Appendix 1 Electric and magnetic quantities

Quantity Symbol Units Dimensions

Electric current I A A Electric charge q C AT Electric dipole moment p Cm ALT Electric field E Vm-I A-IML r3 Electric potential c/J V A-I M L2r3 Capacitance C F A 2 M-I L-2 T" Electrostatic energy U J ML2r2 Polarisation P Cm-2 AL-2T Electric susceptibility Xe none Dielectric constant

(rela tive permittivity) €r none Electric displacement 0 Cm-2 AL-2T Electric charge density p Cm-3 AL-3T Surface charge density a Cm-2 AL-2T Electric current density i Am-2 AL-2

Electrical conductivity a Sm-I A-2M""1 L-3T3

Electrical resistance R n A-2ML2T3

Electromotive force & V A-I M L2r3 Magnetic field B T A-I Mr2 Magnetic dipole moment m Am2 AL2 Magnetic flux <I> Wb A-1ML2r 2

Inductance, mutual M H A-2MLr2

Inductance, self L H A-2ML r Magnetisation M Am-I AL-I

Magnetising field H Am-1 ACI Magnetic 'susceptibility Xm none Relative permeability ~r none Surface current density i Am-1 AL-I

Magnetostatic energy U J ML2r2 Larmor frequency wL

S-1 r l

Magnetomotive force IF A A Magnetic reluctance fJl AWb-1 A2M""1 L-2T2

11 0 Appendix 1

Equation

S.I. unit [4.2]

[2.26] [2.4]

[2.12] [2.19] [2.36] ~p/vol P/€oE

(1 + Xe) EoE + P

q/vol q/area

q/area/s i/E [ 4.7] [ 5.1] [4.8]

[ 4.12] [4.28] [5.11] [ 5.14] ~m/vol

(B/~o) -M M/H

(1 + Xm) [6.3]

[6.15 ] [6.20] [6.29] [6.29]

Appendix 2 Physical constants

Constant Symbol

Electric constant Eo = I /(Po c2)

Magnetic constant Po Speed of light c Electronic charge e Rest mass of electron me Rest mass of proton mp Planck constant h

11 = h/27T Boltzmann constant k Avogadro number NA Gravitational constant G

Bohr radius 47TEO Ii 2

ao= --2-me eh

Bohr magnet on PB=-2me

Electron volt eV Molar volume at S.T.P. Vm Acceleration due to gravity g

Value

8.85 X 1O-12 Fm-1 47T X 10-7 H m- I

3.00 X 108 ms-1 1.60 X 10-19 C 9.11 X 10-31 kg 1.67 X 10-27 kg 6.63 X 10-34 J s 1.05 X 10-34 J s 1.38 X 10-23 J K- I

6.02 X 1023 mol-1 6.67 X 10-11 N m2kg-2

5.29X 10-l1 m

9.27 X 10-24 J II

1.60 X 10-19 J 2.24 X 10-2 m3 mor l

9.81 m s-2

Appendix 2 111

Appendix 3 Vector operators

Scalar and vector fields may be operated on by the differential operators grad, diy, curl and 'il2• Expressions for the results of these operators in three coordinate systems are given below, with reference to the original equation where it is in the text.

Cartesian coordinates (x, y, z)

grad n = 'iln = an i + an j + an k ax ay az

div F = V. F = aFx + ~ + aFz ax ay az

curlF=VXF= j k

a a a ax ay az

Fx Fy Fz

a2 a2 a2 div(grad) = 'il 2 = - + - +-ax2 ay2 az2

Spherical polar coordinates (r, 8, t/!) see Fig. 2.5

an A 1 an A 1 an A

gradn = -r +--8 +-. - --1/1 ar r a8 rsm8 at/!

[2.16]

[3.10]

[3.11]

[3.12]

[3.17]

. _ 1.! 2 _1_ ~. _1_ aF", dlV F - 2 a (r F,.) + . 8 a8 (sm 8Fe) + . 8 a.I, r r rsm rsm ."

112 Appendix 3

curl F --­- r2 sin 0 r r8 rsin 01/J

a a a ar ao a1/J

Cylindrical polar coordinates (r, 0, z) see Fig. 3.2

an. an l\ an. grad n = ar r + rao t1 + az Z

div F = Ii. (rF) + aFe + aF; r ar Y rao az 1

curl F = -r

r r8 z a a a ar ao az Fy rFe Fz

\]2 = li. Ir i.) + ~ ..£:. + ~ [3.18] r ar ~ ar r2 ao2 az 2

Appendix 3 113

Appendix 4 Exercises

(L) indicates University of London question.

Chapter 2

1 Compare the magnitudes of the electrostatic and gravitational forces between two alphas particles (He2+).

2 A charge of 111 pC is uniformly distributed throughout the volume of an isolated sphere of diameter 40 cm. Calculate the electric field at the following distances from the centre of the sphere: (a) zero; (b) 1 Ocm; (c) 20 cm; (d) 50 cm. (L)

3 How much work is done on an electron when it is moved from (3, 2, -1) to (2, 1, -4) in an electric field given by E = (3 i -4j + 2k) V m-1? (Distance in metres) (L)

4 A charge of 31lC is uniformly distributed along a thin rod of ~ength 40cm. Find the electric field at a point 20cm from the rod on its perpendicular bisector.

5 Show from first principles that the dimensions of capacitance are A2 M-1 L-2 T4 on the M.K.S.A. (S.I.) system and find the dimensions of the electric constant, €o-

6 An electron moves from rest through a displacement (I 0-3 i + 10-3 j + 10-3 k) metres within an electric field (105 i + 105 j + 105 k) V m -1. Find: (a) the kinetic energy gained by the electron; (b) its final velocity.

7 Prove that the capacitance of an isolated sphere of radius a is 47T€oa.

8 Eight identical spherical drops of mercury are each charged to

114 Appendix 4

10 V above earth (ground) potential and then allowed to coalesce into a single spherical drop. What is the potential of the large drop? How has the electrostatic energy of the system changed?

(L) 9 A spherical electrode is required to carry a charge of 33 nCo

Estimate its minimum radius if the breakdown field strength of the surrounding air is 3 X 106 Vm-1•

10 Two isolated metal spheres of radii 40 and 90 mm are charged to 0.9 and 2.0 kV respectively. They are then connected by a fine wire. Explain what happens to: (a) the electric charges; (b) the electric potentials; (c) the electrical energy stored in the system. Hence find (d) the stored energy lost as a result of the connection.

11 A 12 pF parallel-plate air capacitor is charged by connecting it to a 100 V battery. How much work must be done to double the separation of the plates of the capacitor: (a) with the battery connected; (b) with it disconnected and the capacitor fully charged? Explain why the answers differ.

12 A large parallel plate capacitor is completely filled by a 2 mm thick slab of a dielectric (fr = 6) and a 1 mm thick slab of another dielectric (fr = 2). The plates are connected to a 1 kV battery, the plate next to the thick layer being positive and the other earthed (grounded). Calculate: (a) the surface charge density on the plates; and (b) the potential at the dielectric interface. (L)

13 A long coaxial cable consists of an inner wire radius a inside a metal tube of inside radius b, the space in between being com­pletely filled with a dielectric of relative permittivity f r . Show that its capacitance is (21Tfrfo)/(lnb/a) F m-1. (Hint: Use Gauss's law for the electric displacement flux.)

14 A parallel plate capacitor consists of rectangular plates of length a and width b spaced d apart connected to a battery of voltage V. A rectangular slab of dielectric of relative permittivity fr that would just fIll the space between the capacitor plates is inserted part way between them, so that it resembles a partly opened matchbox. Show that the force pulling it into the plates is (fr - 1)(fobV2)/(2d) and explain why it is independent of a.

Appendix 4 115

Chapter 3

1 An electric dipole of moment p = qa is placed with its centre at the origin and its axis along 8 = 0, where (r, 8, "') are spherical polar coordinates. Show that the potentiall/J(r, 8, "'), for r »a, is independent of '" and given by (p.r)/(41T€oyl). Hence show that the electric field at a point (r, 8) has amplitude

-p- (l + 3 cos2 8)t 41T€or 3

(L)

2 Using the expressions in exercise 1, plot the equipotentials and field lines for a dipole and compare them with those for a point charge.

3 A pair of electric dipoles are placed end-on in a straight line so that their negative charges coincide and they form a linear quadrupole of length 2a with charges +q at each end and -2q in the centre. Given that the electric field on the axis of a dipole at a distance r »a is (2qa)/(41T€or 3), find the electric field on the axis of the quadrupole at a distance r(r »a) from its centre.

4 Show that a dipole of moment p placed in a uniform electric field E acquires a potential energy -p. E and that a couple p X E acts to align it with the field.

S A conducting sphere of radius R is placed in a uniform electric field Eo. Show that the potential at a point outside the sphere and at (r, 8) from its centre is

and hence find the components Er and Eo of the electric field at that point.

6 An isotropic dielectric sphere of radius R and dielectric constant €I' is placed in a medium of dielectric constant €2 containing a uniform electric field Eo. Show that the potentials at points (r, 8) from the centre of the sphere are

(a) -3€2Eorcos8/(€1 + 2€2)

inside the sphere and

116 Appendix 4

outside it. Hence draw the lines of electric displacement 0 inside and outside the sphere when El < E2 and El > E2. (Hint: Assume that the potentials are of the same form as that for the conducting sphere and apply the boundary conditions to obtain the coef­ficients in each case.)

7 A long cylindrical conductor of radius R is earthed and placed in a uniform electric field Eg with its axis normal to Eo. Show that the potential at a point (r, e, z) is

rp = Eor cos e {I - (a2)j(r~}

(Hint: Show that this solution satisfies Laplace's equation and the boundary conditions.)

8 Show that a positive charge placed at (4, 2) between conducting planes y = 0 and x = y produces the same electric field between the planes as a system of four positive and four negative charges and find the positions of these charges. Will this method work for any angle between the planes?

9 Use the method of images to show that an uncharged, insulated conducting sphere of radius R is attracted to a positive charge ql at a distance r from its centre by the force:

q12 [R R ]

41TEo r3 - r {r - (R2)j(r)} 2

(Hint: Find the image charge for an earthed (grounded) sphere first.)

10 A dipole of moment p is placed at a distance r from the centre of an earthed conducting sphere of radius R < r. The axis of the dipole is in the direction from the centre of the sphere to the centre of the dipole. Prove that the image system for the dipole is a point charge pRjr2 plus a dipole of strength pR3jr3 and find their positions.

Chapter 4

1 The current density in a long conductor of circular cross-section and radius R varies with radius as i = ior2. Calculate the current flowing in the conductor.

2 An earthing (grounding) plate is formed from a hemispherical

Appendix 4 117

spinning of copper sheet placed in the earth with its rim at the earth's surface. Calculate its earthing resistance, given that its diameter is 0.5 m and the conductivity of the earth is 0.01 Sm-I .

3 Calculate the magnitude and direction of the magnetic field at the centre of a short solenoid of radius 3 cm and length 8 cm carrying a current of lOA and having 8 turns per cm. (L)

4 Show that the magnetic field at the ends and on the axis of a long solenoid is half that at its centre.

S A Helmholtz pair consists of two identical circular coils of radius a carrying the same current I in the same direction mounted coaxially an optimum distance b apart to give the maximum volume of uniform magnetic field between them. Show that this is achieved when b = a and find the uniform field. (Hint: If the axis of the coils is the x·axis, show that aB/ax and a2 B/ax2 are zero for B on the axis.)

6 An electron is moving at 2 X 106 m S-I round a circular orbit of radius 5 X 10-11 m. What is the magnetic field at the centre of the orbit?

7 Calculate the force per unit length on each of a pair of parallel wires 5 cm apart carrying the same current of 500 A. What happens when one of the currents is reversed? (L)

8 Calculate the magnetic field at the centre of a square wire loop of sides 10 cm in length carrying a current of 10 A. (Hint: Con· sider each side separately.)

9 A steady current I flows in one direction in the solid inner con· ductor, radius a, of a coaxial cable and in the opposite direction in the outer conductor, of inner radius b and outer radius c. Find the magnetic field at a distance r from the axis in each of the following regions: (a) r ~a; (b) a ~ r ~ b; (c) b ~ r ~ c; (d) r ~c.

10 Find the magnetic dipole moment of an electron moving in an orbit with a Bohr radius ao.

Chapter 5

1 A current loop of area S in the xy plane is placed in a uniform magnetic field Bz = Bosin wt. Find the e.m.f. induced in the coil: (a) when it is fixed; (b) when it rotates at an angular fre­quency w.

118 Appendix 4

2 An aircraft with a wingspan of 50 m flies horizontally at 300m S-1 through a vertical magnetic field of 20J,LT. What is the potential difference between the wing tips?

3 Estimate the kinetic energy in MeV of the electrons in a betatron when they are travelling along an orbit of radius 0.25 m in a magnetic field of 0.5 T. (Hint: High-speed electrons travel at nearly the speed of light.)

4 A thin metal disc of 5 cm radius is mounted on an axle of 1 cm radius and rotated at 2,400 revolutions per minute in a uniform magnetic field of 0.4 T. Brushes, connected in parallel, make contact with many points on the rim of the disc, and another set of brushes make similar contacts with the axle. The resistance measured between the sets of brushes due to the disc is lOOJ,Lil. Calculate the magnitude and direction of the current flowing in the disc. (L)

S A solenoid of 1000 turns is 2 cm in diameter and 10 cm long. Estimate the energy stored in it when it is carrying 100A. (L)

6 Calculate the mutual inductance between a very long solenoid with 200 turns per metre and a circular coil consisting of 50 turns of mean area 20 cm2 placed coaxially inside the solenoid. (L)

7 A large solenoid of radius 5 cm is wound uniformly with 1000 turns per metre and has a secondary winding of 500 turns closely wound over its centre portion. If a sinusoidal current of 2 A r .m.s. and frequency 50 Hz is passed through the solenoid, what is the e.m.f. induced in the secondary?

8 An ignition coil consisting of 16 000 turns wound closely on a solenoid of 400 turns and length lOcm has a radius of 3 cm. If the primary current of 3 A is broken 10000 times each second by the car's motion, what is the voltage produced across the spark plugs?

9 A long transmission line consists of two thin, parallel metal strips each 1 cm wide facing each other across a 5 cm gap. What is the inductance per metre of the line? (Hint: Consider the strips to be segments of an infinite circular, coaxial line.)

Appendix 4 119

Chapter 6

1 A long paramagnetic circular rod of diameter 6 mm is suspended vertically from a balance so that one· end is between the poles of an electromagnet providing a horizontal magnetic field of 1.5 T and the other end is in a negligible field. If the apparent increase in mass of the rod is 5 g, what is the magnetic susceptibility of the material? (Hint: Use the principle of virtual work to relate force to energy .)

2 Calculate the total magnetic energy in the earth's external field by assuming it is due to a dipole at the centre of the earth pro­ducing a magnetic field of 1 gauss at the equator. (Assume the radius of the earth is 6400 km.)

3 State whether the following statements are true or false in the MKSA system. (a) Band H are the same physical quantity but are measured in different units. (b) In free space B = J.Lo H means that B measures the response of a vacuum to the applied field H. (c) In matter H is the volume average of the microscopic value of B, multiplied by a factor independent of the material. (d) In matter if H, Band M are smoothed to remove irregularities on an atomic scale, then B is the sum of Hand M, multiplied by a factor independent of the material. (e) The sources of H are magnetic poles and of B are electric currents.

4 An intense magnetic field pulse is generated by discharging rapidly a bank of 1000 5 J.LF capacitors, wired in parallel, through a short, strong coil of inductance lOmH and volume 10cm3 .

If the capacitors are charged to 1.5 kV before discharge, calculate: (a) the maximum magnetic field produced; (b) the maximum current; (c) the time to reach the maximum field. (Hint: Assume a sinusoidal discharge.)

S An electromagnet consists of a soft iron ring of mean radius 8 cm with an air gap of 3 mm. It is wound with 100 turns carrying a current of 3 A. Given that the B-H curve for soft iron has the following characteristics, determine the magnetic field in the gap:

120 Appendix 4

0.2 0.4 0.6 0.8 1.0 100 170 220 310 500

Appendix 5 Answers to exercises

Chapter 2

1 3.10 X 1035 •

2 (a) 0 (b) 12.5Vm-1 (c)25Vrn-1 (d)4Vm-1 .

3 8 X 10-19 1. 4 4.77 X 105 Vm-l .

5 A2 M-1 L-3 T 4 .

6 (a)4.80X 10-171 (b) 1.03 X 107 ms-l .

8 40 V; increased fourfold. 9 lOmm.

1 0 (a) Total charge of 24 nC is conserved. (b) Potentials equalised to 1.66kV by charge flow. (c) Some energy lost to 10ulean heat in wire and flash of light (spark). (d) 1.9 J.L1.

11 (a)3 X 10-8 1 (b)6 X 10-8 1. In (a) 6 X 10-8 1 work is done charging the battery, but work done on capacitor is -3 X 10-8 1.

12 (a) 10.8J.LC m-2 (b) 600V. 14 Force acts only where the electric field is non-uniform, at the

edge of the plates over the dielectric.

Chapter 3

1 ¢ = (p coSe)j(47T€or~. 3 (3qa2)j(27T€or4). 5 Ey =Eocose {l + 2(R3jr~};Ee = -Eosine {I -(R3jr3)}. 6 Lines of 0 are continuous and uniform within the sphere, but

forced out when €I < €2 and forced in when €I > €2'

Appendix 5 121

8 Positive at (-2, 4), (-4, -2), (2, -4), (4, 2); negative at (2,4), (-4,2), (-2, 4), (4, -2). No, angle must be submultiple of 21T.

10 Both at a point inverse to the centre of the dipole in the sphere.

Chapter 4

1 1TioR4/2. 2 63.7 st. 3 8.04 X 10-3 T. S 8J.loI/(5 vis a). 6 12.8 T. 7 1.00 Nm-1 . Same direction, attractive; opposite direction,

repulsive. 8 1.13 X 10-4 T.

9 (a)J.lolr/(21Ta2) (b)J.loI/(21Tr) (C)i;~(;:=~:) (d)O.

10 1 Bohr magneton = 9.27 X 10-24 J r 1 .

Chapter 5

1 (a)BoSwcoswt (b)BoSwcos2wt. 2 0.3 V. 3 37.5 MeV. 4 1.21 kA. For motion clockwise about B, current flows from

centre to rim. S 1.97 kJ. 6 25.1J.lH. 7 3.14 V r.m.s. 8 6.82kV. 9 6.28J.lH.

122 Appendix 5

Chapter 6 1 1.96 X lO-3. 2 2.83 X lO19 J. 3 (a) False, since they have different dimensions. (b) False; this just

defines the units of H. (c) False; H = (Ea lllo) - Nfii always has finite fii, although it can be very small. (d) True; B = llo(H + M). (e) False; sources of H are conduction currents and of B are con­duction and magnetisation currents.

4 (a) 37.6T (b) 1.12kA (c) 1l.l ms. S 0.71 T.

Appendix 5 123

Index

Acceleration due to gravity g, III

Ampere, 52, 59 ampere, unit of current, 49, 62 Ampere's law, 63, 80,104; ap-

plications, 64 ff angle, solid dn, 10 anisotropic media, 108 antiferromagnetics, 96 Avogadro number, 49, III

betatron, 73 Biot-Savart law, 58; applications,

59 ff Bohr magneton, 91, III Bohr radius, 81, Ill, 118 Boltzmann constant, 91, 111 boundary conditions: for electric

field, 28; for magnetic field, 85

capacitance, C, 20, 107, 110; of coaxial cable, 115; of isolated sphere, 114

capacitor, parallel-plate, 20, 24, 105,115

Cartesian coordinates, 7, 8, 112 Chadwick,2 charge density p, 7, 110; of free

charges Pt, 26; of polarisation charges Pp, 26

charges, electric: conservation, 8,

124 Index

108; invariance, 54; motion in a magnetic field, 55 ff

chemical energy, 22 circulation, of a vector field, 32;

of B, 64; of E, 16,33; of H, 85,97

circulation law, 16,33 coaxial cable: capacitance, 115;

electric field, 37; electrical potential, 37; magnetic field, 118

coercivity, 95 conductivity a, 51,110; of

materials, table, 51 conductor, 18, 107; sphere in an

electric field, 116 ; cylinder in an electric field, 11 7

conservation principles, 108 conservative fields, 16,63 continuity, equation of, 108 coordina tes: Cartesian, 7, 8,

112; cylindrical polar, 35, 113; spherical polar, 9, 34, 112

Coulomb,1 coulomb, unit of electric charge,S Coulomb's law, 1,3,4,106; for

dielectrics, 30 coupled circuits, 78 coupling, coefficient of, 79 Curie point Te , 96 Curie's law, 92

Curie-Weiss law, 96 curl, of a vector, 32, 33, 112 ff;

of B, 106; of E, 33,72, 104, 107; of H, 107, 109

current, electric I, 49; element dl, 58; forces between two current elements, 61

current density j, 49, 110 current loop, 56; magnetic dipole

moment, 57; torque in magnetic field, 57

cylindrical coordinates, 35, 113

del operator \/,33 demagnetising field, 101 diamagnetic, 82, 89 dielectric constant En 23, 26, 28,

110 dielectrics, 23 ff dimensions: of electric quantities,

110, 114; of magnetic quan­tities, 110

dipole, electric: potential, 35,116; field, 116

dipole moment: electric, 25, 110, 116; magnetic, 57,110; of an orbital electron, 81, 118; in diamagnetic, 90

displacement current, 105 divergence, of a vector, 31, 33,

113 ff; of B, 66, 104, 107; ofD,32, 107, 109;ofE,32, 103; of j, 108; of jf, 109

divergence theorem, Gauss's, 32 domain, magnetic, 93 du Fay, 4

Einstein, vii electrical images, 38ff; dipole and

spherical conductor, 117; point charge and plane conductor, 38; point charge and plane dielectric, 40; point charge and spherical conductor, 117

electric constant Eo,S, III

electric current, 48 ff, 110 electric displacement D, 27, 110;

divergence, 32; refraction, 29; for a point charge and plane dielectric, 40; for a dielectric sphere in a dielectric medium, 116

electric energy density, see energy density, of electrostatic field

electric field E: for a conducting sphere in a uniform field, 116; curl, 33; definition, 6; dimensions, 110; for a dis­tribution of charge, 7, 58; divergence, 32; for a line charge 13; for a plane sheet of charge, 13; for a point charge and plane conductor, 38; refraction, 29; for a sphere of charge, 12

electric potentialt/>, 14 ff, 110; examples, 116, 117

electric stress, 22; in capacitors, 23

electric susceptibility Xe, 26, 110 electromagnet, 98 electromagnetic force law, 106 electromagnetism, vii, 67 ff electromotive force 8" 68, 110;

motional e.m.fs, 69ff; trans­former e.m.fs, 72 ff

electron: charge, 111; in copper, 49, 52; drift velocity, 50; relaxation time, 51; spin, 81, 88,91; in vacuum, 48

electron optics, 43ff electron volt, 111 electrostatic energy U, see energy,

electric electrostatic field E, viii; circu­

lation law, 16,72; curl, 33; gradient of potential, 18, 106; hollow conductor, 19; see also electric field E

electrostatic lenses, 45ff; thin lens, 45; quadrupole lens, 47

Index 125

energy U: of earth's magnetic field, 121; electric, 21, 29, 110; of electromagnetic field, 107; magnetic, 79, 87, 110; of magnetic dipole in field B, 92

energy density u: of electro­static field, 21,30; of hysteresis loop, 96; of magnetic field, 88

equipotential surface, 18, 22

farad, unit of capacitance, 20 Faraday, viii, 67 Faraday's disc, 74, 120 Faraday's law, 67ff, 80 Fermi temperature TF , 93 ferrimagnetics, 96 ferromagnetics, 93 Feynmann, R.P., ix, 3 fields, conservative, 14, 63 flux, 8; electric, see Gauss's law;

of electric displacement D, 28; leakage, 99; linkage, 66; of polarisation P, 27; magnetic <1>,65, 98ff, 110

flux-rule, 71 force: between a current and a

charge, 53; between currents, 53; inverse square law, 1; long­range, 2; between a permanent magnet and a current, 53; short-range, 2

Franklin, Benjamin, 4

gauss, unit of magnetic field, 54

Gauss's divergence theorem, 32 Gauss's Law, 8ff, 31, 103;

applications, 12; for electric displacement, 28; for magnetic flux,66,80,97,104

gradient: of a scalar, 33, 112ff; of potential(/>, 18

grains, 93

126 Index

gravitation constant, 1, III gravitational force, 1

Helmholtz pair of coils, 118 henry, unit of inductance, 77 homogeneous materials, 30, 84 homopolar generator, 74 hysteresis, 95

ignition coil, 119 images, electric, 38 ff inductance: mutual M, 76ff; self

L, 77,107,110,119 iron, 93ff, 100, 120 isotropic materials, 30, 84

joule, unit of energy, 110 Joule's law, 71

Laplace's equation, 34, 106; solutions 34

Laplacian operator \/2, 34; in Cartesian coordinates, 34, 112; in cylindrical polar coordinates, 35, 113; in spherical polar coordinates,34,113

Larmor frequency WL, 91, 110 Lenz's law, 71 linear materials, 30, 84 line integral, 14 lines: of B, 60, 64, 65, 99,118;

of D, 42, 117; of E, 9, 12, 39

Lorentz force, 52; applications, 55ff

magnet, permanent, 100 magnetic circuits, 99 magnetic constant Jio, 59, 77,

III magnetic dipole moment, 57,

81ff, 91, 11 0, 118 magnetic domains, 93 magnetic energy density, see

ene'rgy density, of magnetic field

magnetic field B, 53, 110; in a coaxial cable, 119; of a current element, 59; of a current loop, 60; of earth, 120; of electro­magnet, 98; of a Helmholtz pair, 119; of an infinite wire, 59; of iron, 95; and Lorentz force, 53; of an orbital electron, 119; of refraction, 87; of a solenoid, 65, 119; of a square loop, 119; strong, 98; of a torus, 64; transient, 98

magnetic flux <P, 65 magnetic implosions, 98 magnetic monopoles, 65 magnetic permeability /1, 84 magnetic stress, 102 magnetic susceptibility Xm, 84,

90ff, 110 magnetisation M, 81, 110; of

diamagnetic, 89; of ferro­magnetic, 93ff; of para­magnetic, 91; of permanent magnet, 101; spontaneous, 97

magnetising field H, 84, 110; circulation, 85; of electro­magnet, 100

magnetomotive force ~ , 99, 110 magetostatic field B, viii; see also

magnetic field, B mass, rest: of electron, 111; of

proton, 111 Maxwell, vii, viii Maxwell's equations, vii, 3, 103 ff,

107ff Maxwell's law, 106 mean free path: of electrons in

copper, 52; of molecules in air, 52

Millikan, 2 molar volume, 111 monopoles: electric, self-energy,

22; magnetic, 65 multipoles, electric, 35

Neel point, 97 Neumann's theorem, 77 Newton, vii, I non-homogeneous media, 108 non-linear materials, 28, 108 nuclear force, 2

ohm, unit of resistance, 51 Ohm's law, 51 order-disorder transition, 96

paramagnetic, 82, 91ff Pauli spin paramagnetism, 93 permanent magnet, 100; alloy,

95 permeability: differential, 97;

magnetic /1,84; relative /1r, 84,87,110 .

permittivity: of free space eo, see electric constant; relative en see dielectric constant

Planck constant h, fl, III Poisson's equation, 34, 106 polarisation: P, 24ff, 110; of a

dielectric, 25; of an atom, 25 polycrystal, 93 pro ton sta bility, 8

quadrupole, electric, 35, 116

refraction: electric field, 29; magnetic field, 87

relaxation time, 51 reluctance, 99,110 remanence, 95 resistance, 51, 110 right-hand screw rule, 32, 60 Rutherford scattering, 6

siemens, unit of admittance, 110

SI units, viii, 110 skin effect, 78 solenoidal current density it,

83

Index 127

solenoids: Bitter, 98; magnetic field, 65, 118;superconducting, 98

speed of light c, 5, 111 spherical coordinates, 9, 34 Stokes's theorem, 33, 72 superconductors, 87, 98 superposition, principle of, 5 surface charge density a, 110;

free af' 24; of polarised di­electric ap , 24

surface current density: i, 110; of magnetised matter im , 81

susceptibility: electric Xe, 26, 110; magnetic Xm, 84, 90, 92, 110

tesla, unit of magnetic field, 54, 65

128 Index

test charge, 15 Thomson, 1.1., vii torque: on electric dipole, 57,

116; on magnetic dipole, 57, 91

transmission line, 119

uniqueness theorem, 37 units, viii, 110

vector operators, 112ff velocity of light c, 5, II r volt, unit of electric potential, 17

watt, unit of electric power, 71 weber, unit of magnetic flux, 65 Weiss electromagnet, 100 work: electrical, 14,21,79;

virtual, 30, 102

STUDENT PHYSICS SERIES

Series Editor: Professor R. 1. Blin-Stoyle, FRS Professor of Theoretical Physics, University of Sussex

The aim of the Student Physics Series is to cover the material required for a first degree course in physics in a series of concise, clear and readable texts. Each volume will cover one of the usual sections of the physics degree course and will concentrate on covering the essential features of the subject. The texts will thus provide a core course in physics that all students should be expected to acquire, and to which more advanced work can be related according to ability. By concentrating on the essentials, the texts should also allow a valuable perspective and accessibility not normally attainable through the more usual textbooks.

STUDENT PHYSICS SERIES

QUANTUM MECHANICS

Quantum mechanics is the key to modern physics, yet it is notoriously hard to learn. This book is designed to overcome that obstacle. Clear and concise, it provides an easily readable introduction intended for science undergraduates with no previous knowledge of the quantum theory, and takes them up to final-year level.

The emphasis is on clarity, achieved by focusing on the essential features of the subject and excluding much of the technical padding which characterizes more developed discussions in longer books. The style is informal, and the author uses his wide experience as a writer and broadcaster on 'popular science' to explain the many difficult abstract points of the subject in easily comprehensible language and imagery.

P. C. W. Davies

Professor Paul Davies is Professor of Theoretical Physics at the University of Newcastle upon Tyne.

ISBN 0-7100-9962-2 160 pp., 198mm x 129mm, diagrams

STUDENT PHYSICS SERIES

RELA TIVITY PHYSICS

Relativity Physics covers all the material required for a first course in relativity. Beginning with an examination of the paradoxes that arose in applying the principle of relativity to the two great pillars of nineteenth-century physics-classical mechanics and electromagnetism - Dr Turner shows how Einstein resolved these problems in a spectacular and brilliantly intuitive way. The implications of Einstein's postulates are then discussed a:1.d the book concludes with a discussion of the charged particle in the electromagnetic field.

The text incorporates details of the most recent experiments and includes applications to high-energy physics, astronomy, and solid state physics. Exercises with answers are included for the student.

R. E. Turner

Dr Roy Turner is Reader in Theoretical Physics at the University of Sussex.

ISBN 0-7102-0001-3 160pp., 198mm x 129mm, diagrams

STUDENT PHYSICS SERIES

CLASSICAL MECHANICS

A course in classical mechanics is an essential requirement of any first degree course in physics. In this volume Dr Brian Cowan provides a clear, concise and self-contained introduction to the subject and covers all the material needed by a student taking such a course. The author treats the material from a modern viewpoint, culminating in a final chapter showing how the Lagrangian and Hamiltonian formulations lend themselves particularly well to the more 'modern' areas of physics such as quantum mechanics. Worked examples are included in the text and there are exercises, with answers, for the student.

B.P.Cowan

Dr Brian Cowan is in the Department of Physics, Bedford College, University of London

ISBN 0-7102-0280-6 128 pp., 198 mm x 129 mm, diagrams