Lecture Notes EMAPP

8/2/2019 Lecture Notes EMAPP

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Abstract

In these notes I present an overview of electrodynamics, quantum mechanics andstatistical mechanics which are the foundations of most of todays technology:electronics, chemistry, communication, optics, etc.


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CONTENTS

1 Introduction: the Unity of Science 4

2 Electromagnetism 52.1 Electrostatics 6

2.1.1 Gausss Law 102.1.2 Examples 122.1.3 Solving Poissons Equation 172.1.4 Electric field in matter: dielectrics 21

2.2 Magnetostatics 24

2.2.1 Electrostatics, magnetostatics and relativity: the Lorentz-force 26

2.2.2 Amperes Law 312.2.3 Examples 322.2.4 Magnetic fields in matter: permeability and perma-

nent magnets 362.3 Electromagnetic induction 38

2.3.1 Faradays Law 382.3.2 Examples 40

2.4 Maxwells equations 452.4.1 Simple radiating systems 502.4.2 Optics and diffraction 56

2.4.3 Electromagnetic waves in matter: refraction 603 Appendix 70

3.1 Vector calculus 703.2 Fourier Transforms 74

3.2.1 Properties of the Fourier transform 743.3 Laplace and Helmholtz Equation 76

3.3.1 Solutions in a Box 763.3.2 Solutions for a cylindrical geometry 773.3.3 Solutions for a spherical geometry 78

3.4 Special relativity 84

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INTRODUCTION: THE UNITY OF SCIENCE

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ELECTROMAGNETISM

Electromagnetism is the study of electric and magnetic phenomena. In presenting thissubject I shall usually follow the historical narrative. First static fields were investigated.The electric field by Coulomb, Cavendish, Gauss, Franklin and others and the magneticfield by Ampere, Biot, Savart, Volta, Galvani and others. One of the great discovery ofthat time was that electricity and magnetism were actually related phenomena. Sincethe equivalence of electric and magnetic fields is more clearly displayed by considering

their description in different moving frames using the special theory of relativity, inpresenting static magnetic fields I shall adopt that point of view even though it differsfrom the historical narrative. We shall then study the electromagnetic fields generated bytime dependent currents (accelerated charges). That area or study is associated with theworks of Faraday and Maxwell. The later in particular was the first to formulate a fulltheory of electromagnetism which unified electricity, magnetism and optics. It predictedthe existence of electromagnetic waves which were indeed observed by Hertz and gaverise to modern communication by radio waves. Maxwells equations were the reasonthat forced Einstein to reconsider Galilean invariance and propose his special theoryof relativity. These equations and the problems they raised were also at the foundationof Quantum Mechanics. Studying and understanding their enormous predictive poweris thus of paramount importance for the comprehension of modern Technology and

Science. Since that is such a vast field with so many ramifications, we will only study itsfoundations, use those to explain various natural phenomena (lightning, aurora borealis,the rainbow, etc.) and illustrate their practical applications in our everyday life (motors,generators, electrical circuits, optics, radio, etc.).

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6 ELECTROMAGNETISM

F. 2.1. (a) Schematic drawing of Cavendish torsional balance used to test Newtonstheory of attraction between massive body and Coulombs theory of interaction be-tween charged bodies. The force between the sphere(s) on the pendulum and thefixed sphere(s) exerts a torsion on the pendulum which results in an angular rotationby an angle . (b) Original drawings of Coulombs pendulum.

2.1 Electrostatics

Electricity is a phenomenon known for thousands of years (for a detailed historicalreview see en.wikipedia.org/wiki/History of electromagnetism). People had direct ex-

perience of it through contacts with electric fish (called by the ancient Egyptians thun-derer of the Nile). By rubbing certain materials, man also knew that a force could begenerated between them. In fact the word electricity comes from the greek word foramber (elektron) which when rubbed with fur can attract or repel other objects. In par-ticular it repels an other piece of rubbed amber, but it attracts a piece of glass rubbedwith silk. The charged left on amber is defined as negative (then called resinous electric-ity), that left on glass as positive (then called vitreous electricity). Thus it was knownlong before it was understood that like-charged objects repel each other, whereas op-posite charge objects attract each other. This action at a distance of what we now callcharged objects was very intriguing and mysterious to the pre-scientific world.

In 1687 Newton published his master piece: Mathematical Principles of NaturalPhilosophy (known from its original latin title as Principia). This work opened thearea of scientific inquiry. In it Newton exposes his mathematical formulation of the mo-tion of bodies and the law of gravity (to that purpose he invented calculus!). Influencedby this novel approach (like many of his contemporaries) Coulomb decided to similarlyinvestigate the force between charged objects. At that time (18th century) the only wayto generate charges was by an elctrostatic generator that used the friction of a rubberband to transport charges and store them in a so-called Leyden jar (a primitive form of


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ELECTROSTATICS 7

capacitor, see below). People new that some materials conducted electricity (like met-

als) and some didnt (they were insulators like amber). They could charge a Leyden Jarby connecting it with metalic wires to a charged one. To evidence the charge in the jarpeople would use an electroscope, which consisted of a small sphere of non-conductingmaterial (a pith ball) which would be attracted to charged surfaces. Alternatively onecould use the repulsion between two charged gold leafs to demonstrate the presence ofcharges on a conducting material brought into contact with the leaves.

Even though there was no understanding of electrical phenomena, these observa-tions led in 1774 to the first invention of a telegraphic machine. It consisted of 24 in-sulated wires, each representing a letter of the alphabet connected at their distant endto an electroscope. To send a message, a desired wire was connected momentarily to acharged Leyden jar, whereupon the electroscope at the other end would get charged (thegold leaves or a pith ball would fly out) and in this way messages were transmitted at a

distance!Intrigued by these observations, Coulomb decided in 1784 to measure the force Fbetween two charged spheres using a torsional pendulum similar to the one used atabout the same time by Cavendish to study Newtons Law, see Fig.2.1. It consistedof two balanced spheres (of mass m, one of them conducting) a distance R from theaxis of rotation. The force between the charged conducting sphere on the pendulumand a nearby similarly charged sphere yields a torque = FR = , which twists thependulum by an angle measured by the deflection of a light ray incident on an attachedmirror. The torsional stiffness of the pendulum is deduced (when no force is exerted)from the oscillation frequency 0 = /Iof the pendulum of moment of inertia I = 2mR2

:

Id2

dt2+ = 0

The electrostatic force exerted on one of the spheres on the pendulum by a nearbyone is then: F = /R. Coulomb observed that the force was proportional to the productof the charges (q1, q2) on the spheres, inversely proportional to the square of the distancebetween their centers (|r1 r2|) and directed along the vector r12 = (r1 r2)/|r1 r2|linking these centers:

F = kq1q2

|r1 r2 |2 r12

The value of the proportionality constant ksets the units of the charge Q. In the CGSsystem, which we will mostly adopt here, k = 1 and the charge is measured in so-calledelectrostatic units (esu) or statcoulomb. In the MKS system k = 1/40 = 9 109 Nm2/C2 (0 is known is the dielectric constant of the vacuum) and the unit of charge isthe Coulomb (C): 1 C = 3 109 esu.

The charge of an electron is e = 4.8 1010esu = 1.6 1019C (see below how itwas measured). The neutrality of atoms (the fact that matter is rarely charged and eventhen very weakly) is ensured by the fact that protons have equal and opposite charge tothat of electrons.

The force between an electron and a proton in a hydrogen atom (radius r 0.1nm= 108cm) is F = e2/r2 (4.8 1010)2/(1016) = 23 104 dynes = 23 109N = 23


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8 ELECTROMAGNETISM

nN (this set the scales of the strength of covalent bonds between atoms (order nN)).

Notice that this is of the same magnitude as the gravitational force between two massesm1 = m2 = 2kg held a distance r = 10cm apart Fg = Gm1m2/r2 28 nN! (whereG = 6.67 1011 N (m/kg)2 is the gravitational constant).

Charge is a conserved and relativistic invariant (i.e. velocity independent) quantity.If this was not the case, the misbalance of charges between the moving electrons and themore static protons in an atom would generate forces that overwhelm the gravitationalattraction between bodies.

If there are many charges ({qi}) acting on a charge q at position r, the total force onthe charge q is the vectorial sum of the individual forces:

Ftot =

iqqi(r ri)|r ri|3

(2.1)

That property of the electrostatic (and also gravitational) forces known as linear super-position is what allows one to understand and predict their action (this is not the casewith hydrodynamic forces, which is why the weather is so unpredictable). The electricfield Eof a distribution of charges {qi} is defined by the combined force on a test chargeq located at position r:

E Ftotq

=

i

qi(r ri)|r ri|3

(2.2)

= d3r

(r)(r r)

|r

r

|3

(2.3)

This equation can be further developped to yield a function often used in electrostaticproblems, the potential :

E =

d3r

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

= r

d3r

(r)|r r| (2.5)

r

(2.6)

Where the electrostatic potential due to a charge distribution is defined as:

(r)

i

qi

|r ri|=

d3r

(r)|r r| + Const (2.7)

The constant (which has no physical consequences) is usually chosen such that thepotential at infinity is zero. The electrostatic potential is measured in units of statvolts


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ELECTROSTATICS 9

in CGS units and in volts (V, in MKS units): 1 statvolt = 300 V. Since E =

, by

standard vector calculus identities:

E = = 0 (2.8)Therefore by Stokestheorem the work W exterted by an electrostatic force on a

closed path is zero:

W =

F dr = q

E dr = q

E dS = 0 (2.9)

The work performed by moving a charge between two distinct points (r1,r2) is thussimply proportional to the difference in electrostatic potential between those points:

W = 21

F dr = q21

E dr = q21

dr (2.10)= q[(r2) (r1)] = q (2.11)

Where is the potential (or voltage) difference. No work is done displacing acharge on an equipotential surface, i.e. a surface for which (x,y,z) = const. Conse-quently the electric field E is always perpendicular to the equipotential surfaces. Due tothese properties equipotential surfaces play an important role in electrostatic problems.

The work done to ionize an hydrogen atom (bring its electron a large distance awayfrom the proton ( r2 = ) is: W = e2/r 23 1012erg = 23 1019J = 14 eV (electron-Volt). That energy (a few eV) is the typical energy of a covalent bond.

Lightning. From these considerations we can also understand the mechanism oflightning. During a lightning storm strong air-currents in the clouds results in collisonsbetween supercooled water drops and ice particles, that leave the drops slightly nega-tively charged and the ice particles slightly positively charged (the same way amber ischarged when rubbed with fur or a silk shirt is charged when taken offand rubbed overthe hair). The lighter ice particles are advected towards the top of the cloud leaving thebottom of the cloud negatively charged. When the electric field between the top andbottom of the cloud is large enough to ionise the surrounding air an electric dischargeoccurs which is the lightning bolt. Like the explosion of an atom bomb, lightning is anavalanche process, Fig2.2: a spontaneously ionised electron is accelerated by the elec-tric field in the cloud. It hits an atom in the air and ionises it; the two electrons are thenaccelerated again until the next collison with two atoms, generating 4 electrons, etc. For

this avalanche to occur the energy gained by the electron between two collisons must belarge enough to ionise an atom (a few eV), i.e. to generate an other free electron. Oncethe avalanche has been initiated a path is traced in air where the ionised gas channel theelectrons and where the lightning current can pass.

To be more precise, let the gas density be = NA/VA 3 1019 particles/cm3 (whereNA = 6 1023 is Avogadros number and VA = 22.4 l is the molar volume of a gas in


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F

. 2.2. (a) Schematic drawing of dielectric breakdown: the process where an electron(small red dot) accelerated by the electric field E, hits a particle (atom or molecule,blue dot) in the air and ionises it. The two electrons are accelerated again untill thenext collison with two atoms, generating 4 electrons, etc. (b) Ligthning on the EiffelTower (the charges are initially accelerated in the vicinity of the towers tip wherethe field is maximal).

standard conditions). If the ionization cross section S = r2 1 2, then the mean freepath between collisons lc = 1/S = 3m. For a typical ionisation energy ofWion 3eVthis corresponds to an electric field ofEb = Wion/e lc = 106 V/m, which is known as thedielectric breakdown field in air. Hence the potential difference in a cloud or between itand the ground can reach many million volts!

2.1.1 Gausss LawElectricity is of course fundamental to modern civilization and the control of staticelectric fields is an essential part of modern technology (for example in electron micro-scopes or TV tubes, see below). To that purpose Eq.2.7 is all one needs to compute theelectric field of a given charge distribution. There are however various ways to computethat sum (or integral) which we will now review.

Consider a charge q and a surface S surrounding that charge at a distance r(, ).Since the electric field E is directed along the radius and decays as 1/r2 (a property alsoshared by the gravitational field), Gauss proved that:

E dS = 4q

To convince oneself of the validity of Gausss theorem one may write EdS = E ndS =Ecos dS , see Fig.2.3. However cos dS is the area perpendicular to the radius vectorr which defines the solid angle d spanned by dS : cos dS /r2 = d and therefore:

E dS =

Er2d = q

d = 4q. Because of linear superposition, the field due tomany charges {qi} will obey:


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ELECTROSTATICS 11

F. 2.3. A surface element dS whose normal n makes an angle with the electric fieldE originating from a charge q at point O. The solid angle d spanned by dS isgiven by d = dS cos /r2. Therefore: E dS = EdS cos = Er2d = qd, whichintegral gives Gausss law.

E dS = 4

i

qi (2.12)

As we shall see below, Gausss theorem is often used to compute the electric fieldwhen the charge distribution is constant on some symmetric surfaces (planes, spheres,

cylinders, etc.). Using the divergence theorem from vector calculus:

4q =

E dS =

E dV (2.13)

Since q =

dV, one obtains Coulombs Law:

E = 4 (2.14)

or using the relation E = , Poissons equation:

2 = 4 (2.15)

The solution of this equation is in fact Eq.2.7, as can be checked using the identity:

2r1

|r r | = 4(r r) (2.16)

Using Eq.2.15, the work Wdone to generate a certain charge distribution (qi or(r)),i.e. the energy stored in the electric field generated by that charge distribution, is:


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W =

1

2ij

qiqj

|ri rj| (2.17)=

12

(r)(r)d3r = 1

8

2d3r (2.18)

= 18

dS

()2d3r

(2.19)

Since the fields vanish at infinity, the first integral on the right is zero. We are thereforeleft with:

W =1

8

()2d3r = 1

8

| E|2d3r (2.20)

The local energy density of a static electric field is thus: w(r) = 18 | E(r)|2.

2.1.2 Examples

The field of a dipole. Consider two opposite charges (q, q) a distance d apart. Thischarge configuration is known as a dipole (its orientation defines in the following thez-axis). To compute the electric field of a dipole at a distance r d consider first thepotential it generates :

(r) =q

|r dz/2| q

|r+ dz/2| (2.21)

Expanding the denominators:

|r dz/2|1

=

1/

x

2 +y

2 +

(z d/2)2

(2.22)= r1(1 zd/r2)1/2 (2.23)= r1(1 zd/2r2) (2.24)

We obtain:

(r) =qdz r

r3=

p rr3

=p cos

r2(2.25)

The dipole moment is defined as: p qd z. Using the expression of the operatorin polar coordinates (r, , ):

= r

r+1r

+

1rsin

(2.26)

we can compute now the electric field:

E = = 2p cos r3

r+p sin

r3 (2.27)

Notice that the attractive (or repulsive) component of the field (the component along r)is maximal when = 0 (or ), i.e. along the dipole axis.


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ELECTROSTATICS 13

The energy of a dipole in an external electric potential (or field E) is:

W = q(r) q(r d) = qd = p E = pEcos (2.28)Thus to minimize their energy, dipoles tend to align parallel to the external field E or(in abscence of an external field) to each other. The torque aligning the dipole is then: = r F = p E.

Piezo-electricity. The previous small exercise with electric dipoles has interestingpractical consequences. Certain materials known as ferroelectric (for example Bariumtitanate, BaTiO3) have small permanent electric dipoles that according to our precedinganalysis will tend to align parallel to each other generating a strong electric field in thematerial and an electric potential difference across it. Small mechanical deformation

of the crystalline material affect the dipole moment (the distance dbetween the charges)and thus the potential across the material. This effect known as piezo-electricity is usedeither to generate electricity (in electric gas lightners or wrist-watches) or to preciselymove objects by applying an electric field on a piezo-electric material thus affectingits extension. Because changes in temperature cause a crystal to expand or contract(thus altering the distance between charges), ferro-electric materials also display pyro-electricity, i.e. a change in the voltage across the material as a function of temperature.This property is being used to make very sensitive infrared cameras, where changes intemperature on a pixel (picture element) as small as a millionth of a degree is convertedinto a detectable voltage change on that pixel.

Application of Gausss theorem: the capacitor.When the distribution of charges is symmetric and uniform Gausss theorem is of-

ten the simplest way to compute the electric field. In particular in charged conductingmaterials, since like charge repel each other they will distribute themselves on the sur-face so as to cancel the average force on them, i.e. Ftot = 0 which implies that insidea conductor E = 0. Since E = the surface S of a conductor is an equipotentialsurface |S =Const.

For example consider two large flat conducting planes a small distance dfrom eachother bearing a constant but opposite surface charge density = Q/S , see Fig.2.4.Since the charge is opposite on both plates, there is no field outside them. Betweenthe two planes however application of Gausss theorem in a box enclosing one of thesurfaces yields: ES = 4S , i.e. E = 4 and the potential difference between the

plates is: = Ed = 4Qd/S = Q/C where C = S/4d is known as the capacitanceof the plates (notice that in the CGS system capacitance is measured in cm; in the MKSsystem capacitance is measured in Farad: 1 farad = 1 C/V). A capacitor is an elementthat stores electric energy:

We =

dQ =

Q2

2C=

(ES /4)2

2C= S d

E2

8(2.29)


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F. 2.4. A capacitor consists of two conducting surfaces of area S separated by a smallgap d. The surfaces are charged with equal and opposite charge densities and .Application of Gausss Law in the element KLMN (which encloses a charge equalto zero) shows that the field outside the capacitor is null. Similarly application of

Gausss Law in the element ABCD shows that E = 4.

Thus the energy stored per unit volume is E2/8. As we shall see this is a general resultfor the energy density of electric fields.

The two flat parallel plates configuration is the simplest example of a capacitor,but any two conducting plates not necessarily flat separated by a small gap will forma capacitor which can store charge. The Leyden jar mentioned earlier consisted of aglass bottle whose inner and outer surfaces was covered with a conducting sheet. Theinner surface was charged by an electrostatic generator while the outer surface wasgrounded (thus ensuring that its charge counter-balances the charge deposited insidethe jar). Capacitors are a ubiquitous components of every electrical circuit. By storingcharge they buffer the circuit against sudden changes in the current.

Millikans measurement of the electron charge. The fact that the electric fieldbetween the two plates of a capacitor is a constant has allowed Robert Millikan toprecisely measure in 1909 the charge of an electron. He proceeded in the followingway: he sprayed small charged oil-droplets and let them fall between the plates ofa capacitor, see Fig.2.5. Due to friction with the air (of known viscosity and den-sity air) the droplets (of density oil) reached a constant drift velocity v (upward ordownward) that he could measure. Equating the viscous drag on a droplet (the Stokesforce: FStokes = 6rv) with the balance of electrostatic (qE) and gravitational force(mg = (4/3)(oil air)r3g) yields: FStokes = qE mg. Studying the same dropletwith different values of electric field E and thus final drift velocity v allowed Millikanto deduce the droplet radius r and its charge q. Repeating the experiment on differ-ent droplets, Millikan noticed that the droplets charges were quantized: they werean integer multiple of an elementary value, see Fig.2.5, the charge of the electrone = 1.6 1019 C, which he got with only a 1% error-bar!


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ELECTROSTATICS 15

F. 2.5. Left: scheme of Millikans oil drop experiment. Charged oil-droplets are in-troduced between the plates of a capacitor and their movement is observed with amicroscope. Millikan deduced the charge on the droplet from the dependence ofits final velocity on the electric field . Right: histogram of the charge of a colloidaldroplet in a recent experiment similar to Millikans original study (see PRL 100,218301 (2008)).

The electrostatic lens. The simple capacitor can also be used to focus electronbeams in applications such as cathode ray tubes (TV sets) or electron microscopes. Theprinciple of an electrostatic lens is the following. Electrons emitted by some hot cathodeare accelerated in a voltage difference V0, to a velocity v1 such that: mv21/2 = eV0. Asthey enter a capacitor consisting of two planar grids separated by a small distance danda voltage difference V their velocity increases so that their velocity at the exit from thatcapacitor v2 satisfies: mv22/2 = mv

21/2+eV = e(V+V0), see Fig.2.6(a). As a consequence

of the longitudinal acceleration of the electron (the acceleration along the capacitor fieldE), their propagation direction is altered (like the light rays that we will study later inthis course they are refracted). Since the transverse component of the velocity is notaltered by the capacitor field: v = v1 sin = v2 sin and therefore:

sin sin

=v2

v1=

1 + V/V0 (2.30)

If the capacitor is a thin spherical shell of radius r d and in the small angle limit(, 1 it is easy to show that a parallel electron beam will be focused at a distancef such that: f sin( ) = rsin , i.e. f = r/(1 /) 2rV0/V. This is the simplestimplementation of an electrostatic lens, see Fig.2.6(b). But in fact many electric fieldconfigurations whose equipotential surfaces (surfaces on which =constant) are curvedcan be used to focus an electron beam, see for example Fig.2.6(c).

In a TV set capacitors are not only used to focus the electron beam on a fluorescentscreen but also (by applying a transverse electrostatic force) to steer it horizontally and

vertically across the screen. Finally by modulating the voltage difference between theelectron emitting hot cathode and a nearby grid, the amount of electrons emitted fromthe cathode (the beam intensity, i.e. the brightness of the image) can be controlled.This is also the working principle of vacuum tubes (triodes), which served as voltageamplifiers in many electronic devices (radio, radars, TV sets, etc.) before the inventionof the transistor, see Fig.2.7.


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F. 2.6. a): Change in the direction of propagation of an electron beam as it is ac-

celerated between the plates of a capacitor. (b): Examples of an electron lenses: athin spherical capacitor which can focus the electron beam at a distance f satisfy-ing: f sin( ) = rsin . (c) Example of a field configuration which equipotentialsurfaces are curved and that can also be used to focus an electron beam.

The method of images.When point charges are located in the vicinity of grounded conducting surfaces the

so-called method of images is a trick often used to solve for the electric field. The ideais to introduce a charge of opposite sign on the other side of the conducting surfacesuch as to ensure that the potential on the surface is null (it is grounded). For exampleconsider finding the field E due to a charge q located on the z-axis a distance d from aflat infinite grounded plane located at z = 0. Introducing a charge q at z = d ensuresthat the potential on the plane is indeed zero, as can be verified:

(0,y,z) = qx2 +y2 + d2

qx2 +y2 + d2

= 0 (2.31)

Since the two charges (q and its image charge q) form a dipole of moment p = 2qdz,the field in the upper half space (z > 0) is simply that of a dipole, Eq.2.27.

The method of images can also be used if a charge q is placed a distance rfrom thecenter of a grounded sphere of radius r0, see Fig.2.8. In that case we have to determinethe charge q and position r of the image charge so that:

(r0, , ) =q

r20 + r2 + 2rr0 cos

+q

r20 + r2 + 2rr0 cos

= 0 (2.32)

Which we may rewrite as:

(r0, , ) =q

r

1 + (r0/r)2 + 2(r0/r)cos +

q

r0

1 + (r/r0)2 + 2(r/r0)cos = 0

(2.33)The potential is null on the sphere if an image charge q = qr0/ris placed at a distancer = r20/rfrom the center of the sphere along the line joining it to the charge q.


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ELECTROSTATICS 17

F. 2.7. (a-b) The vacuum tube was used as a voltage amplifier before the invention of

the transistor. (a) A hot cathode emitted a current of electrons that were collected ona cold anode. The voltage difference between the cathode and a grid placed betweenit and the anode controlled the intensity of the current (and voltage) between cathodeand anode. In abscence of a grid the device works as a diode which allows currentto flow in only one direction: from the hot cathode to the cold anode. To preventcollisons with the air and corrosive reactions at the electrodes the tube (b) was heldin vacuum, hence its name. (c) Advances in the understanding of the conductanceof semiconductors allowed the development in 1925 of the Field E ffect Transistor(FET) which working is similar to that of a vaccum tube: a voltage on the gate con-trols the density of charge carriers beneath it and thus controls the current flowingbetween the source and the drain.

2.1.3 Solving Poissons Equation

Often in electrostatic problems, the potential (or charge) on certain surfaces is controlledand one needs to determine the resulting electric field. That is the case for example whendesigning an electrostatic lens, see above and Fig.2.6. The problem then is to solve theLaplace (or homogenous Poisson) equation:

2 = 0 (2.34)


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18 ELECTROMAGNETISM

F. 2.8. A charge q is placed a distance r from the center O of a grounded sphere ofradius r0. The potential outside the sphere is the same as would be generated by thecharge q and an image charge q = qr0/rplaced a distance r = r20/rfrom O.

When the potential is given on a surface |S = S , the problem is known as aLaplace problem with Dirichlet boundary counditions. When the charge is given on thesurface, by Gausss theorem this is equivalent to setting the electric field perpendicularto the surface, i.e. the normal derivative of the potential n|S . This type of boundarycounditions are known as Neumann boundary conditions.

In cases where the boundary counditions depend on all three coordinates, one needsto solve Eq.2.34 in 3D. In most cases that is done numerically. There are however afew geometries where Eq.2.34 can be solved analytically. It is interesting to considerthese cases for various reasons. First they can be used as test grounds for numericalalgorithms. Second, the Laplace equation and its generalization, the Helmholtz equa-tion (2 = ) are encountered in a great variety of contexts (electromagnetic radi-ation, quantum mechanics, heat equation, etc.), so that their solutions in one field canbe relevant in a totally different area. Because of the relevance of the solutions of theseequations in various fields, a whole Appendix is devoted to a study of this equation inrectangular, cylindrical and spherical coordinates (geometries).

2.1.3.1 Solution of Laplace equation in two dimensions If the field depends ononly two coordinates, say x and y (which is the case if the boundary conditions areuniform along the vertical coordinate) there are powerful methods to solve the Laplace

equation based on the knowledge of analytical functions. These are functions f(z) of thecomplex variable z = x + iy also written as f(z) u(x,y) + iv(x,y) (notice that in theseproblems z is a complex number and does not refer to the vertical coordinate). Sinced f/dz = x f = xu + ixv and d f/dz = iyf = iyu + yv, the real and imaginaryparts of f(z) satisfy:

xu = yv


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ELECTROSTATICS 19

F. 2.9. (a) The electric field E generated by a wire carrying a uniform charge densityper unit length can be computed by Gausss theorem on a co-axial cylinder ofradius r0. (b) The electric field lines and equipotential surfaces ( = Const.) inthe plane perpendicular to the wire. (c) The electric field lines and equipotentialsurfaces ( = Const.) for the problem conjugated to the uniformly charged wire, i.e.the problem where the potential is null for = 0 and fixed at V0 for = .

yu = xvFrom which one readily shows that both u(x,y) and v(x,y) satisfy the 2D-Laplace equa-tion: 2u = uxx + uyy = 0 and vxx + vyy = 0. Therefore all analytical functions (allfunctions f(z)) are solutions of Laplace equation in 2D. One simply(!) needs to find the

one satisfying the given boundary condition.

Consider for example the case of a very long straight wire with uniform chargedensity per unit length: , see fig.2.9. This is clearly a 2D problem as the potential doesnot depend on the position along the wire but on the distance from it. Gausss Law on acylinder of radius rcentered on the wire, yields:

E dS = 2rLE = 4L (2.35)

Namely E = 2/r and therefore the potential away from the wire is (r) = 2 ln r/r0.The potential is therefore a component (here the real part) of an analytical function

which in this case is simply f(z) = 2 lnz/r0 = 2 ln r/r0 + i2(with z = rei) .Notice that the imaginary part of f(z) (v() = 2) also satisfies the Laplace equa-

tion and is thus also a solution of a 2D electrostatic problem. It is in fact the solutionof the problem where the positive half of the y = 0-plane (i.e. = 0) is grounded((x > 0,y) = 0) while the negative half of that plane (i.e. = ) is kept at a po-tential (x < 0,y) = 1 = 2. Notice that while the equipotential surfaces obey


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20 ELECTROMAGNETISM

= const, the field lines are given by = const. As expected they are orthogonal to

each other. Consequently this solution also describes the field inside a wedge of angle one side of which is grounded ((, = 0) = 0) the other side being held at potential V((, = ) = V (i.e. constant along an equipotential line of the previous problem). thesolution is: = V/.

An extension of that problem is the case where the potential along the x-axis is 1for x < 1, 2 for 1 < x < 2 and 3 for x > 2. By superposition the potential at point(x,y) is:

= A11 + A22 + A3 (2.36)

Where i is the angle between points (x,y), (i, 0) and the x-axis: tan i = y/(x i).The coefficients Ai are determined by the boundary conditions:

3 = A3 (2.37)

2 = A2 +A3 (2.38)1 = (A2 + A1) +A3 (2.39)

So that Ai = (i i+1)/. By extension for a boundary condition on the x-axis givenby a potential b(, 0) the potential at a point (x,y) in the plane is:

(x,y) =1

(db

d)()d =

1

b(, 0)

d

dd =

y

b(, 0)

(x )2 +y2 d (2.40)

The wedge we considered previously had one side held at potential V while the

other was grounded. Let us now consider the potential inside a wedge both sides ofwhich are held at constant voltage V0, see Fig.2.10. Let us first consider the case wherethe wedge is flat, i.e. the case where in the w-plane ( w = u + iv) the u = 0 plane isheld at constant voltage V0 (i.e. (u, 0) = V0). The potential throughout is then simply: = V0 + Av = Im{iV0 + Aw}. The solution for a wedge of angle held at constantpotential V0 can then be found by mapping the wedge boundary into the u-axis bythe conformal transformation w = z/ = /ei/. The solution of Laplace equationbetween the wedge is then: (, ) = V0 + A/ sin / (this function satisfies theboundary conditions on = 0, and being the imaginary part of an analytical functionit also satisfies the Laplace equation). Notice that for > as 0 the field (and thusthe electrostatic force) diverges E . In particular for a half plane held at constantvoltage, i.e. = 2: Er( 0, ) = A1/2/2. This is a general result for sharptips: the electric field (and thus the force applied on charges) diverges near the tip. This

result sets the theoretical foundation for the working of a lightning rod (its tip attractscharges of opposite sign present in the air), even though at the time of its invention(1752) Franklin didnt know it (though he did show that lightning was an electricalphenomena by using it to charge a Leyden jar)!

The use of conformal mapping (w = f(z)) to transform the Laplace equation ina given geometry (in the complex z-plane) into a simpler geometry (in the complex


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ELECTROSTATICS 21

F. 2.10. Example of the solution of Laplace equation in 2D using conformal maps.The solution of Laplace equation in the plane w = u + iv where the potential alongthe horizontal axis u is given: (v = 0) = V0 is simply (u, v) = Im(iV0 + Aw)(the equipotential lines are shown as dashed lines). In the z-plane the poten-tial is V0 along a wedge of angle . The problem in this plane can be solvedby mapping the z-plane and its boundary conditions into w: w = z/. Noticethat by this transformation the wedge boundary is mapped into the u-axis andits interior into the upper w-plane. The solution of the potential is therefore:(, ) = Im(iV0 +Az/) = V0 +A/ sin /.

w-plane) where the solution is known is a very powerful tool for solving the Laplaceequation in 2D. This is however beyond the scope of this course, where the purpose ofthe previous exercises were simply to suggest the power of these analytical methods.

2.1.4 Electric field in matter: dielectrics

In the previous discussions we have considered the electric field arising from isolatedcharges in vacuum. Often though it is of interest to know the field arising from chargesembedded in a material environment (liquid, gas or solid) which itself maybe respon-sive to the electric field. Indeed mater is made of atoms and molecules which althoughelectrically neutral can polarize in the presence of an external electric field, namely their

charge distribution may become anisotropic in the presence of an electric field due tothe opposite forces felt by positive and negative charges. As a result they develop aninduced dipole moment P = E (where is the electric susceptibility of the material).We have seen previously that the potential due to a single dipole (of dipole momentp) was = p r/r3. For a distribution of dipoles with density P and in the presenceof charges with density the potential is the sum of the contributions from the charges


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22 ELECTROMAGNETISM

(which define the field E) and the dipoles (permanent or induced by the field E):

(r) =

d3r(r)

|r r| +

d3rP(r) (r r)

|r r|3

=

d3r[

(r)|r r| +

P(r) 1|r r| ] (2.41)

=

d3r

(r) P(r)|r r| (2.42)

Since E = and using Eq.2.16 one gets: E = 2 = 4( P)

which can be rewritten as: D = 4 (2.43)

where the displacement field D is defined as:

D E+ 4P = E (2.44)

where 1 + 4 is known as the dielectric constant of the material. At the interfacebetween two media of different dielectric constants, Gausss law for Eq.2.43, yields:

( D2 D1) n21 = 4 (2.45)

where n21 is the unit vector normal to the interface directed from medium 1 to medium

2. Similarly from Eq.2.8: 0 =

E dS = E dl . Choosing a path l close to andparallel to the interface, one obtains:

( E2 E1) n21 = 0 (2.46)

The work W necessary to change the charge distribution by (r) in presence of adielectric medium is:

W =

d3r(r)(r) =

14

d3r D = 1

4

d3r D = 1

4

d3r E D

Thus the energy contained in the electric field of a charge distribution is:

W =1

8

d3r D E (2.47)The previous considerations are particular relevant to the case of a capacitor where

the medium between the charged plates has dielectric constant > 1. By Gausss lawthe displacement field is then: D = 4 = 4Q/S . The potential between the plates is:V = Ed = 4Qd/S . Therefore in comparison with the case where the medium between


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ELECTROSTATICS 23

the plates is air, the capacitance C = Q/V = S /4d is increased by a factor which

can be very large. The energy stored in such a capacitor:

W =1

8

d3r D E = S d

8E2 =

S

8dV2 =

CV2

2

is also increased by a factor . Increasing the dielectric constant between the plates of acapacitor is a very practical way of storing more charge and buffering against changesin voltage.


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24 ELECTROMAGNETISM

2.2 Magnetostatics

Magnetism like electricity has been known since ancient times, when people noticedthat magnetite (lodestone) has the power to attract iron. Aristotle attributes to Thales(6th century BC) the first scientific discussion of magnetism. Mention of magnetism canalso be found as early as the 4th century BC in the chinese literature. The chinese wereindeed the first to use the property of magnets to point to the North in order to developin the 11th century the lodestone compass. It took about a century for the invention toarrive in Europe were it revolutionized shipping and sea-trade, and was instrumental inthe discovery of America. The unusual properties of the lodestone led the ancients toattribute a spirit to the stone. Indeed a full scientific explanation of magnets wouldhave to wait the development of Quantum and Satistical Mechanics to explain howatoms could carry a magnetic moment (through their spin - see below) and how anensemble of interacting spins could give rise to a global magnetic moment.

The understanding of magnetism came not from a study of magnetite, but fromthe discovery in 1819 by Oersted that current passing through a wire could deflect anearby magnetic needle. This observation was revolutionary: it showed that two distinctphenomena, electricity and magnetism, were actually related. It immediately led to thedevelopment of an instrument (the Galvanometer) that measured the amount of currentflowing in a given coil from the deflection of a small permanent magnet in its vicinity.The following investigations by Ampere of this phenomenon led him to formulate histheory of electro-dynamics relating magnetic fields and forces to the currents passingin interacting wires. These experiments were made much easier by the discovery ofchemically generated electricity and the development of the battery.

Untill the end of the 18th century, currents were generated by the momentary dis-charge of a capacitor (the Leyden jar). In 1800, Alessandro Volta observed that whentwo plates (one made of copper (Cu) and the other of zinc (Zn)) were immersed in asolution of brine or sulphuric acid a constant current would flow in a wire connectingthese two plates. The current flow implied the existence of a force, i.e an electric poten-tial difference between the two plates, the unit of which (the Volt) has been named in thehonor of its discoverer. Nowadays we understand the functioning of Voltas battery asbeing an example of an oxido-reduction reaction. An oxidation reaction is taking placeat the Zn anode: Zn Zn++ + 2e and a reduction reaction catalyzed by Cu is happen-ing at the cathode: 2H+ + 2e H2. Thus electrons are transfered from the anode tothe cathode, i.e. an electrical current is generated.

For its development of the first battery, Volta washonored throughout Europe. Napoleonwho was present at the demonstration by Volta of his battery at the French Academy,was so impressed that he made him knight of the Legion of Honor, Count of Lombardyand gave him a gold medal and a lifetime allowance to celebrate his contibutions to

Science! At that time scientists were admired by politicans who attended sessions of theAcademy!!.

Voltas invention of the battery launched the field of electrochemistry (the use ofelectricity to catalyze chemical reaction): within a few months of his invention, using avoltaic cell as a current source Carlisle and Nicholson discovered the phenomenon ofwater electrolysis by which they showed that water is made of hydrogen and oxygen.


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MAGNETOSTATICS 25

The battery also allowed for the first investigations of electricity: the behavior of con-

ducting materials. It led Georg Simon Ohm to formulate in 1827 his famous law thatthe current I (which MKS units are Amperes (1A = 1 C/s)) is related to the voltage Vacross a conductor by:

V = IR (2.48)

Where R is known as the resistance. Its units in the MKS system are named after Ohm:1 = 1V/1A, while in the CGS system the unit of resistance is sec/cm: 1 = 1011/9sec/cm.

The electrical resistance of a material is now understood as resulting from the scat-tering of the charge carriers (e.g. the electrons) by the atoms of the material (by theirthermal vibrations (known as phonons) or by impurities). Let be the mean time be-tween collisions and < v > the mean velocity of the electrons of mass m, then by New-tons Law: q E = F = dp/dt

m < v > /. For a wire of given length l, cross-section S

and a density of free electrons ne, the current is:

I = neqS < v >=neq

2

mS E =

neq2

m

S

lV

which is Ohms law. The quantity R = neq2/m is known as the conductivity of thematerial. In CGS units it has dimensions of sec1, while in MKS it is (m)1. The re-sistance R = V/I = l/RS increases with the length and decreases with S and R (itincreases with the resistivity R 1/R). On can intuitively understand Ohms law: thelonger the wire the greater the probability of scattering, while the larger its cross sec-tion the smaller the probability of encountering an impurity. As a result of that scatteringelectric energy is dissipated: the material is heated (this is how electric heaters work andwhy lamps are hot). The power dissipated is: Pdiss = V I = I2R = V2/R. At fixed voltage

(which is usually the case with electrical appliances that are powered by batteries or bythe electrical network) the smaller the resistance the larger the dissipated power. Nowa-days, Ohms law is appreciated as a particular example of a more general principle (thedissipation-fluctuation theorem) which states that the response (the current) of a systemnear thermodynamic equilibrium is proportional to the perturbation (the voltage), thecoefficient of proportionality (the resistance) being related to the dissipation of energyby the system and the fluctuations of the current at equilibrium.

For a system of charges in motion, one defines the current density as: J = v (where is the local charge density and v the local velocity of the charges). The total currentthrough a closed surface S is: I =

J dS . Charge conservation implies:

I = dQdt

=

dV

t=

J dS =

dV J (2.49)

from which one obtains the equation of continuity for the charge density :

t+ J = 0 (2.50)


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26 ELECTROMAGNETISM

The equation of continuity is the stronger local form of a conservation law. It states that

any local change in the conserved quantity is due to its transport. Such equation can beformulated in any context where a quantity is conserved (mass, energy, probability, etc.).For example in hydrodynamics, where mass is conserved, the equation of continuity isvery similar to Eq.2.50: m/t+ mv (where m is the fluid density). If the fluid isincompressible (m = const), the equation of continuity is v = 0: there are no fluidsources or sinks (in the volume studied).

In magnetostatics, the charge density does not depend explicitly on time (/t =0) and thus: J = 0, i.e. the total current trough a closed surface is nul: whatevercurrent gets in must get out. This is the type of current configurations that Amp erestudied. These investigations relied on Voltas invention of the battery, which delivereda constant current I for extended times. In particular Ampere studied the force betweentwo very long parallel wires a distance rapart and bearing currents I1,I2. He observed

that the force per unit length between the wires dF12/dl was given by:

dF12/dl = k2I1I2

r(2.51)

In the MKS system k = 0/4 = 107 N/A2. In the CGS system k = 1/c2 (where c is thespeed of light). To understand how the speed of light enters into a magnetostatic prob-lem, one has to resort to the special theory of relativity. Even though, this connectionwas made by Einstein about 80 years after Amperes discovery, we shall deviate fromthe historical narrative and adopt his point of view since it reveals a profound connectionbetween electrostatics and magnetostatics.

2.2.1 Electrostatics, magnetostatics and relativity: the Lorentz-force

Consider a charge q a distance rfrom a long wire of length L bearing a charge Q (chargedensity = Q/L). We have seen above, see Eq.2.35 that the electric field of such a wireis: E = 2/r. The force on the charge is therefore F = qE = 2qQ/Lr. Now consider thedescription of that system by an observer in a frame moving parallel to the wire withvelocity v. In that frame, both the wire and the charge appear to move in the oppositedirection with velocity v. From Einsteins special theory of relativity (see Appendix),we know that for this observer the length of the wire L is shorter than its rest framelength L by : L = L

1 v2/c2. On the other hand the distance between the charge

q and the wire is unchanged, being perpendicular to the direction of motion: r = r.Similarly the force on the charge (perpendicular to its motion) is smaller that in its rest

frame: F = F

1 v2

/c2

. Since the charge is a relativistic invariant (independent ofthe inertial frame, i.e. q = q, Q = Q):

F =2qQLr

1 v2/c2 = 2qQ

Lr(1 v2/c2) (2.52)

F =2q(Q/L)

r q v

c

2(Qv/L)rc

= qE q vc

2Irc

(2.53)


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MAGNETOSTATICS 27

The first term on the right represent the electrostatic repulsion (measured by an observer

in the moving frame) between a wire with charge density = Q/L and a charge q .The second term on the right corresponds (for the moving observer) to the magneticattraction between a charge q moving with velocity v parallel to a wire bearing a currentI = Qv/L (along the z-axis) and generating at a distance r from it a magnetic field:

B =2Irc

(2.54)

so that the magnetic or Lorentz force, Fm becomes:

Fm = qv

c B (2.55)

Electric and magnetic fields are thus ways of describing in different moving frames thesame interaction between charged particles. Notice the relation between the magneticand electric field in the moving frame: B = vE/c. In the CGS system they have thesame units, even though these are called Gauss (G) for the magnetic field and statvolt/cmfor the electric field. In the MKS system the units of B are Tesla: 1 T = 104 G. Whilethe values of the electric and magnetic fields will be different for observers in differentinertial frames, the observed physical phenomena (e.g. charge attraction or repulsion)remain the same: their interpretation only differs. However, even though electric andmagnetic fields are relative description of the same phenomena, there is a fundamentaldifference between them: while electric charges are commonly observed no one has everobserved a magnetic charge, a magnetic monopole. In Nature magnets always come asdipoles, with a north and a south pole. In that sense magnetic fields are derived fromthe more fundamental electric fields. The consequences of this fact are multiple. First,

since there are no magnetic charges according to Gausss law:

B = 0 (2.56)

In other words, the magnetic field lines must close on themselves. The second conse-quence is that magnetic fields do no work: Wm = 0. From Eq.2.55, one readily derives:

Wm =

Fm dr = q

c

(

dr

dt B) dr = q

c

(

dr

dt dr) B = 0 (2.57)

The Lorentz force experienced by a moving charge in a magnetic field is a gener-alization of Amperess force law for the interaction between electric currents, Eq.2.51.Indeed, the force on a wire carrying a current I2 in the magnetic field B1 generated by a

wire carrying a current I1 can be computed from Eq.2.55 by noticing that qv = vdl =I2dl:

dFm =1c

I2dl B1 = 1c

2I1I2rc

dl = 2I1I2rc2

dl r (2.58)

where we used Eq.2.54 for the magnetic field B1 a distance r from a wire carrying acurrent I1. This equation is Amperes force law, Eq.2.51.


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28 ELECTROMAGNETISM

F. 2.11. Curtains of Aurora Borealis over an Alaska sunset.

Aurora Borealis. The Lorentz force provides an explanation for the beautiful phe-nomena of the Aurora Borealis observable at the Earths poles, see Fig.2.11. This phe-nomena is due to the excitation of atoms in the upper atmosphere (80km above ground)by charged particles emitted from the sun, the so-called solar wind. These particlesmoving at a speed of about 400km/s are deflected upon their encounter with the Earthmagnetic field to the poles where they are decelerated by their collisons with the atomsin the atmosphere (mostly Oxygen and Nitrogen). As we shall see later on, the colors of

the Aurora are due to the light emitted by the atoms as they return to their ground state(green for oxygen, red for nitrogen).

To study the motion of a particle of charge q and velocity v in a magnetic field B, weshall align the z-axis along the magnetic field lines (a valid local approximation), i.e.B = B0 z. The Lorentz-force, Eq.2.55, acting on the particle is then: F = (q/c)v B =(qB0/c)(vy x vx y). From Newtons law, F = ma, we therefore obtain:

m x = (qB0/c)y (2.59)

my = (qB0/c) x (2.60)mz = 0 (2.61)

Defining the cyclotron frequency c = qB0/mc the solution of these equations is: vx(t) =v0 sin(ct+ ); vy(t) = v0 cos(ct+ ); vz = Const. From this solution it is clear that:

i) the energy of the particle is unaffected by the magnetic field (mv2/2 = Const), asexpected since the field does no work;

ii) the velocity of the particle along the field line is constant, hence the deflection ofthe solar wind to the poles;

iii) the particle spirals about the magnetic field line at a frequency c and with aradius = v0/c.


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MAGNETOSTATICS 29

F. 2.12. The original drawings of Lawrences patent for the cyclotron. A conductingdisk-like chamber was placed between the poles of a magnet. Electrons emitted at

the center of the chamber were accelerated by a voltage on each half of the chamberalternating at twice the orbital (cyclotron) frequency of the electrons.

The confinement of moving charged particles in the vicinity of magnetic field lineswhich explains the observation of Aurora Borealis at the poles only, is a property thatis being used to confine the highly energetic ionized matter (the plasma of electronsand nuclei, but not the neutrons) generated during nuclear fusion in an effort to sustainthe reaction and prevent the particles from coming in contact with the walls of theircontainer (in so-called tokamak reactors).

Particle accelerator. For non-relativistic velocities (v

c) the cyclotron frequency

is independent of the velocity of the particle. This has important practical applications.For example the first particle accelerator (built in 1932 by Lawrence at UC Berkeley)consisted of a chamber in the form of disk each half of which was held at an oppo-site alternating potential (see Fig.2.12). A magnetic field perpendicular to the chamberwas used to bend the trajectory of the electrons emitted at the center of the chamberand accelerated between its electrodes. The fact that the cyclotron frequency did notdepend on the velocity of the particle allows to accelerate them coherently at twice thecyclotron frequency (i.e. each time they pass the gap between the two half of the disk).As they are accelerated the radius of their motion around the magnetic field lines in-creases untill they escape from the chamber. This very simple particle accelerator isused today to generate high energy proton or ion beams for cancer radiation therapy.In modern particle accelerators where magnetic fields are used to bend the trajectory ofthe particles into a more or less circular orbit, relativistic effects are important (v

c).

The cyclotron frequency varies with the velocity which requires the synchronization ofthe pulse of electric acceleration with the frequency of the particles in the accelerator,hence the name of these accelerators: synchrotrons.


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30 ELECTROMAGNETISM

F. 2.13. The Penning trap is a configuration of electric and magnetic fields that can

trap particles in an orbital motion about the magnetic field lines so that their mass tocharge ratio can be estimated accurately by a precise measurement of their cyclotronfrequency.

The Lorentz force is also used in electron microscopes and TV sets (together withelectrostatic lenses) to control (bend) the path of electron beams. In analytical chem-istry magnetic fields are used in mass spectrometer to separate ionized molecules ac-cording to their mass to charge ratio (m/q). In so called MALDI-TOF spectrometer,the molecules to be analyze (organics molecules, proteins, sugars, etc.) are vaporizedwith their solvent (the matrix, hence the acronym MALDI for Matrix Assisted LaserDesorption Ionization) by a laser pulse that charges the solvent and the molecules to beanalyzed. They are then accelerated by a combination of electric and magnetic fields.Their time of flight (TOF) can then be measured and related to their velocity v which

by energy conservation satisfies: mv2/2 = qV. Thus knowing the acceleration voltageV the ratio m/q is determined. Alternatively their bending by a magnetic field can bemeasured and related again to their mass to charge ratio.

The Penning trap. However the most accurate method to measure that ratio isthrough the use of a Penning trap, see Fig.2.13, an instrument that earned his inven-tor, H.G. Dehmelt, the Nobel prize in 1989 for its applications in precision atomicclocks. This trap consists of a combination of electric and magnetic fields that forcecharged particles into circular orbits the frequency of which is related to the cyclotronfrequency. The measurement of their orbital frequency allows for a very precise de-termination of their mass to charge ratio. As a simple approximation of the Penning

trap consider the following configuration of charges: charges +2Q on the z-axis at po-sition d and charges Q on the x and y-axes at position d. One can show that theelectric field E generated in the vicinity of the origin by this charge configuration is:E = (6Q/d3)(x + y 2z). In a Penning trap this charge configuration is supplementedwith a constant magnetic field along the z-axis: B = B0 z, which role is to confine themotion of the particles in the xy plane, while the confinement (of a positively charged


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MAGNETOSTATICS 31

particle) along the z-axis is due to the repulsion from the positive charges at (0 , 0,

d).

In the xy-plane the motion of the particle obeys:

m x =6Qd3

x +qB0

cy (2.62)

my = qB0c

x +6Qd3

y (2.63)

Which solution is an epicycle (like the trajectory of the moon about the sun) character-ized by two frequencies the sum of which is the cyclotron frequency qB0/mc (check foryourself! Hint: x + iy = A1ei(1t+1) +A2ei(2t+2) ). Therefore by measuring these fre-quencies, one can determine very precisely the ratio m/q. The Penning trap can also beused to trap ions at very low temperatures in order to investigate their atomic transitions(a subject we shall study later in this course) and use them as time standards (atomic

clocks).

2.2.2 Amperes Law

From our previous derivation of the magnetic field of a current line, Eq.2.54, we seethat:

B dr = 4c

I =4c

J dS (2.64)

i.e. the integral of the magnetic field around a closed loop is proportional to the currentnormal to that loop. That equation known as Amperes law is to magnetostatics whatGausss law is to electrostatics. Since from Stokes theorem:

B dr =

B dS (2.65)

One may cast Amperes law into a differential form (valid for any current distribution):

B = 4c

J (2.66)

This equation is the equivalent of Gausss equation for electrostatic fields, Eq.2.14. Itsgeneral solution is:

B(r) =1c

d3r J(r) r r

|r

r

|3

=I

c

dl (r r)

|r r|3 (2.67)

where the last equation, known as the law of Biot-Savart is obtained when the current isflowing in a wire and the 1D integration (dl) is along the wire. The magnetic field B inmagnetostatics plays the role of the electric field E in electrostatics (compare Eq.2.67


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32 ELECTROMAGNETISM

with Eq.2.3). In analogy with the introduction of the electric potential in electrostatic

problems, it is usefull to rewrite Eq.2.67 differently such as to introduce the magneticvector potential A:

B =1c

d3r J(r) r r

|r r|3 (2.68)

=1c

d3rJ(r)

|r r| A (2.69)

The magnetic field is thus obtained as the rotational of the vector potential: B = A.The vector potential A plays a similar role in magnetostatics as the potential plays inelectrostatics ( E = ). The real measurable fields, the ones acting on charged particlesare the electric and magnetic fields ( E and B). However, the potential fields ( and A)

are often simpler to compute. Notice though that while is defined up to a constant,A is defined up to the gradient of a function (since the rotational of a gradient is zero, = 0). That freedom of choice (known as gauge invariance) plays a major rolein Quantum mechanics (see below). From these considerations we may write:

A =1c

d3r

J(r)|r r| +

(2.70)

Notice that this equation is valid only in cartesian coordinates. Since A = 0,Eq.2.70 is consistent with Eq.2.56 expressing the abscence of magnetic charges. Usingthe relation B = A Eq.2.66 can be cast as an equation for the vector potential A:

B = A = ( A) 2

A =

4c J (2.71)

In magnetostatics charge conservation implies J = 0 and therefore A = 0, whichgiven a time independent current distribution of sources J leads to a Poissons equationfor the vector field A :

2 A = 4c

J (2.72)

Thus given a distribution of currents the three components of the vector potential canbe computed (analytically or more often numerically) just like the electrostatic potentialcan be computed for a given charge distribution (see Appendix).

2.2.3 Examples

The magnetic field of a current segment.Let us use Biot-Savarts Law, Eq.2.67 to compute the contribution to the magnetic

field at a distance r from the axis of a segment of length l carrying a current I, seeFig.2.14:


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MAGNETOSTATICS 33

F. 2.14. Current I is flowing through a segment of length l. The magnetic field atposition r = x tan is computed using Biot-Savarts law

B(r) =I

c

dl (r r)

|r r|3 (2.73)

=I

c

dx

sin x2 + r2

= Ic

d

dx

d

sin3 r2

= Irc

21

d sin (2.74)

=I

rc(cos 2 cos 1) (2.75)

One can check that for an infinite wire (1 = , 2 = 0) one recovers our previous result:B = 2I/rc. The previous result can be used to compute the magnetic field at the centerof a square loop (1 = 3/4, 2 = /4): B = 4

2(I/rc) and other variants (octogonal

loop, etc.).

The magnetic field of a current loop.To demonstrate the usefullness of the vector potential A in computing the magnetic

field B, consider the field of a current I flowing in a loop of radius a, lying in thexy-plane. In polar coordinates the current density is: J = I(cos())(r a)/a. Incartesian coordinates the current density is thus: J(r) = J sin x + J cos y. Thevector potential A is computed from Eq.2.70 (in cartesian coordinates):

A(r) =1c

d3r

J(r)|r r| (2.76)

Since the problem has azymutal symmetry, we may choose the observation point in thexz-plane. In that case the denominator becomes:

|rr| = {r2 + r2 2rr(cos cos + sin sin cos }1/2 =

r2 + a2 2arsin cos (2.77)


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34 ELECTROMAGNETISM

where we used the fact that r = a and = /2. Since the denominator is symmetric

with respect to , in the component of J along x will not contribute (it changessign as ) and thus we are left with:

Ay(r, , = 0) =1c

r2drdcos

dJ cos r2 + a2 2arsin cos

(2.78)

= Ia/c

2pi0

d cos r2 + a2

[1 +arsin cos

r2 + a2+ ....] (2.79)

Ia2

c

sin r2

(2.80)

Where we have assumed that r a (far field approximation). Because of the azymutalsymmetry: A(r, , ) = Ay(r, , = 0) and thus we may compute the components of the

magnetic fields B:

Br =1

rsin

(A sin ) = 2

Ia2

c

cos r3

2m cos r3

(2.81)

B = 1r

r(rA) =

Ia2

c

sin r3

m sin r3

(2.82)

where m = (IS /c)z is the magnetic dipole moment of a small current loop of areaS = a2. In atoms the magnetic dipole is often the result of the motion of electronsabout the nucleus with frequency = 2/T. We can relate the magnetic dipole to theangular momentum of the electron by writing:

m = (qa2/T c)z = qma2/2mcz = (q/2mc)Lz (2.83)

where the angular momentum is: L = r p.Notice that the vector potential can be written as:

A =m r

r3=

m sin r2

(2.84)

A current flowing through a small loop will therefore generate a magnetic dipole whichwill tend to align magnetic dipoles (e.g. permanent magnets) in its vicinity. As we shallsee below, this property has found many applications in our day to day life.

Application of Amperes Law: the solenoid.

When the distribution of currents is symmetric and uniform Amperes law, Eq.2.64,is often the simplest way to compute the magnetic field. Consider for example an in-finitely long solenoid of radius a carrying a current I and characterized by n coilsper unit length. Because of symmetry the magnetic field B will be aligned along thesolenoid (z-)axis. Outside the solenoid, there being no currents, the field along theclosed loop (a) shown in Fig.2.15 satisfies: Bout(r1)L Bout(r2)L = 0 which implies


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MAGNETOSTATICS 35

F. 2.15. A) a solenoid is a made by wrapping a long wire around a cylinder. Thecurrent flowing through a solenoid creates a magnetic field that is very large in thesolenoid and very weak outside. (B) In the limit of infinitely long solenoid, themagnetic field B is aligned along the solenoid z

axis. Its value inside and outside

the solenoid (Bin,Bout) can be computed by applying Amperes law on the loops a,b and c. (C) A voice coil is a solenoid which interacts with a permanent magnetwhena current is flowing through it. In the present example it is used to move themembrane of a speaker as it gets in and out of the permanent magnet.

that Bout(r) = const = 0 (since as r the field must go to zero). For a loop (c) insidethe solenoid by the same argument Bin(r) = const, while for a loop (b) enclosing nLcoils Amperes law states that : (Bin Bout)L = 4InL/c, hence: Bin = 4nI/c.

The magnetic field generated by a solenoid can be used as an actuator to attractor repel a magnetic object along its axis. Since the current in a solenoid can be tunedwithin a very short time, it has many applications such as a pneumatic valve, the startersolenoid of a car or the voice coil of a speaker. In the later case, a current through the coilproduces a magnetic field which reacts whith a permanent magnet fixed to the speakersframe, thereby moving the cone of the speaker. By applying an audio waveform tothe voice coil, the cone will reproduce the sound pressure waves corresponding to theoriginal input signal. The fast response time of a voice coil is also used in order torapidly position the read/write head of hard-disk drives.

DArsonvals Galvanometer.An other common use of the field generated by a solenoid is in the measurement

of the current flowing through the solenoid in a so-called DArsonval Galvanometer,see Fig.2.16. This instrument consists of a flat solenoid (with n coils of area S ) placedbetween the poles of a permanent magnet, parallel to the field lines. When a currentflows in the solenoid a magnetic dipole m = (nIS /c)z is generated along the solenoidaxis (z) that tends to align with the field of the permanent magnet. A torsional springlinked to the solenoid counter-balances the torque = m B generated by the current inthe solenoid. The distortion of the balancing spring is indicated by an appropriate dialwhich deflection is thus proportional to the current in the solenoid.


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36 ELECTROMAGNETISM

F. 2.16. DArsonvals galvanometer consists of a flat solenoid placed between thepoles of a permanent magnet parallel to the field lines. When current flows in thesolenoid, the magnetic torque tends to align it perpendicular to the magnetic field.This torque is balanced by the deformation of a torsional spring coupled to thesolenoid and measured by the deflection of a dial, which is proportional to the cur-rent flowing in the loop

2.2.4 Magnetic fields in matter: permeability and permanent magnets

In the previous discussions we have considered the magnetic field arising from themovement of charges (currents) in vacuum (or in very thin conducting wires). Just aswe discussed the interaction of static electric fields with dielectric materials we willnow discuss the behaviour of magnetic fields in permeable and magnetizable materials.The major difference between magnetic and electric fields however is that the former

do no work on moving charges. Therefore they cannot induce a magnetic dipole in amaterial the way the electric field does. However magnetic fields do interact with ex-isting magnetic dipoles in the materials. These microscopic magnetic dipoles m are apurely quantum phenomena: as we shall see below (see Chap.???) they arise from thespin and angular momentum of the particles. We have seen that the vector potential Adue to a magnetic dipole m is: A = m r/r3. Thus the vector potential due to a currentdistribution Jand a dipole density M is:

A(r) =1c

d3r

J(r)|r r| +

d3r M(r) 1|r r|

=

d3r

J/c + M

|r

r

|

(2.85)

We have seen that: B = A = ( A) 2 A = 2 A, where the lastequation equation is a result of charge conservation J = 0 and the mathematicalidentity M = 0. Using Eq.2.16, one then obtains:

B = 4c

J+ 4 M


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MAGNETOSTATICS 37

which we can rewrite as:

H = (B 4 M) = 4c

J (2.86)

The field H is sometimes called the magnetic field strength in contrast with the fieldB which is then called the magnetic induction (or flux density). In most circumstances(e.g. non-magnetic materials) the average magnetization in the sample is small and pro-portional to the field (B or H) and it is customary to write:

B = H+ 4 M = H (2.87)

(with 1). In ferromagnetic materials on the other hand, which can have a largemagnetic dipole even in abscence of a magnetic field, the relation between the fields H

and

B is non-linear:

B =F(

H). At large fields the magnetization saturates

M

Mmax(all the microscopic dipoles are aligned with the field). This is how a strong magnet is

usually made: by aligning its intrinsic microscopic dipoles in a strong external field andthen turning offthe external field and relying on the internal dipole field in the materialto keep the dipoles aligned against the thermal agitation which will tend to disorientthem. Alternatively to demagnetize a given material one may heat it so as to disorganizeits internal dipoles and create an average internal field that is close to zero.


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38 ELECTROMAGNETISM

2.3 Electromagnetic induction

Untill now we have dealt with time independent distributions of charges and currents.When these distributions become implicitely dependent on time new phenomena areobserved that have many fundamental as well as practical consequences. We will seethat these new phenomena lead to a formulation of electromagnetism by Maxwell thatunified electricity, magnetism and optics and predicted the existence of radio-waves (orhertzian waves as they were called in the 19th century). The unification of so disparatephenomena (the first in physics) was an intellectual revolution that spurned the theoryof relativity, quantum mechanics and quantum electrodynamics, the unification of elec-tromagnetism and the weak and strong interaction (in the so-called standard model ofparticle physics) and is still the driving force behind the efforts to develop a theory ofeverything that would unify these forces with gravity.

2.3.1 Faradays Law

Electromagnetic induction was dicovered independently by Michael Faraday and JosefHenry in 1831. They observed that electric currents could be generated in two di fferentways: either by moving a conducting loop in an inhomogenous magnetic field or bymoving a magnet across a stationary conducting loop. The remarkable equivalence ofthese two phenomena was one of the driving forces that led Einstein to develop thetheory of special relativity.

Consider first the displacement at velocity v of a square conducting loop in a inho-mogenous magnetic field perpendicular to the loop, see Fig.2.17. The Lorentz force oncharges in the conducting wire is : F = (q/c)v B. Since the field B is inhomogenousthe force on the charges in the right leg of the loop will be different from the force onthe charges in the left leg of the loop. As a consequence there is a force di fference F:

F =q

cv (B(x + w/2) B(x w/2)) =q

c wvx

dB

dx (y) (2.88)which results in a movement of charges around the loop, i.e. a work:

qEem f = Fl = qc

wlvxdB

dx= q

c

dB

dt(2.89)

Where B =

BdS = wlB is the magnetic flux through the loop. Eem f = (1/c)dB/dtis known as the electromotive force. Notice that the current induced in the loop flowssuch as to generate a magnetic field that counters the change in the inducing magneticflux. In the example shown in Fig.2.17, if the field increases as the loop moves to theright the current induced in the loop flows clockwise which results in a magnetic fieldwhich opposes the increasing flux. If on the other hand the field decreases as the loopmoves to the right, the induced current runs counter-clockwise and generates a magnetic

field in the same direction as the decreasing flux. This is a general stability principle,sometimes called the Le Chatelier principle: a system responds to a perturbation in sucha way as to reduce its effect.

To demonstrate the usefullness of his discovery, Faraday invented a continuous cur-rent generator (at that time only batteries could supply a constant current). To that pur-pose he spun a disk of copper between the poles of a permanent magnet, see Fig.2.17.


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ELECTROMAGNETIC INDUCTION 39

F. 2.17. Schematics of Faradays first observation: by moving a conducting loopacross a magnetic field one can generate a current in the loop. A conducting disk(such as the one used by Faraday) rotated in a magnetic field also generates a cur-rent due to the differential of the Lorentz force on the charges in the spinning disk.Wires in contact with the disk at its periphery and its center close the current loop.

The field B on the charges in the disk is (to a first approximation) constant, but the ve-locity of the charges varies with their distance from the axis: v = r. Hence the Lorentzforce on a charge q at a distance r from the axis is in this case: F = qBr/c. Theelectromotive force (work per unit charge) due to this force is therefore:

Eem f =1q

Fdr =12cBr2 (2.90)

In the experiments just described the magnetic field is fixed and a current is inducedin a moving circuit by the action of the Lorentz force on free charges in the circuit.Faraday also noticed that an induced current would be generated in a fixed solenoid ifa magnet was moved across it or if the current in a nearby solenoid was varied. In thatcase there is no Lorentz force on the charges since the solenoid is not moving, however(as Einstein pointed out later) the force that is exerted on them is a result of the electricfield induced in their reference frame by the moving magnetic field. Faraday noticedthat just as in the previous experiments, the electromotive force was given by:

Eem f =

1

c

dB

dt

(2.91)

So it is the relative motion of magnet and conducting loop that matters for the generationof an induced current, not their absolute velocity. It is this remarquable observationof Faraday that spurred Einstein to develop the special theory of relativity. Indeed wehave already used that theory to demonstrate the equivalence of magnetic and electricfields when deriving the equation of magnetostatics from the electrostatics field of a


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40 ELECTROMAGNETISM

moving charged wire. In the context of Faradays law, this equivalence is clearer if we

rewrite Eq.2.91 in a differential form. On the one hand the electromotive force drivingthe current in a closed loop, i.e. the work per unit charge, is:

Eem f =

E dl =

E dS (2.92)

On the other hand: B =

B dS . Thus Eq.2.91 can be written as a differential equationrelating time varying electric and magnetic fields:

E = 1c

B

t(2.93)

As an example consider an observer moving with the loop shown in Fig.2.17. For thisobserver, while the loop is at rest the magnetic field varies with time: B = B(x + vt)z.While for him the charges in the wire are at rest and there is no Lorentz force due to the

magnetic field, there is however an electric field, Ey satisfying Eq.2.93:

xEy = 1c

tB = vc

xB (2.94)

Namely Ey = (v/c)B (notice that for a moving charged wire we derived a similarrelation: B = (v/c)E). For this observer the differential in electric field on the right andleft legs of the loop will result in an electromotive force:

Eem f = (Ey(x + w/2) Ey(x w/2))l = wlxEy = 1c

tB (2.95)

which is the same result as the one reached by an observer at rest (for which the mag-netic field is constant and inhomogenous and the loop is moving).

As we shall now show Faradays law have had an enormous impact on our daily life,

which is fitting since when Faraday was asked shortly after the formulation of his law:What is the use of it?, he answered: what is the use of a new-born baby?.

2.3.2 ExamplesApplication of Faradays Law: the inductor.

Consider a solenoid made of N loops (whose length l a, where a is the radius)and carrying a time varying current I. According to Amperes law (see above) the mag-netic field inside the solenoid is: Bin = 4nI/c (n = N/l), while according to FaradaysLaw the variation in magnetic field induces an electromotive force (i.e. potential dif-ference) in one loop: V1 = Eem f = (1/c)dB/dt. The potential across the N loops ofthe solenoid is thus: VN = NV1. Therefore a varying current will result in an inducedvoltage at the solenoid ends given by:

VN = Nc

dB

dt= NS

c

dBin

dt= S

l

4N2

c2dI

dt= L dI

dt(2.96)

L = (4N2/c2)(S /l) is known as the inductance of the solenoid. Its units in CGS aresec2/cm. In the MKS system inductance is measured in Henry (1Hy is the inductance


41/88


that results in a voltage of 1V for a change of current of 1A/s). Just like a capacitor

stores electric energy, an inductor stores magnetic energy. Consider the energy stored ina solenoid carrying a current If (generating a field Bf):

Wm =

Vd Q = L

If0

Id I = LI2

f

2= L

lcBf

4N

2

= S lB2

f

8(2.97)

Since S l is the volume of the solenoid the energy stored per unit volume is B2/8. Aswe shall see this is a general result for the energy density of magnetic fields. Notice thesimilarity with the energy density of electric fields, Eq.2.29.

Application of Faradays Law: the electrical motor and alternative current(AC) generator.We have seen that Faraday invented a DC current generator by spinning a conduct-

ing disk between the poles of a permanent magnet. It is interesting that the first alterna-tive current generator and motor was invented by an Hungarian physicist, Anyos Jedlik,four years before Faraday could explain their working principle! In the first dynamo hegenerated a current by spinning a conducting loop between the poles of a permanentmagnet, see Fig.2.18(a). As the magnetic flux through the loop varies periodically, theinduced electro-motive force, Eq.2.91 also varies cyclically giving rise to an alterna-tive current. This application is of immense relevance today as all the current fed intoour homes is still generated in essentially the same way! Jedlik also invented the firstDC motor, see Fig.2.18(b), by using a current in a fixed loop (the stator) to generate amagnetic field, a second current in a rotating loop (the rotor) that experiences a torque

due to the magnetic field of the stator and a commutator to switch the current in therotating loop so that the torque always points in the same direction turning the rotorcontinuously.

Application of Faradays Law: the electric transformer.As mentioned above when Faraday moved a solenoid carrying a DC current in-

side a fixed one, he observed in the later the generation of a measurable current, seeFig.2.19(a). This observation is the core principle of todays electric transformers. Theseare built by wounding two solenoids around a common core of iron (which is used tocanalize the magnetic field and limit losses), see Fig.2.19(b). An AC (primary) currentflowing in the primary solenoid (with Np windings) generates an electromotive force

(primary voltage) across it: Vp = Npd/dt while across the secondary solenoid (withNs windings) the induced electromotive force (secondary voltage) is: Vs = Nsd/dt =(Ns/Np)Vp (the ratio of secondary to primary voltage is equal to the ratio of secondaryto primary windings). If the losses are minimal (in practice on the order of 2%) thepower input to the transformer is equal to the power output, i.e. VpIp = VsIs and thusIp/Is = Ns/Np. The losses themselves are mostly due to Faradays law: the changing


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42 ELECTROMAGNETISM

F. 2.18. a) Schematics of an AC-generator: a loop rotates between the poles of apermanent magnet, the alternating current generated in the loop is picked up bybrushes in contact with conducting slip rings. (b) Schematics of a DC motor: acurrent is flowing in a loop set in a constant magnetic field. The direction of thecurrent flow is alternated such as to ensure that the torque on the loop is alwayspointing along the same direction. (c) the first DC motor build by Jedlik (and stillworking!): an external loop carrying a constant current generates the magnetic field.A freely rotating loop carrying a constant current is spun by the magnetic torque anda commutator ensures that the torque is always directed along the same direction.

flux through conducting surfaces creates induction currents that heat the surfaces be-cause of their intrinsic electric resistivity (i.e. Ohms law). To reduce these dissipativecurrents the surface perpendicular to the magnetic field is sliced by stacking layers of

thin steel laminations, so that the changing flux (and thus the induction current) througheach layer is minimal.

Electrical circuits.

With batteries, generators and transformers as sources of electric energy, capacitors


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F. 2.19. a) The original drawings of Faradays inductor: by displacing a solenoid A

carrying a constant current (supplied by a battery) inside a fixed solenoid B, heobserved a current flowing in solenoid B (measured by the galvanometer G). (b)The modern inductor principle: two solenoids are wound around an iron core (tocanalize the magnetic field). The AC voltage across one loop is related to the ACvoltage across the other as the ratio of the number of loops in the two solenoids.

and inductors to store it, resistors to dissipate it, diodes to rectify it and amplifiers (vac-uum tubes or transistors) to amplify the signals, all of todays electrical circuits can beunderstood! This is of course well beyond the scope of that course, but a few examplescan be worked out to give a glimpse of the field of electrical engineering.

Consider first the RC-circuit shown in Fig.2.20(a). When it is connected to the bat-tery the voltage across the resistance VR = IR and across the capacitor VC sum up toequal the battery voltage V0. Since I = dQ/dt = CdVC/dt one obtains:

VR + VC = RCdVC

dt+ VC = V0

Which solution is: VC = V0(1 exp(t/RC)). The capacitor is charged within a typicaltime = RC.

If the input voltage is oscillating V0(t) = Vosceit then using Fourier transforms(see Appendix), one obtains:

(iRC+ 1)VC() = VoscDefining the cutofffrequency c = 1/RCyields

|VC()| =Vosc

(/c)2 + 1

At low frequencies ( < c): VC() = Vosc whereas at high frequencies VC() =Vosc/

2. When the output of that circuit is the voltage across the capacitance Vc, itfunctions as a low-pass frequency filter: it damps signals whose frequency is above cand let signals pass undisturbed if their frequency is below c.


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44 ELECTROMAGNETISM

Next consider the RLC circuit shown in Fig.2.20(b). When it is connected at t = 0

to a constant current source I0, the current is split between the capacitor branch IC =CdVC/dt and the inductor-resistor branch with amplitude IL = I0 IC. The equality ofthe voltage across both branches: VC = VR + VL = ILR + LdIL/dt yields:

LCd2VC

dt2+RC

dVC

dt+ VC = RI0

Introducing the frequency 0 = 1/

LC and damping = R/2L, one can write thesolution of the previous equation as:

VC = RI0(1 et cos t) + Bet sin t

with 2 = 20 2 and B = I0(1 RC)/ chosen such that at t = 0: VC = 0 and IL = 0(the voltage on a capacitor and the current in an inductor cannot change abruptly).If the current I0 is oscillating (as it is in a radio receiver, see below): I0(t) = Ireceit,then using Fourier transforms (see Appendix), one obtains:

Vc =RIrec

1 LC2 + iRC =RIrecQ

Q(1 (/0)2) + 2i/0 A()RIrec

Where we define the quality factor Q = 0/and the transfer function A() = |A|eiawith:

|A()|2 = Q2

Q2(1 (/0)2)2 + 4(/0)2 (2.98)

tan a =2Q/0

1 (/0)2(2.99)

At the resonance of the circuit : = 0, the amplitude of the transfer functionis maximal |A| = Q/2 (its phase a = /2). If is slightly away from resonance,the transfer function decreases rapidly |A()|2 reaching half its maximal value when(/01) = /0 = 1/Q. A circuit with high Q has low damping, i.e. low dissipation.Such a circuit responds maximally to frequencies that are tuned close to its resonancefrequency 0. In a radio receiver, given the frequency of the emitting station, thereceiving RLC circuit is tuned to it by changing the value of the capacitor C.


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MAXWELLS EQUATIONS 45

F. 2.20. (a) Schematics of a RC circuit charged by a battery. (b) A RLC circuit.

2.4 Maxwells equations

By the mid-nineteenth century, much was known about electric and magnetic fields.One knew of C

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