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Appendix 1 Electric and magnetic

Quantity Symbol Units Dimensions Equation I A A S.I. unit q C AT [4.2] p Cm ALT [2.26] Electric E Vm-I A-IML r3 [2.4] c/J V A-I M L2r3 [2.12] C F A 2 M- I L-2 T" [2.19] Electrostatic U J ML2r2 [2.36] Polarisation P Cm-2 AL-2T ~p/vol Electric susceptibility Xe none P/€oE constant (rela tive ) €r none (1 + Xe) Electric displacement 0 Cm-2 AL-2T EoE + P Electric p Cm-3 AL-3T q/vol Surface charge density a Cm-2 AL-2T q/ Electric i Am-2 AL-2 q/area/s Electrical conductivity a Sm-I A-2M""1 L-3T3 i/E Electrical resistance R n A-2ML2T3 [ 4.7] & V A-I M L2r3 [ 5.1] B T A-I Mr2 [4.8] Magnetic dipole moment m Am2 AL2 [ 4.12] Magnetic Wb A-1ML2r 2 [4.28] , mutual M H A-2MLr2 [5.11] Inductance, self L H A-2ML r [ 5.14] Magnetisation M Am-I AL-I ~m/vol Magnetising field H Am-1 ACI (B/~o) -M Magnetic 'susceptibility Xm none M/H Relative permeability ~r none (1 + Xm) Surface current density i Am-1 AL-I [6.3] Magnetostatic energy U J ML2r2 [6.15 ] S-1 Larmor wL r l [6.20] Magnetomotive force IF A A [6.29] fJl AWb-1 A2M""1 L-2T2 [6.29]

11 0 Appendix 1 Appendix 2 Physical constants

Constant Symbol Value

Electric constant Eo = I /(Po c 2) 8.85 X 1O- 12 Fm-1 Magnetic constant Po 47T X 10-7 H m-I Speed of c 3.00 X 108 ms-1 Electronic charge e 1.60 X 10-19 C Rest of electron me 9.11 X 10-31 kg Rest mass of proton mp 1.67 X 10-27 kg Planck constant h 6.63 X 10-34 J s 11 = h/27T 1.05 X 10-34 J s Boltzmann constant k 1.38 X 10-23 J K-I Avogadro number NA 6.02 X 1023 mol-1 Gravitational constant G 6.67 X 10-11 N m2kg-2 47TEO Ii 2 Bohr radius ao= --2- 5.29X 10-l1 m me eh Bohr on PB=- 9.27 X 10-24 J II 2me Electron eV 1.60 X 10-19 J Molar at S.T.P. Vm 2.24 X 10-2 m3 morl Acceleration due to gravity g 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 'il 2• 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 [2.16] ax ay az

div F = V. F = aFx + ~ + aFz [3.10] ax ay az

curlF=VXF= j k [3.11] a a a ax ay az

Fx Fy Fz

a2 a 2 a 2 div(grad) = 'il 2 = - + - +- [3.12] 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 [3.17] ar r a8 rsm8 at/!

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

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 r8 r z a a a ar ao az

Fy rFe Fz

\]2 = li. Ir i.) + ~ ..£:. + ~ [3.18] r ar ~ ar r2 ao 2 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 (He 2 +). 2 A charge of 111 pC is uniformly distributed throughout the volume of an isolated sphere of diameter 40 cm. Calculate the at the following distances from the centre of the sphere: (a) zero; (b) 1Ocm; (c) 20 cm; (d) 50 cm. (L) 3 How much 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 ) (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 gained by the electron; (b) its final . 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 stored in the system. Hence find (d) the stored energy lost as a result of the connection. 11 A 12 pF parallel-plate air 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 com• pletely 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 a and width b spaced d apart connected to a battery of 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 (L) 41T€or 3 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 -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 coef• ficients 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 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:

2 q1 [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 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 fre• quency 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 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 each 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 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 pro• ducing 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 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 , 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 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: 0.2 0.4 0.6 0.8 1.0 100 170 220 310 500

120 Appendix 4 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 10-191. 4 4.77 X 105 Vm-l . 5 A 2 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. 10 (a) Total charge of 24 nC is conserved. (b) Potentials equalised to 1.66kV by charge flow. (c) Some energy lost to 10ulean 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 (3qa 2)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 con• duction 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, 108; invariance, 54; motion in III a magnetic field, 55 ff , 52, 59 , 22 ampere, unit of current, 49, 62 circulation, of a vector field, 32; Ampere's law, 63, 80,104; ap- of B, 64; of E, 16,33; of H, plications, 64 ff 85,97 angle, solid dn, 10 circulation law, 16,33 anisotropic media, 108 coaxial cable: capacitance, 115; antiferromagnetics, 96 electric field, 37; electrical Avogadro number, 49, III potential, 37; magnetic field, 118 betatron, 73 coercivity, 95 Biot-Savart law, 58; applications, conductivity a, 51,110; of 59 ff materials, table, 51 Bohr magneton, 91, III conductor, 18, 107; sphere in an Bohr radius, 81, Ill, 118 electric field, 116 ; cylinder in Boltzmann constant, 91, 111 an electric field, 11 7 boundary conditions: for electric conservation principles, 108 field, 28; for magnetic field, conservative fields, 16,63 85 continuity, equation of, 108 coordina tes: Cartesian, 7, 8, capacitance, C, 20, 107, 110; of 112; cylindrical polar, 35, coaxial cable, 115; of isolated 113; spherical polar, 9, 34, sphere, 114 112 capacitor, parallel-plate, 20, 24, Coulomb,1 105,115 coulomb, unit of electric charge,S Cartesian coordinates, 7, 8, 112 Coulomb's law, 1,3,4,106; for Chadwick,2 , 30 charge density p, 7, 110; of free coupled circuits, 78 charges Pt, 26; of polarisation coupling, coefficient of, 79 charges Pp, 26 Curie point Te , 96 charges, electric: conservation, 8, Curie's law, 92

124 Index Curie-Weiss law, 96 electric current, 48 ff, 110 curl, of a vector, 32, 33, 112 ff; electric displacement D, 27, 110; of B, 106; of E, 33,72, 104, divergence, 32; refraction, 29; 107; of H, 107, 109 for a point charge and plane current, electric I, 49; element dielectric, 40; for a dielectric dl, 58; forces between two sphere in a dielectric medium, current elements, 61 116 current density j, 49, 110 electric energy density, see energy current loop, 56; magnetic dipole density, of electrostatic field moment, 57; in magnetic electric field E: for a conducting field, 57 sphere in a uniform field, 116; cylindrical coordinates, 35, 113 curl, 33; definition, 6; dimensions, 110; for a dis• operator \/,33 tribution of charge, 7, 58; demagnetising field, 101 divergence, 32; for a line charge diamagnetic, 82, 89 13; for a plane sheet of charge, dielectric constant En 23, 26, 28, 13; for a point charge and plane 110 conductor, 38; refraction, 29; dielectrics, 23 ff for a sphere of charge, 12 dimensions: of electric quantities, electric potentialt/>, 14 ff, 110; 110, 114; of magnetic quan• examples, 116, 117 tities, 110 electric , 22; in capacitors, dipole, electric: potential, 35,116; 23 field, 116 electric susceptibility Xe, 26, 110 dipole moment: electric, 25, 110, electromagnet, 98 116; magnetic, 57,110; of an electromagnetic force law, 106 orbital electron, 81, 118; in , vii, 67 ff diamagnetic, 90 electromotive force 8" 68, 110; , 105 motional e.m.fs, 69ff; trans• divergence, of a vector, 31, 33, former e.m.fs, 72 ff 113 ff; of B, 66, 104, 107; electron: charge, 111; in copper, ofD,32, 107, 109;ofE,32, 49, 52; drift velocity, 50; 103; of j, 108; of jf, 109 relaxation time, 51; spin, 81, divergence theorem, Gauss's, 32 88,91; in vacuum, 48 domain, magnetic, 93 electron optics, 43ff du Fay, 4 electron volt, 111 electrostatic energy U, see energy, Einstein, vii electric electrical images, 38ff; dipole and electrostatic field E, viii; circu• spherical conductor, 117; point lation law, 16,72; curl, 33; charge and plane conductor, of potential, 18, 106; 38; point charge and plane hollow conductor, 19; see also dielectric, 40; point charge and electric field E spherical conductor, 117 electrostatic lenses, 45ff; thin electric constant Eo,S, III lens, 45; quadrupole lens, 47

Index 125 energy U: of earth's magnetic gravitation constant, 1, III field, 121; electric, 21, 29, gravitational force, 1 110; of , 107; magnetic, 79, 87, 110; Helmholtz pair of coils, 118 of magnetic dipole in field B, henry, unit of inductance, 77 92 homogeneous materials, 30, 84 energy density u: of electro• homopolar generator, 74 static field, 21,30; of hysteresis hysteresis, 95 loop, 96; of magnetic field, 88 ignition coil, 119 equipotential surface, 18, 22 images, electric, 38 ff inductance: mutual M, 76ff; self farad, unit of capacitance, 20 L, 77,107,110,119 Faraday, viii, 67 iron, 93ff, 100, 120 Faraday's disc, 74, 120 isotropic materials, 30, 84 Faraday's law, 67ff, 80 Fermi temperature TF , 93 , unit of energy, 110 ferrimagnetics, 96 Joule's law, 71 ferromagnetics, 93 Feynmann, R.P., ix, 3 Laplace's equation, 34, 106; fields, conservative, 14, 63 solutions 34 flux, 8; electric, see Gauss's law; Laplacian operator \/2, 34; in of electric displacement D, Cartesian coordinates, 34, 112; 28; leakage, 99; linkage, 66; in cylindrical polar coordinates, of polarisation P, 27; magnetic 35, 113; in spherical polar <1>,65, 98ff, 110 coordinates,34,113 flux-rule, 71 Larmor frequency WL, 91, 110 force: between a current and a Lenz's law, 71 charge, 53; between currents, linear materials, 30, 84 53; inverse square law, 1; long• line integral, 14 range, 2; between a permanent lines: of B, 60, 64, 65, 99,118; magnet and a current, 53; of D, 42, 117; of E, 9, 12, short-range, 2 39 Franklin, Benjamin, 4 , 52; applications, 55ff gauss, unit of magnetic field, 54 magnet, permanent, 100 Gauss's divergence theorem, 32 magnetic circuits, 99 Gauss's Law, 8ff, 31, 103; magnetic constant Jio, 59, 77, applications, 12; for electric III displacement, 28; for magnetic magnetic dipole moment, 57, flux,66,80,97,104 81ff, 91, 11 0, 118 gradient: of a , 33, 112ff; magnetic domains, 93 of potential(/>, 18 magnetic energy density, see grains, 93 ene'rgy density, of magnetic field

126 Index magnetic field B, 53, 110; in a Neel point, 97 coaxial cable, 119; of a current Neumann's theorem, 77 element, 59; of a current loop, Newton, vii, I 60; of earth, 120; of electro• non-homogeneous media, 108 magnet, 98; of a Helmholtz non-linear materials, 28, 108 pair, 119; of an infinite wire, nuclear force, 2 59; of iron, 95; and Lorentz force, 53; of an orbital electron, ohm, unit of resistance, 51 119; of refraction, 87; of a Ohm's law, 51 solenoid, 65, 119; of a square order-disorder transition, 96 loop, 119; strong, 98; of a torus, 64; transient, 98 paramagnetic, 82, 91ff

Index 127 : Bitter, 98; magnetic test charge, 15 field, 65, 118;superconducting, Thomson, 1.1., vii 98 torque: on electric dipole, 57, speed of light c, 5, 111 116; on magnetic dipole, 57, spherical coordinates, 9, 34 91 Stokes's theorem, 33, 72 transmission line, 119 superconductors, 87, 98 superposition, principle of, 5 uniqueness theorem, 37 surface charge density a, 110; units, viii, 110 free af' 24; of polarised di• electric ap , 24 vector operators, 112ff surface current density: i, 110; velocity of light c, 5, II r of magnetised matter im , 81 volt, unit of electric potential, 17 susceptibility: electric Xe, 26, 110; magnetic Xm, 84, 90, 92, watt, unit of electric , 71 110 weber, unit of magnetic flux, 65 Weiss electromagnet, 100 tesla, unit of magnetic field, 54, work: electrical, 14,21,79; 65 virtual, 30, 102

128 Index STUDENT PHYSICS SERIES

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