1 Lecture contents Review: Few concepts from physics Electric field Coulomb law, Gauss law, Poisson equation, dipole, capacitor Conductors and isolators 1 Electric current Dielectric constant
Overview of Electromagnetics Fundamental laws of classical electromagnetics Maxwell s equations Special cases Electrostatics Magnetostatics Electrodynamics Geometric Optics Statics: t Transmission Line Theory Wave Optics Circuits Kirchoff s Laws d
Few concepts from Electrostatics: Coulomb law 3 Charge Charges interact Quantized e = 1.6x1-19 C Conserved {material dielectric constant e will be discussed later, for now e =1 for vacuum} qq r Coulomb s law: electric force 1 F e 1 qq 4e e r 1 SI F 1 F C 4 c m Nm 7 1 8.851 Electric Field from a charge force that would act on a charge q Linear field F 1 q V N q 4e e r m C E r Field line map of pair of + and - charges Charge motion in the field: d x F q E dt m m
Gauss law Few concepts from Electrostatics: Gauss law E n ds Q ee Integration over the closed surface 4 Note: Coulomb law can be deduced from Gauss law and symmetry considerations Using divergence theorem (also known as Gauss theorem in mathematics) V divadv Can derive Gauss law in in differential form or more accurately: In dielectrics, it is useful to introduce electric displacement D(r): S An ds dive E the del or nabla operator ee div e e( r) E( r) ( r) div D( r) ( r)
Few concepts from Electrostatics: Potential 5 Potential Change in potential energy equals to work done by the electric field over a charge (electric field is conservative) If zero potential at infinity: Or in differential form: Poisson equation P1 1 U W F dl q q q In a medium with no charge density ( r) Laplace s equation E P P E dl P P 1 E dl ee
Circulation of Electric field 6 Electric field is conservative: work done depends on start and finish of the trajectory. Circulation equals zero. E dl 1 1 Applying Stokes s theorem C E dl E ds S Because the above must hold for any surface S, we must have E curle
Few concepts from Electrostatics: Dipole 7 Electric dipole Dipole moment p lq In an electric field net force is zero, but torque pe Electric field from dipole in vacuum at r >>l Potential due to dipole in vacuum at r >>l E 1 3 4e p r r r p r 5 3 q 1 1 1 p cos 1 p r 3 4e r1 r 4e r 4e r () r r 1 r Potential energy of dipole in uniform field varies between: Umin p E and Umax p E
Capacitor 8 Let s apply the Gauss law to a cylinder: The only contribution to the integral is the bottom surface of the cylinder Uniform field between the plates gives the potential difference (voltage ) between the plates E n ds Q ee ee EdS ee EA A f E f ee f d d Q V Ed Q ee ee S C 4 f E CGS e With capacitance C ee d S
Fundamental Laws of Electrostatics in Differential Form 9 Conservative field E Gauss s law D q ev D e E Constitutive relation (we ll discuss it in just a few slides)
Conductors and Insulators at a glance 1 Conductors Charges (electrons) are more or less free to move free electrons Electrical properties described by electron motion Insulators Mostly no long-range mobility of electrons Electrical properties described by electrical dipoles Polarization:
Current: drift velocity 11 Electric current = charge flow through a cross-section in a time interval I Q, t t or introducing n -concentration of electrons, q -electron charge v d drift velocity Current density In many materials current is proportional to the applied potential difference: Ohm s law : with specific resistivity,, or conductivity, (material parameters) q x I naq naqv t t J A m nqvd R d I R x 1 x A A It is also useful to introduce local Ohm s law J I Ex A R R J E
Current: mobility In the Ohm s law regime, the drift velocity should be proportional to the electric field J nqv d J E 1 Similar to viscous hydrodynamic flow with friction force proportional to velocity of an object Friction is related to scattering with time Proportionality coefficient is called mobility: vd E q m cm qn Vs qn
Insulators (dielectrics): Polarisability 13 Three basic mechanisms of polarization: Dipolar (molecular) polarisability due to reorientation (most significant in liquids and gases) Ionic polarisability due to displacements of the positive and negative ions Results in lattice distortions May give rise to ferroelectricity Atomic polarisability due to redistribution of charge in any atom
Polarization 14 Assume macroscopic neutrality, but solids are composed of positively and negatively charged entities Displacements of charges generate dipole moment and polarization (electric dipole moment per unit volume) V can be a unit cell volume in crystals in the uniform field In general, polarization can be written as a series of the electric field (through susceptibility, c ) If electric field is much smaller than crystal fields, linear response is good enough : Otherwise nonlinear terms ( nd and 3 rd order susceptibility) are employed nonlinear optics P V udv V i n q u V i i i P e c E c E (1) ()... (1) () P c E c E CGS P ece P ce CGS.. { }
Charge and Polarization - I Polarization may be thought of as a bulk movement of the positive charges relative to the negative charges resulting in the bound charge density ρ b. Consider three cases: We ll show: P b 15 No polarization. Charge density (ρ b ) in the medium is zero since the positive (ρ + ) and negative (ρ - ) distributions overlap. Uniform polarization. The relative shift of the charge densities leads to the appearance of surface charge densities (σ) The positive and negative charge densities in the bulk still cancel. Nonuniform polarization. The positive charge density is stretched out as well as displaced to the right. The charge density on the positive surface is greater than that on the negative surface. The polarisation increases to the right.
Charge and Polarization - II 16 Consider a small volume within the dielectric In the unpolarized dielectric the net charge density is If non-uniform displacement is u(x), at the left face it is u(x), whereas at the right face it is u(x + δx) Positive charge enters at the left face: Positive charge leaves at the right face: The net charge appearing in the volume is no longer zero: Or Including the other two dimensions, the total charge appearing in δv is This gives a bound charge density, ( x) u( x) A P ( x) A ( x x) u( x x) A Px ( x x) A Px Q Px( x) Px( x x) A x y z x Q P P x y Pz b( x, y, z) divp x y z x y z x P P P x y z x y z Q x y z x y z x y z P b
Dielectric constant Polarization contributes an amount b( r) P( r) to the charge density at r: We can rewrite Gauss law in differential form as E f e b 17 This enables us to restate the Gauss law as: e E P f Electric displacement vector, D, and dielectric constant e can be introduced within linear medium response: D e E P e E e ce 1 c e E ee E In many cases e is a complex scalar material parameter depending on frequency of electric field. Good news: that s the only parameter (dielectric function) which define electrical and optical properties of a medium D e E CGS ee E f
Dielectric properties of some materials 18 Dielectric Dielectric constant Breakdown field, kv/cm Air 1.6 3 Glass (pyrex) 5.6 14 PMMA (Plexiglass) 3.4 4 Polystyrene.6 5 PTFE (Teflon).1 6 Aluminum oxide 8.4 67 Silicon 11.9 Tantalum oxide 6 5 Barium titanate ~3 Water 8