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From Maxwell's Equations to the Polarization Ellipse

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From Maxwell's Equations to the Polarization Ellipse 2130-1 Preparatory School to the Winter College on Optics and Energy 1 - 5 February 2010 ELECTODYNAMICS: FROM MAXWELL’S EQUATIONS TO THE POLARIZATION ELLIPSE I. Ashraf Zahid Quaid-i-Azam University Pakistan ELECTODYNAMICS: FROM MAXWELL’S EQUATIONS TO THE POLARIZATION ELLIPSE Imrana Ashraf Zahid Quaid-i-Azam University, Islamabad Pakistan 9-Feb-10 Preparatory School to the Winter 1 College on Optics and Energy Layout • Electrostatic : Revisited • Magneto- static : Revisited • Introduction to Maxwell’s equations • Electrodynamics before Maxwell • Maxwell’s correction to Ampere’s law • General form of Maxwell’s equations • Maxwell’s equations in vacuum • Maxwell’s equations inside matter • The Electromagnetic wave • Energy and Momentum of Electromagnetic Waves • Polarization of Light • Polarization Ellipse 9-Feb-10 Preparatory School to the Winter 2 College on Optics and Energy Nomenclature • E = Electric field • D = Electric displacement • B = Magnetic flux density • H = Auxiliary field • ρ= Charge density • j = Current density -7 • μ0 (permeability of free space) = 4π×10 T-m/A -12 2 2 • ε0 (permittivity of free space) = 8.854×10 N-m / C • c (speed of light) = 2.99792458×108 m/s 9-Feb-10 Preparatory School to the Winter 3 College on Optics and Energy Introduction • Electrostatics Electrostatic field : Stationary charges produce electric fields that are constant in time. The theory of static charges is called electrostatics. Stationary charges Constant Electric field; 9-Feb-10 Preparatory School to the Winter 4 College on Optics and Energy Electrostatic :Revisited Coulombs Law G Q r Test G 1 qQ Charge F = rˆ 4πε r 2 q 0 Source Charge 2 −12 C ε = 8.85×10 Permittivity of free space 0 N − m2 9-Feb-10 Preparatory School to the Winter 5 College on Optics and Energy The Electric Field y G G F = QE ri q q2 P n i G Field 1 qi qn ˆ ′ Point E(P) = ∑ 2 ri r i r 4πε0 i=1 ri E- the electric field of the source charges. x Physically E(P) Is force per unit charge exerted on a test charge z placed at P. 9-Feb-10 Preparatory School to the Winter 6 College on Optics and Energy The Electric Field: cont’d P r G 1 rˆ E(P) = λdl ∫ 2 4πε0 Line r λ is the line charge density P r G 1 rˆ E(P) = σda ∫ 2 4πε0 Surface r σ is the surface charge density 9-Feb-10 Preparatory School to the Winter 7 College on Optics and Energy The Electric Field: cont’d r P G 1 rˆ E(P) = ρdτ ∫ 2 4πε0 Volume r ρ is the volume charge density 9-Feb-10 Preparatory School to the Winter 8 College on Optics and Energy Electric Potential The work done in moving a test charge Q in an electric field from point P1 to P2 with a constant speed. W = Force• dis tan ce P2 G G W = −∫QE • dl p1 negative sign - work done is against the field. For any distribution of fixed charges. G G ∫ E • dl = 0 The electrostatic field is conservative 9-Feb-10 Preparatory School to the Winter 9 College on Optics and Energy Electric Potential: cont’d Stokes’s TheoremG G gives G∇× E =G 0 E = −∇V where V is Scalar Potential The work done in moving a charge Q from infinity to a point P2 where potential is V W = QV V = Work per unit charge = Volts = joules/Coulomb 9-Feb-10 Preparatory School to the Winter 10 College on Optics and Energy Electric Potential : cont’d Field due to a single point charge q at origin ∞ qdr q V = = ∫ 2 r 4πε0r 4πε0r 1 F ∝ r 2 1 E ∝ r 2 1 V ∝ r 9-Feb-10 Preparatory School to the Winter 11 College on Optics and Energy G G 1 Gauss’s Law E • da = Q ∫ enc ε 0 G G ρ Differential form of Gauss’s ∇ • E = Law ε 0 ρ Poisson’s Equation ∇2V = − ε 0 Laplace's Equation ∇2V = 0 9-Feb-10 Preparatory School to the Winter 12 College on Optics and Energy Electrostatic Fields in Matter Matter: Solids, liquids, gases, metal, wood and glasses - behave differently in electric field. Two Large Classes of Matter (i) Conductors (ii) Dielectric Conductors: Unlimited supply of free charges. Dielectrics: • Charges are attached to specific atoms or molecules- No free charges. • Only possible motion - minute displacement of positive and negative charges in opposite direction. • Large fields- pull the atom apart completely (ionizing it). 9-Feb-10 Preparatory School to the Winter 13 College on Optics and Energy Polarization A dielectric with charge displacements or induced dipole moment is said to be polarized. E=0 + E + + + + + + Induced Dipole Moment p = αE The constant of proportionality α is called the atomic polarizability P ≡ dipole moment per unit volume 9-Feb-10 Preparatory School to the Winter 14 College on Optics and Energy The Field of a Polarized Object Potential of single dipole p is G 1prˆ• r dV = 2 4rπε 0 G 1 P • rˆ V = dτ ∫ 2 4πε0 volume r 11⎡⎤GGG 1 VPdaPd=•−∇•⎢⎥τ 4 πε ∫∫rr() 0 ⎣⎦⎢⎥surface volum e Potential due to dipoles in the dielectric 9-Feb-10 Preparatory School to the Winter 15 College on Optics and Energy The Field of a Polarized Object: cont’d G σ = P•nˆ Bound charges at surface b G G Bound charges in volume ρb = −∇ • P 11⎡ 1⎤ Vdad=+⎢ σ ρτ⎥ 4 πε ∫∫rrbb 0 ⎣⎢ surface volume ⎦⎥ The total field is field due to bound charges plus due to free charges 9-Feb-10 Preparatory School to the Winter 16 College on Optics and Energy Gauss’s law in Dielectric • Effect of polarization is to produce accumulations of bound charges. • The total charge density G G ρ = ρ + ρ D • da = Q f b ∫ fenc From Gauss’s law Q G G fenc -Free charges enclosed ε 0∇ • E = ρ = ρb + ρ f Displacement vector G G G G G ∇ •=D ρ f D = ε 0 E + P 9-Feb-10 Preparatory School to the Winter 17 College on Optics and Energy Magnetostatics : Revisited • Magnetostatics Steady current produce magnetic fields that are constant in time. The theory of constant current is called magnetostatics. Steady currents Constant Magnetic field; 9-Feb-10 Preparatory School to the Winter 18 College on Optics and Energy Magnetic Forces Lorentz Force G G G G F = q[E + (v × B)] • The magnetic force on a segment of current carrying wire is G G F = I × B dl mag ∫ ( ) G G F = I dl × B mag ∫ ( ) 9-Feb-10 Preparatory School to the Winter 19 College on Optics and Energy Equation of Continuity The current crossing a surface s can be written as G K G G I = ∫ J • da = ∫ (∇ • J )dτ s v G G d ⎛ ∂ρ ⎞ ∫ ()∇ • J dτ = − ∫∫ρdτ = − ⎜ ⎟dτ v dt ⎝ ∂t ⎠ Charge is conserved whatever flows out must come at the expense of that remaining inside - outward flow decreases the charge left in v G G ∂ρ ∇ • J = − This is called equation of continuity ∂t 9-Feb-10 Preparatory School to the Winter 20 College on Optics and Energy Equation of Continuity 1 In Magnetostatic steady currents flow in the wire and its magnitude I must be the same along the line- otherwise charge would be pilling up some where and current can not be maintained indefinitely. ∂ρ = 0 ∂t In Magnetostatic and equationG G of continuity ∇• J = 0 Steady Currents: The flow of charges that has been going on forever - never increasing - never decreasing. 9-Feb-10 Preparatory School to the Winter 21 College on Optics and Energy Magnetostatic and Current Distributions G I Biot and Savart Law dl G G G G μ0 I × r G dB B()p = G 3 dl r 4π ∫ r dl is an element of length. G p r vector from source to point p. μ0 Permeability of free space. Unit of B = N/Am = Tesla (T) 9-Feb-10 Preparatory School to the Winter 22 College on Optics and Energy Biot and Savart Law for Surface and Volume Currents G G G μ0 K × r For Surface Currents B = G 3 da 4π ∫ r G G G μ0 J × r For Volume Currents B = G 3 dτ 4π ∫ r 9-Feb-10 Preparatory School to the Winter 23 College on Optics and Energy Force between two parallel wires The magnetic field at (2) due to current I is 1 I1 I 2 μ I 0 1 Points inside dF dl B1 = B 2πd 1 Magnetic force law d G G dF = I dl × B ∫ 2 ( 2 1 ) G (1) (2) ⎛⎞μ01I ˆ dF=× I22⎜⎟ dl k ∫ ⎝⎠2πd 9-Feb-10 Preparatory School to the Winter 24 College on Optics and Energy Force between two parallel wires μ I I dF = 0 1 2 dl 2πd 2 The total force is infinite but force per unit length is dF μ I I = 0 1 2 dl2 2πd If currents are anti-parallel the force is repulsive. 9-Feb-10 Preparatory School to the Winter 25 College on Optics and Energy Straight line currents The integral of B around a circular path The current is out of the page of radius s, centered at the wire is G G μ I Bdl•=0 dl =μ I vv∫∫2π s 0 For bundle of straight wires. Wire that passes throughG G loop contributes only. B •dl = μ I ∫ 0 enc Applying Stokes’ theorem G G G ∇× B = μ0 J 9-Feb-10 Preparatory School to the Winter 26 College on Optics and Energy Divergence and Curl of B Biot-Savart law for the general case of a volume current reads G G G G μ J (r′)× r r B = 0 dτ ′ 4π ∫ r 3 G r G G G G G and ∇ • B = 0 ∇× B = μ0 J 9-Feb-10 Preparatory School to the Winter 27 College on Optics and Energy Ampere’s Law G G G ∇× B = μ0 J Ampere’s law Integral form of Ampere’s law Using Stokes’ theorem G G G G G G G ∇× B • da = B • dl =μ J • da ∫ ( ) ∫ 0 ∫ G G B •dl = μ I ∫ 0 enc 9-Feb-10 Preparatory School to the Winter 28 College on Optics and Energy Vector Potential The basic differential law of Magnetostatics G G G ∇× B = μ0 J G G ∇ • B = 0 B curl of some vector field called vector potential A(P) G G G Coulomb’s gauge B (PAP) =∇× ( ) G G G G G ∇ • A = 0 ∇× ∇× A = μ J 2 ( ) 0 ∇ A = μ0 J 9-Feb-10 Preparatory School to the Winter 29 College on Optics and Energy Magnetostatic Field in Matter ¾ Magnetic fields- due to electrical charges in motion.
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