube in the x-direction isthen:

(Px(x+ x)yz Px(x)yz) = Pxx

xyz. (3.42)

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We can write similar equations for the y and z directions, and find that the total charge is given by:(Pxx Py

y Pz

z

)xyz (3.43)

If we divide this charge by the volume element xyz then we find an effective polarization charge density:

P = P. (3.44)In uniform, bulk dielectrics this will tend to zero as the number of charges entering and leaving will be the same; nearsurfaces or areas where the density varies rapidly then charges accumulate. These two effects are seen more clearly next.

[NOTE This derivation has already appeared in Section II on Macroscopic Fields, but its repeated here for complete-ness; it may well not be repeated in the lectures.]

For the more rigorous demonstration, we start by finding the potential at a point r due to a small volume of polarizedmaterial at a point r. We will then integrate this over the entire piece of dielectric material. We write, using Eq. (3.17):

(r) =p (r) (r r)

4pi0 |r r|3(3.45)

=vP (r) (r r)

4pi0 |r r|3(3.46)

When we take the limit v 0 and sum over the elements, we get an expression for the total potential:

(r) =

V

dvP (r) (r r)4pi0 |r r|3

(3.47)

We use the gradient of 1/ |r r| to transform this:

(r) =1

4pi0

V

P (r) (

1

|r r|)dv (3.48)

Using the formula for (F) from the Mathematical Identities, and rearranging (we want F ) we can write:

(r) =1

4pi0

V

{

(P (r)|r r|

) 1|r r| P (r

)}dv (3.49)

Finally, we use the divergence theorem on the first term to give the potential outside a polarized dielectric object:

(r) =1

4pi0

S

P (r) n|r r| da

+1

4pi0

V

P (r)|r r| dv

(3.50)

Polarization Charge Densities

The surface polarization charge density is defined:

P = P n (3.51)

The volume polarization charge density is defined:

P = P (3.52)

We can write the potential as:

(r) =1

4pi0

(S

P|r r|da

+V

P|r r|dv

)

(3.53)

=1

4pi0

dqP|r r| (3.54)

2011 35


Does this mean that the dielectric is now charged? To understand this, consider a polarized dielectric, with volume V0.We must consider the outside surface S0, and its associated polarization charge density. Using the divergence theorem wecan write:

QP =

V0

P dv +

S0

P da (3.55)

=

V0

Pdv +S0

P nda (3.56)

= S0

P nda+S0

P nda = 0. (3.57)

So the overall dielectric is electrically neutral (which we assumed at the start). However, the field can be non-zero: inparticular, if there is an applied external field inducing the polarization, then the dielectric itself will affect that field. Forcompleteness, note that we can write:

E (r) =1

4pi0

(S

P (r r)|r r|3 da

+V

P (r r)|r r|3 dv

)

(3.58)

Polarization arises from alignment of microscopic dipoles These give surface and volume polarization charge densities We looked at potential inside and outside dielectric

3.2 Magnetic DipoleWe will now consider the magnetic field due to a circular current loop at the origin.

Geometry

Current loop, radius a, current I at origin We consider the magnetic induction at a point P Using cylindrical polar coordinates, P = (R,, z) Small element of loop dl at P = (a, , 0) Vector from dl to P is rThere are two ways to do this derivation: first, using a multipole expansion and approximating the loop by its dipole

moment (which will be given briefly now); second, more fully and slowly, leading to a full expression for the vectorpotential in terms of elliptic integrals. The approximation for the elliptical integral leads to the same result as the dipolemoment.

2011 36


[Derivation of vector potential for arbitrary current loop]In the Macroscopic Fields part of the course, we showed that, using the Coulomb gauge, the vector potential at a point

r due to a current density J(r) distributed over a volume at r could be written:

A(r) =04pi

J(r)|r r|dv

(3.59)

For a constant current I in an arbitrary loop, with Jdv Idl, we can rewrite the volume integral as a line integral.Re-introducing the vector R = r r, we write:

A(r) =0I

4pi

dl

|R| (3.60)

As with the potential of a charge distribution, we now need to write |R| in terms of r and r. Using Legendre polynomialsagain, and recalling that the angle between r and r is , we find by substituting into Eq. (3.60) that:

A(r) =0I

4pi

0

1

rn+1

(r)nPn(cos )dl (3.61)

=0I

4pi

[1

r

dl +

1

r2

r cos dl +

1

r3

(r)2

(3

2cos2 1

2

)+ . . .

](3.62)

Now notice that the first term (the monopole term) is multiplied by a closed loop integral with integrand 1, whichis identically zero (it is not surprising that the monopole term disappears as we started from the assumption that thereare no monopoles). So the first non-zero term in the expansion is a dipole term; we will use the identity

r cos dl =

(r r)dl = r da to simplify this. Then the dipole term only is:Adip(r) =

0I

4pir2

r cos dl (3.63)

=0I

4pi

da rr2

=04pi

m rr2

, (3.64)

where m = Ida = Ia is the magnetic dipole moment. This derivation allows a full expansion to be made for an

arbitrary current loop; far from this loop it behaves like a dipole. This vector potential will be seen again below.[End of multipole expansion for arbitrary current loop]

We now return to the derivation of the vector potential for a circular current loop. Let us consider first how to writethe vector for the small element of loop, dl:

dl = d a ( sini+ cosj) (3.65)

Here weve used the standard two-dimensional formula for arc length, and projected it onto Cartesian vectors. Note thatwe want the direction of dl to be tangential to the current loop; so at = 0 it lies along y-axis, at = pi/2 it lies alongthe x-axis but in the opposite direction etc. Weve basically taken the gradient of the position on the unit circle.

We now consider the magnetic induction due to the current loop. One approach would be to use the Biot-Savart law,and integrate around the current loop, but this quickly becomes very complicated. Instead, we will use the vector potential;this isnt trivial, but its easier, and allows us to get further.

First we notice that we can change Jdv Idl for a constant current through the loop.

Vector Potential

We can write for the vector potential at P (a point, not polarization!):

A (P) =04pi

V

J (P)|PP|dv (3.66)

=0I

4pi

dl

|r| (3.67)

2011 37


The vector potential is in the direction of the current elementUsing our knowledge of the geometry of the system, we can deduce that the vector potential only has an azimuthal

component (i.e. one around the loop), A. We can write A = A i with:i = sini+ cosj (3.68)

So we can write:

A = A i (3.69)=

0I

4pi

dl i|r| (3.70)

Remembering from trigonometry that cos ( ) = sin sin+cos cos, we expand the dot product, using Eqs. (3.65)and (3.68), as:

dl i = d a [sin sin+ cos cos] (3.71)= d a cos ( ) (3.72)

A =0Ia

4pi

2pi0

cos ( ) d|r| (3.73)

This now seems to be quite an easy integral to evaluate. Using the cylindrical symmetry of the system, we know that Amust be independent of the value of , so we can evaluate the integral for any convenient choice of ; well take = 0.However, we mustnt forget that |r| must be written in terms of R, z, a, and :

|r| =

(PP) (PP) (3.74)=

(R cos a cos)2 + (R sin a sin)2 + z2 (3.75)

A =0Ia

4pi

2pi0

cos ( ) d{(R cos a cos)2 + (R sin a sin)2 + z2

} 12

(3.76)

Setting = 0, we can simplify somewhat:

|r| =

(R a cos)2 + (a sin)2 + z2 (3.77)

=

R2 2aR cos + a2 cos2 + a2 sin2 + z2 (3.78)

=R2 + a2 + z2 2aR cos (3.79)

A =0Ia

4pi

2pi0

cosd

{R2 + a2 + z2 2aR cos} 12(3.80)

This expression, while somewhat complex, can be written in terms of special mathematical functions called ellipticalintegrals which have been tabulated, and are implemented in packages such as Mathematica. However, with one approx-imation we can find simpler expressions.

|r| =R2 + a2 + z2

(1 2aR cos

R2 + a2 + z2

) 12

(3.81)

Let us make the approximation that 2aR cos < R2 + a2 + z2. This is fulfilled if

R2 + z2 > a2: far from the current loop a2 + z2 > R2: close to the axis

These are two important cases. Now we can write:

|r|1 = (R2 + a2 + z2) 12 (1 2aR cosR2 + a2 + z2

) 12(3.82)

' (R2 + a2 + z2) 12 (1 + aR cosR2 + a2 + z2

)(3.83)

A =0Ia

4pi

2pi0

cosd

(R2 + a2 + z2)12

(1 +

aR cos

R2 + a2 + z2

)(3.84)

2011 38


But we can simplify this, using some basic integrals: 2pi0

cosd = 0 (3.85) 2pi0

cos2 d =

2pi0

1

2(1 + cos 2d) =

[1

2

]2pi0

= pi (3.86)

Form of A

Far from the dipole (or near the axis), we can write:

A =0I

4

a2R

(R2 + a2 + z2)32

(3.87)

But spherical polars are easier R = r sin and r = R2 + z2

A =0Ia

2

4

R

(R2 + z2)32

(3.88)

=0Ia

2

4

sin

r2(3.89)

Valid if R2 + z2 > a2

Now that we have the vector potential, we can recover the magnetic induction, B. We write:

B =A = 1r2 sin

ir ri r sin ir

Ar rA r sin A

(3.90)But we know that A only has a component in the direction, which simplifies things considerably! We can write thedifferent components of B as follows:

Br =1

r2 sin

{

(r sin A)

}(3.91)

=1

r

A

+1

r2 sin r cos A (3.92)

=0Ia

2

4

{cos

r3+

cos

sin

sin

r3

}(3.93)

B = 0 (3.94)

B =r

r2 sin

{ r

(r sin A)

}(3.95)

=1

r sin

{ sin A r sin A

r

}(3.96)

=0Ia2

4

{sin

r3 2sin

r3

}(3.97)

2011 39


Form of B

We find for the components of B far from the dipole:

Components of B:

Br =0Ia

2

2

cos

r3(3.98)

B = 0 (3.99)

B =0Ia

2

4

sin

r3(3.100)

These are identical to the electric dipole far from the dipole

Close to the dipole, the fields differ

The electric dipole consists of two charge aligned along the dipole axis, while the magnetic dipole consists of a currentloop lying in the plane perpendicular to the dipole axis. Field lines for the electric dipole start and end on the charges,while the field lines for the magnetic dipole form closed loops (see Fig. 3.6).

Figure 3.6: Illustration of field lines for an electric dipole (left) and for a magnetic current loop (right). The plane of thecurrent loop is perpendicular to the page, so that it would be coming out and going into the page.

3.3 Magnetic Dipoles and Magnetization

As we saw with the electric field, E, the introduction of a medium other than vacuum results in changes to Maxwellsequations. These changes can be handled by using an alternative field which includes the effects of the medium implicitly.We derived various equations for this displacement using a dipole model of the polarizability of the atoms (or molecules)making up the medium. We will now apply the same ideas to magnetic media. A word of caution: non-linear magneticmedia are much more common than non-linear electric media; we will deal with these rather interesting materials in thenext chapter (Chapter 4) on Ferromagnetism.

2011 40


Magnetization

We introduced the polarizability of a dielectric material, P E

Similarly, we introduce a quantity, proportional to the magnetic induction B

This is the magnetization, M

It describes the response of a material to the magnetic induction

Electrons move in loops around atoms: we can use the magnetic dipole to model the response

Microscopic Origin

Consider a small piece of a material of volume V

If mi is the dipole due to the ith atom in V , we define:

M = limV0

1

V

i

mi (3.101)

This is analogous to the polarization (electric dipole moment per unit volume)

With no field, the directions are random and M = 0

We will now consider magnetization currents arising from dipoles

Approach

We will consider a given body made up from adjacent current loops

We find three simple limits:

1. Uniformly magnetised bulk: No net magnetization current

2. Non-uniformly magnetised bulk: volume magnetization current

3. Uniformly magnetised slab: surface magnetization current

Surface Magnetization Current

Consider a slab thickness t, surface area S shown in Fig. 3.7

Uniformly distributed (small) magnetic dipolesIf there is a net magnetization of M, then given that the volume of the sample is St, the magnetic field (at large

distances) is the same as would come from a dipole of size StM . The magnetization is perpendicular to the surface of theslab.

Now consider a small strip of current loops, ABCD, shown in the bottom of Fig. 3.7. For each loop with a componentto the right, there is an equal and opposite loop with a component to the left; for the strip PQRS, for each current goingup there is an equal and opposite current going down. From a distance which is large compared to the current loops, thesewill cancel out (this can be shown with the Biot-Savart law).

However, at the boundary of the material there are no loops to cancel out the edge loops. This will give us a surfacemagnetization current density jM, analogous to surface polarization charge density in a dielectric.

We know that the dipole moment is StM. The surface magnetization current density is equivalent to a current ofmagnitude jMt. But a current loop has dipole moment of magnitude IS, so we know that StM = IS, and jM = I/t = M .As jM is perpendicular to the magnetization, we can write:

jM = M n (3.102)

2011 41


Figure 3.7: Uniformly magnetised slab

Volume Magnetization Current

Consider the strip shown with dashed lines in Fig. 3.8

The net downward current is larger than the net upward current

There will be a net downward magnetization current Jm This will give a non-uniform magnetization

Let us make a more rigorous derivation of the dependence of the magnetization on the volume magnetization currentdensity.

Figure 3.9 shows two small volumes in a piece of non-uniformly magnetised material. They have dimension dx, dy, dz,and are located at (x, y, z) and (x, y + dy, z). The magnetization in the first element will be taken as M(x, y, z), and weassume that the materials behave linearly. Then the magnetization in the second material can be written:

M(x, y + dy, z) = M(x, y, z) +M

ydy + . . . (3.103)

As indicated in Fig. 3.9, we will concentrate on the x-component of magnetization, Mx, which arises from small circu-lating currents Ic and I c. Now, the magnetic moment of the first small element is Mdxdydz, which can be written asIcdydz (since m = 12I

r dl, we need the current multiplied by the surface area and 2a = r dl). So we can write

for both elements:

Mxdxdydz = Icdydz (3.104)(Mx +

Mxy

dy

)dxdydz = I cdydz (3.105)

Now the net current flowing in the z-direction on the boundary between the small elements is Ic I c. UsingEqs. (3.104) & (3.105) we can write:

2011 42


Figure 3.8: Non-uniform magnetic material

Figure 3.9: Two small volumes of magnetised material

Ic I c = Mxdx(Mx +

Mxy

dy

)dx

= Mxy

dydx (3.106)

There is another contribution to the current flowing in the z-direction in the first element (i.e. the one at (x, y, z)) thatcomes from a similar consideration involving another small volume element, this time at (x+ dx, y, z); this would be infront of the first element in Fig. 3.9. Here we write:

Mydxdydz = Ic dxdz (3.107)(

My +Myx

dx

)dxdydz = I c dxdz (3.108)

This time were considering the current flowing around the face perpendicular to the y-direction. The net current in thez-direction this time is given by I c I c , which gives:

I c I c =Myx

dydx (3.109)

2011 43


From these two contributions to the current, and the knowledge of the surface area of the small element, we can write thenet current density:

JMz =Myx Mx

y(3.110)

Now this is easily recognised as part of the formula for a curl. We can perform similar calculations for the other twodirections.

Full Result

The final result we obtain is, as before:

JM =

(Mzy My

z

)i+

(Mxz Mz

x

)j

+

(Myx Mx

y

)k (3.111)

JM = M (3.112)

Again, notice the similarity to P = P We have already considered another way of reaching this result using the vector potential[NOTE This derivation has already appeared in Section II on Macroscopic Fields, but I repeat it for completeness; it

may well not be repeated in the lectures.]Let us consider the vector potential at a point r1 due to a small volume of magnetised material at a point r2. This

small volume will have magnetic moment m = M (r2) V2. Then we can write:

A (r1) =04pi

V

m r12|r12|3

(3.113)

=04pi

V

M (r2) r12|r12|3

dV2 (3.114)

=04pi

V

M (r2)2 1r12

dV2, (3.115)

where weve used a standard result to get from Eq. (3.114) to Eq. (3.115). Now we use the expansion of (F), withF = M and = 1r12 to write:

F = F (F) (3.116)A (r1) =

04pi

V

{2 M (r2)r12

2 (M (r2)

r12

)}dV2 (3.117)

Now we use the theoremV FdV =

Sn Fda to write:

A (r1) =04pi

V

2 M (r2)r12

dV2 04pi

S

nM (r2)r12

da2 (3.118)

=04pi

V

2 M (r2)r12

dV2 +04pi

S

M n (r2)r12

da2 (3.119)

Magnetization current densities

We formally define:

Magnetization current densities:

JM = M (3.120)jM = M n (3.121)

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JM is the volume magnetization current density

jM is the surface magnetization current density

We move on to considering how linear magnetic media behave

3.4 Diamagnetism and Paramagnetism

Magnetic Susceptibility

For a linear, isotropic material, we assert (based on experimental observations):

M = mH (3.122)

m is the magnetic susceptibility

We can write B = 0 (1 + m)H = 0rH

If m > 0 we have a paramagnetic material

If m < 0 we have a diamagnetic material

Note that m can depend on temperature, but is generally small for these materials (less than 105)

Diamagnetism

No intrinsic moments (no unpaired electrons): current loops

With H = 0, the loops are unexcited and M = 0

As |H| increases, |B| increases, so there is more flux through each loop

From Faradays and Lenzs laws, we can show that a voltage develops to oppose the change in flux

This results in m < 0 and B < 0H

These arguments are purely electromagnetic: no T dependence

Care: magnetism is properly quantum mechanical!

Figure 3.10: M as a function of H and as a function of T for a diamagnet and paramagnet.

2011 45


Paramagnetism

Some free intrinsic moments (e.g. unpaired electrons) With no external field, these are randomly aligned, M = 0 (thermal fluctuation) As H increases, they experience a torque aligning them with H So m > 0 and B > 0H, where m0 is the magnetic moment on each component of the system However, this is opposed by random thermal fluctuations Thermodynamic analysis gives Curies law:

m =Nm200

3kT(3.123)

There is also often competition between diamagnetic and paramagnetic effects, so m = Nm200/3kT + dia At low T , paramagnetic effects dominate

2011 46

PHAS3201: Electromagnetic Theory CHAPTER 4. FERROMAGNETISM

Chapter 4

Ferromagnetism

A wonder of such nature I experienced as a child of 4 or 5 years, when my father showed me a compass. Thatthis needle behaved in such a determined way did not at all fit into the nature of events which could find aplace in the unconscious world of concepts (effects connected with direct touch). I can still remember - or atleast believe I can remember - that this experience made a deep and lasting impression upon me. Somethingdeeply hidden had to be behind things.

- A. Einstein

Ferromagnetism represents the earliest discovery of a phenomenon which results from quantum phenomena: lode-stones were used in navigation by the Phoenicians several thousand years ago, while the detailed understanding of ferro-magnetism was not worked out until 1928 (by Heisenberg). We will cover the details at a qualitative level only.

4.1 Atomic-level Picture

We start by considering the effect of the unpaired electron in the 3d shell.

Intrinsic Moments

There are intrinsic moments at the atomic level

Unpaired electron spins give the direction of the moments

There is a strong short range force between neighbouring atoms

The atoms will align in the lowest energy configuration

In ferromagnetic materials, the configuration which has the lowest potential energy is with the spins aligned parallelto each other. This is not what would be expected from a simple picture of bar magnets, for instance. A full understandingof the phenomenon requires a careful quantum mechanical treatment of the problem, which turns out to arise from anexchange integral and the Pauli exclusion principle. An approximate, classical understanding was first put forward byWeiss who postulated the existence of an unspecified field (now known as the Weiss, or mean, molecular field) such that:

Hm = M. (4.1)

In other words, there is some field due to the magnetic moments at the atomic level. For a ferromagnetic material, thevalue of M must arise to a large extent from the individual moments, Nm0 (where m0 is the individual moment). Thisrequires a value of to be around 1,000; a simple derivation similar to that used for dielectrics and polarization wouldpredict 13 . Nevertheless, if this theory is followed through, it predicts a change of magnetization with temperature whichis in approximate agreement with experimental measurements, including a prediction that there is a temperature at whichthe spontaneous magnetization vanishes (the Curie temperature).

2011 47


Figure 4.1: Examples of ferromagnetic ordering

FM Orientations

Ferromagnetic ordering can take different forms The defining characteristic is a local, parallel ordering Ordering depends on temperature Ordering may only be local

Figure 4.2: Examples of antiferromagnetic ordering

Other Orientations

Anti-ferromagnetic ordering has anti-parallel local ordering Ferrimagnetic ordering shows both spin components but a net moment Also known as ferrite materials Important materials (more later)It is important to note that the ordering only applies when the energy gained from aligning the spins in certain ways is

more than the random thermal energy available to the atoms; once this condition fails to hold, all these effects are washedout and the materials behave as ordinary paramagnetic materials. The temperatures are known as the Curie temperature(ferromagnetic) and the Nel temperature (anti-ferromagnetic).

2011 48


Figure 4.3: Examples of ferrimagnetic ordering

What will happen as a ferromagnetic material is cooled through this transition temperature without an external field?The system starts in a non-ordered state, where the random thermal motion has a large enough amplitude to overcomethe short-range forces between spins. There is a symmetry to the individual spins, in that there is no preferred direction(hence a spherical symmetry). As the temperature passes through the Curie temperature for the material, local atomicordering in some arbitrary direction (there is no external field, remember) will appear. This is an example of a rathergeneral phenomenon called spontaneous symmetry breaking (suddenly there is a preferred direction for the alignment ofthe moment, and the spherical symmetry is broken).

Domains Locally, there will be a tendency for atoms to align, due to the short-ranged interaction between them. Thiswill lead to the formation of small groups (called domains) of aligned atoms. On a larger scale, these domains will nothave any relationship to each other initially. However, the circulation of the electrons (whose unpaired spins on the atomsgive rise to the local moments) also lead to a magnetic field. Locally, this is much smaller than the moments, and haveno effect. However, at long ranges, the total magnetic field of a domain can lead to a significant field. At this level, it isbetter for domains to align in an opposed manner (by analogy to a bar magnet). In a ferromagnetic material, domains canextend across tens of microns.

Figure 4.4: Ferromagnetic domains

Domains are a consequence of the conflict between the short-range exchange interaction and the long-range mag-netic force

The magnetic sample breaks up into small regions, or domains, typically 0.001 - 0.01 mm across

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The long-range magnetic energy is minimised and the exchange energy is sacrificed only in the region of the wallsbetween the domains. In this region the loss of exchange energy is minimised by the dipoles twisting gently overso that each dipole is very nearly parallel to its neighbour. This structure is called a Bloch wall

Figure 4.5: The structure of the Bloch wall separating domains. In Fe, the thickness of the transition region is about 300lattice constants.

The magnetic properties of a ferromagnet are not determined by the intrinsic interactions between the atoms, butby the ease or difficulty with which the domain walls can move through the solid: very easy in a perfect crystal, butpinned by crystalline defects

When cooled with no external field, the domains are disordered With an external field, they align The resulting magnetization is large (strong moments) B = 0 (H+M) gives B H Ferromagnetism amplifies magnetic effects strongly

4.2 B & H: Macroscopic EffectsIf we want to investigate magnetic properties of different materials, its useful to remember that H arises from freecurrents only (i.e., those flowing in wires or coils), so that we can always impose a value of H on any sample (particularlya ferromagnetic one). The resulting induction B will depend on H and M. As we change H, the magnetization willchange and we can detect the results using Faradays law to detect changes in B (we will discuss a circuit for this later).

Hysteresis First, we note that the response of a paramagnetic material would be almost invisible in Fig. 4.6; the responseof B essentially parallels the H axis.

The normal magnetization curve (dotted line in Fig. 4.6) traces out the value of B reached for a given value of Hstarting from an unmagnetised sample. So initially, we would measure a value of B which followed the normal magne-tization curve. Now, when point 1 (H1,B1) is reached, imagine that the field H is reversed. Initially, the magnetizationis not affected, and the normal curve is not traced; the B field remains nearly constant, and cuts the H = 0 axis at point2. If we continue to reverse H (with negative values) then the magnetization responds, and drops, so that at point 3 theB field is zero (with a finite H field). As the H field is decreased further, the magnetization continues to respond, and atpoint 4 we reach (H1,B1). If H is again reversed (and brought back to zero) then as before, the magnetization doesnot respond until the H field is opposed to it (to a good approximation). As the H field is increased back towards thevalue H1, the B field drops to zero (at point 5) and then increases until we return to point 1. This entire process is calleda minor hysteresis loop.

If we now increase the H field beyond point 1, we will follow the normal magnetization curve until the materialcannot be magnetised any further (all domains are aligned). At this point (point 6) the magnitude of the magnetization

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Figure 4.6: B-H curves for a ferromagnetic material

|M| Ms, the saturation magnetization. To the right of this point (i.e. for larger values of the H field) the B fieldincreases slowly, due only to the term 0H.

If we reverse H from any point to the right of point 6 (whose coordinates are (Bs,Hs), the saturation values of B andH) then we trace out the major hysteresis loop. The value of B when it crosses the H = 0 axis on this loop is calledthe remanence, Br. The (negative) value of H required to reduce B to zero on the major loop is called the coercivity,Hc. When point 7 is reached, then the reverse saturation has been reached, and the loop continues back through point 8to point 6.

Definitions

Saturation magnetization: value of M when domains are fully aligned Saturation intensity, Hs: magnetic intensity required to produce saturation Saturation induction, Bs: magnetic induction at saturation Rema

full notes for electromagnetism course at ucl

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