Microscopic constituents of matter All matter except for that created in big accelerators is made of electrons , protons & neutrons Of these 3 particles, two have charge, and all have “spin” (intrinsic magnetic moment that produce); moreover, as they move they generate an extra magnetic field: all matter is made of tiny charges and magnets In fact charge and spin are in some sense more fundamental than mass: mass is not quantized, varies depending on the reference frame (special relativity), and is the effect of an interaction with a field (Higgs particle); on the contrary, charge and spin are quantized and invariant. The spin is described classically as a magnetic dipole moment (a little magnet with a south pole and a north pole right next to one another, topic 3). This is the magnetic version of an electric dipole moment (topics 1&2), which is a configuration of opposite plus and minus charges right next to one another. This configuration of charges (electric dipole) also exist in nature, not in particles by themselves but in their agglomerates: the water molecule is an electric dipole. The microscopic constituents of matter give rise to E-fields and B-fields (and are subject to external E- and B- fields). The simplest description could be with the following equations: ' ' ' ' ' ' ' ' q rˆ ' ' 1 § 3p r ' p · ' ' + ' P0 v u rˆ ' P0 § 3mr ' m · E(r ) 2 E(r) ¨ r ¸ B(r) r 5 3 B(r) q 2 ¨ 5 3 ¸ 4SH 0 r 4SH 0 © r r ¹ 4S r 4S © r r ¹ E-field of (point)charge & electric dipole B-field of slowly moving charge & magnetic dipole ' ' ' ' ' ' ' * pu E plus the Lorentz force F q ( E v u B ) and the torques on the dipoles ' ' ' * mu B NOTE: Charges are depicted as points, dipoles (both electric and magnetic) are depicted as arrows. So in the first three topics it will be all about points, arrows, spheres, cylinders… - Why do we settle for “less”? ' ' ' ' E da Q H ' ' 1 ' r rc ³ inside 0 E(r) dW c U(rc) ' ' vs 4SH ³ | r r |3 wVol ' ' 0 ³ E d# 0 wArea ' ' ' ' ' ' B da 0 ' P0 ' r rc ³ B(r) dW c J (rc)u ' ' vs wVol 4S ³ | r rc |3 ' ' ³ B d# P0 I wArea 1) Because these expressions cannot be generalized to time-varying fields , while these can; In fact, the equations for the fluxes (Gauss law) are correct even for time-dependent fields 2) Because field equations (with fluxes and line integrals) have a profound meaning (see next slide) 3) They simplify the solution of certain problems: V En 3.1) high-symmetry charge distributions H 0 3.2) fields near interfaces (“boundary conditions”) Æ 4) Helmholtz theorem states that a field that only depends on r (and not time) is completely specified if both its curl and divergence are known . Hence we are not really settling for less! Physical meaning of Gauss’s and Ampère’s law for E & B (electro- & magneto-statics, física 2) ' ' Q ³ E da inside means that E-field lines start at positive wVol H 0 charges and end at negative charges ' ' Example: electric dipole Æ E U H 0 ' ' ³ B da 0 means that there exist no magnetic monopoles (no wVol ' ' positive and negative “charges” or poles exist that B 0 are source of B). B field lines never start nor end Example: bar magnet Æ ' ' ³ E d# 0 means that E-field lines never draw a closed contour , wArea ' ' they never make a full circle around any point u E 0 ' ' means that B-field lines draw closed contours ³ B d# P0 I wArea ' ' ' around currents , they “curl” around currents u B P J 0 Example: current flowing in a ring Æ Differential & integral laws & boundary conditions ' ' ' ' U dWU Qinside _Vol E ³ dW E ³ Vol Vol H 0 H 0 H 0 ' ' ' ' ' ' Q Gauss’s integral theorem ³ d W C ³ C d a , therefore ³ E da inside Vol wVol wVol H 0 If charge is present at the boundary between 2 media: nˆ1 ' ' ' Q VA )(E) E1 nˆ1 A E2 nˆ2 A H 0 H 0 ' ' V whence E n E n 1 ˆ1 2 ˆ2 nˆ H 0 2 Hence: Differential form Integral form Boundary form ' ' ' ' U Qinside ' ' V E ³ E da E1 nˆ1 E2 nˆ2 H wVol H 0 0 H 0 Differential & integral laws & boundary conditions ' ' ' ' ' ' ' ' u B P0 J ³ u B da P0 ³ J da P0 Ithrough _ Area Area Area ' ' ' ' ' ' ' B d# P I Stokes’ integral theorem ³ u C d a ³ C d # , therefore ³ 0 Area wArea wArea If a current is present at the boundary between 2 media: ' ' B d# B d# B d# (B B ) d# ³ t 2 2 t1 1 t 2 t1 P0 Ida P0 K d# ' ' ' whence B1 u nˆ1 B2 u nˆ2 P0 K Hence: Differential form Integral form Boundary form ' ' ' ' ' ' ' ' ³ B d# P I B u n B u n P K u B P0 J 0 1 ˆ1 2 ˆ2 0 wArea Gauss’s integral theorem Take a little volume of sides dx, dy, dz, centered at position r = (x, y, z) We want to calculate the flux of the vector field C = ( Cx , C y , Cz ) through the volume’s boundary Consider the flux through the two vertical facets (parallel to the xy plane). The flux through the right facet is C y(x, y + dy/2, z) dxdz; the flux through the left facet is –Cy(x, y – dy/2, z) dxdz . The minus sign comes from the fact that the flux of a vector is by definition the outward flux. By making a Taylor expansion to first order, we get that the total flux through both xz facets is: the total outward flux is then: looking at the figure to the right, we see that the outward flux from an inner elementary volume cancels out with the outward flux from its neighbors. Whence we get Gauss’s theorem: Theoretical tools The fundamental quantities in electromagnetism are fields (electric field, magnetic field, scalar potential field, etc.), that is, scalar or vector functions of the position r (and, when we consider time varying fields, also of time t). To describe these quantities, we will need few concepts and tools from classical field theory, which we will introduce and use during the course: - Rectangular (cartesian), cylindrical, and spherical spatial coordinates - Gradient, divergence, curl, time derivative - Dirac’s delta - Gauss’s and Stokes integral theorems - Boundary conditions - Derivative rules for products (scalar & vector) - integration by parts ¾ MACROSCOPIC DESCRIPTION Maxwell’s equations (average) (in terms of ɸr , ʅr , g, n) Connection with: circuit theory, electromagnetic waves, and boundary conditions geometric and wave optics (PEF1), statistical physics Atomic/solid state description: ¾ MICROSCOPIC DESCRIPTION microscopic constituents, dipoles, Connection with: chemistry, quantum physics, inter-particle forces; microscopic solid state physics, physical electronics, photonics origin of values of ɸr , ʅr , g, n Coordinates : Finite variations: ' ' NOTATION Derivatives: d, w ,w , infinitesimals (with d or ɷ): - rectangular: x, y, z t x ' ' - distance: Ěы (dx , dy , …), - spherical: r,T,M d# rd vectors and vector fields : ' ' ' - cylindrical: z, s,T Position: r ; distance d - surface: da, da nˆda ' - unit vectors: xˆ, yˆ, zˆ,rˆ Normal direction ˆ ; velocity ' n' v - volume: dʏ ; solid angle: d: (also iˆ, sˆ,Tˆ,...) Force: F , torque: * ' scalars and scalar fields Mass: m ; Time: t Force per unit charge or volume:' f Number/number density: n, N Temperature: T Electric dipole moment p ' Dirac’s delta function: G Length: L, d Magnetic dipole' moment m Electric potential/voltage: V or I Speed of light: c Electric field E ' Field (or potential) energy: U Generic constant: k, N Polarization field P ' Total energy: E Surface : S, Area (electric line dipole density:' ) (field) energy density: u Volume: Vol Displacement field' D Charge: q, Q Capacitance: C Magnetic field B ' Proton/electron charge: +/– e Resistance: R Magnetization field M ' Charge density: (free f, bound b) Current: I ; Power: Auxiliary field H ' - volume U Conductivity: g A Vector potential - surface V Mobility: P Current density:' (free f, bound b) - line O Resistivity: K - volume J ' Susceptibility: F Polarisability: D - surface' K Dielectric permittivity: H Inductance: L I - line ' Magnetic permeability: P Electromotance: E S Poynting vector: ' Magnetic scalar potential: ; Magnetomotance: M Generic vector field: G ' Flux: ĭ Irradiance: Reluctance: Constant vector (field) : C Complex refractive index: ñ complex permittivity: H~ Summary of electrostatics (topic 1) ' ' 1 r Coulomb’s law 1 9 2 2 F q1q2 3 with | 9u10 Nm C 4SH 0 r 4SH 0 ' ' ' F ' q rˆ q r E(r) lim Æ Field of a point charge: E(r) 2 3 qo0 q 4SH 0 r 4SH 0 r ' ' ' ' N q r r Æ Field of a set of point charges: E(r) k k ¦ ' ' 3 k 1 4SH 0 | r rk | SI unit of q: Coulomb (C) Æ Field of a continuous distribution of charges: (1 C = 1 A s ; ' ' e 1.6 · 10 –19 C) ' ' 1 r rc ' E(r) dW c ' ' U(rc) 3 dq UdW ; 6 o ³ ³ 3 SI unit of U : C/m 4SH 0 | r rc | ؄ ' ' ' r ' § 1 · r rc ' 1 ' ' ' ' By the identities: ¨ ¸ and ' ' ' ' , we see that E(r) V (r) r 3 © r ¹ | r rc |3 | r rc | ' ' q 1 ' 1 N q ' 1 U(rc) with V (r) or: V (r) ¦ ' k ' or: V (r) ³ dW c ' ' 4SH 0 r 4SH 0 k 1 | r rk | 4SH 0 | r rc | ' ' ' ' ' ' ' ' E(r) V (r) implies u E uV 0 SI unit of V: Volt (V) ' ' ' ' ' ' (1 V = 1 J/C = 1 J/As) It also implies: V (r) E(r)d# and U (r) qV (r) ³ q SI unit of E: V/m Dirac’s delta Instead of having 2 ' ' ' ' ' ' N q r r ' different types of k k ' 1 r rc ' E(r) ' ' 3 E(r) dW c ' ' U(rc) ¦ 4SH | r r | ³ 3 formulas, for the k 1 0 k 4SH 0 | r rc | or: ' continuous and ' 1 N q ' 1 U(rc) V (r) ' k ' V (r) dW c ' ' discrete cases, we’d 4SH ¦ | r r | 4SH ³ | r rc | like to have only 1 0 k 1 k 0 For this purpose we would need the «continuous» charge density associated with a point charge.