Electromagnetic. G. A. Krafft Jefferson Lab Jefferson Lab Professor of Physics Old Dominion University TAADI Electromagnetic Theory

Transcription

1 TAAD1 Electromagnetic Theory G. A. Krafft Jefferson Lab Jefferson Lab Professor of Physics Old Dominion University

2 Classical Electrodynamics A main physics discovery of the last half of the 2 th century is that electrodynamics is a part of an overarching quantum theory of nature called the standard model In the standard model, electromagnetism is closely linked with the weak interaction force, through the so-called electroweak unification, in a non-abelian gauge theory that relies on and utilizes the usual phase invariance of quantum mechanics As part of the standard model, electromagnetic forces are distinguished in that they couple to electric charge, and they are long (?) range because to the best of present knowledge, the photon (EM force carrier) is (nearly?) massless This fact explains their prevalence in our day-to-day life

3 Range of Validity of Classical Description Large numbers of quanta (quantum mechanical fluctuations are relatively small, leading to classical results) Low photon energy N photons >> E = ω ω << m c photons photons electron Higher energies can lead to (quantum mechanical) effects like Compton Effect. Not sufficient (photoelectric effect!) Macroscopic charge, i. e., large numbers of individual electrons. Electron discreteness not important. Does not work for atoms and new nano-scale devices. Need Quantum Mechanics for them! 1 2

4 Line and Surface Integrals Line integral b dl ( τ ) V dl V ( L ( τ ) ) dτ dτ L a for any parameterization of L (invariant to choice!) Surface (Flux) integral ( ( )) S S V nda V S, ( S (, )) d d σ τ S S S (, ) (, ) (, ) lim S σ + δσ τ S σ τ σ τ σ δσ δσ again independent of parameters up to sign (Jacobian!). Can usually integrate easily in high-symmetry situations. σ τ σ τ σ τ

5 Line and Surface Integrals Line integral parametrization invariant b b ( ( )) τ ( ) ( τ ( ) ) dl s dl ( ( s) ) d τ V dl V ( L( s )) ds = V ( L ( s) ) τ τ ds ds dτ ds L a a ( b ) τ dl ( τ ) = V ( L( τ )) dτ by change of variables formula dτ τ ( a ) Surface (Flux) integral parametrization invariant S S V nda V ( S ( σ ( σ, τ ), τ ( σ, τ ))) ( S ( σ ( σ, τ ), τ ( σ, τ ))) dσ dτ σ τ S P S S S σ S τ S σ S τ S S σ / σ τ / σ = + + = σ τ σ σ τ σ σ τ τ τ σ τ σ / τ τ / τ S S V nda V ( S ( σ, τ) ) ( S ( σ, τ) ) dσ dτ by 2-D change of variables σ τ S P When evaluating flux integrals, normal must be chosen. You choose correct vector order in product to get normal.

6 Flux for a Radial Field V x y z f r r (,, ) = ( ) ˆ xxˆ + yyˆ + zzˆ r = x + y + z rˆ = r Φ = V nda = V nda + V nda + V + S S S a V nda = f ( a) dxdy = f ( a) 2π a a x y 2 2 V nda = f a 2 π a Φ = f a 4π a S x y a S ( ) ( ) V 2

7 Stokes Theorem Vector Calculus Theorems A nda = A dl S L= S Divergence Theorem AdV = A nda V S= V

8 Electromagnetic Quantities Charge [C], Charge Density [C/m 3 ] q ρ x, t Electric Field Vector [V/m] E ( x, t) Electric Displacement Vector [C/m 2 ] D x, t Magnetic Field Vector [A/m] H ( ) ( x, t ) ( ) Magnetic Induction Vector [V sec/m 2 ] B x, t ( )

9 Maxwell s Equations D = ρ B E = t B = D H = J + t Electric charge generates electric displacement Faraday s Law No magnetic charges Magnetic field generated by real or displacement current density Maxwell s Equations are linear in the source terms ρ and J. In general will generate linear (partial) differential equations to solve. Superposition valid!

10 Constitutive Relations Relationships between electric quantities D = ε E Relationships between magnetic quantities B = µ H For most of course assume purely linear relationship exists. As you go on, you ll find there are non-linear dependences and other complications Better approximation for static phenomena In vacuum D = ε E B = µ H C ε µ π Nt m Nt sec C = =

11 Lorentz Force Force on a (quantized) charge q F = q E + v B ( ) From dimensions, see that force is coupled to magnetic induction, not magnetic field As will see later, valid at relativistic energies too when interpret force as change of relativistic momentum with time d ( γ mv ) q( = E + v B) dt

12 Maxwell Equations (Integral Form) Applying Divergence and Stokes Theorems D = ρ D n da = ρ dv B B E = E dl = n da t t L S B = B n da = S S D D H = J + H dl n da J n da t t = V L S S

13 Boundary Conditions Gaussian Pillbox Argument, σ is surface charge D n da D D nˆ A = σ A D D nˆ = ( ) ( ) G G II I S II I S B nda = B B n A = B n B n ( ) ˆ ˆ ˆ II I S II S I S Discontinuity of normal component of D given by surface charge and no change in the normal component of B A DII, BII G D I, B I ˆ n S σ

14 Gaussian Loop Argument, K is surface current [A/m] B E dl = n da E II dl E I dl = t S D H dl n da= J n da H ˆ II l H I l = K n S l t Discontinuity of tangential component of H given by surface current and no change in the tangential component of E l E II, H II L E, H L L S S I I

15 Maxwell Equations plus the boundary conditions provide a complete description of all classical electromagnetic phenomena Given source functions and constitutive relations these equations have unique solutions A wide variety of numerical solutions exist; we want you to be able to understand the solutions that you derive from codes in your future working life!

16 Electrostatics Maxwell Equations simplify enormously if the fields and sources do not depend on time Electric field of a static charge q 1 at location x given by 1 Coulomb slaw E q x x 4 πε x x 1 1 ( x) = 1 3

17 Notation Coulomb Interaction v v + v + v x y z x x = x x1 + y y1 + z z1 Force on test particle with charge q at location x qq1 x x1 F ( x ) = qe ( x ) = 4πε x x + y y + z z (( ) ( ) ( ) ) 3/2 ( ( ) 2 ( ) 2 ( ) 2 ) 3/ Dependence on the distance between charges F = qq 1 1 4πε + + r ( x x1 ) ( y y1 ) ( z z1 )

18 Gauss s Law Easy proof: x x 1 x x = x x x x x x 3 ( ) ( y y ) ( z z ) x 5 3 = 5 x x x x x x x x 3 Evaluate ( x x ) / x x nda on a sphere surrounding x x x 1 2 nda = R d cosθ dφ = 4π 3 2 x x R S x Independent of R! Gauss s Law follows from this!

19 Formal Statement Gauss s Law: for an arbitrary surface S (doesn t need to be a sphere!) E nda = S q inside Pf: If a charge is outside, calculation 1 shows it doesn t contribute to the integral. If a charge is inside, calculation 2 shows it contributes to the integral as given. For any V and S = V q inside 1 Edxdydz = E nda = = ρ x dxdydz V S V E = ρ ε ε ε ε ( ) 1st Maxwell Eqn. in vacuum

20 Magnetostatics: Ampere s Law In conventional vector notation is H dl = I L where I is the total current enclosed by the current loop. Define current density vector, and current density J = J, J, J ( ) x y z For time-independent independent charge density ρ J = =. tt Integrating over a surface bounded by L H dl = H nda = J nda = I L= S S S

21 Electromagnetic Potentials Poincare s lemma plus Div B = implies there is a vector A so that B = A This defines the usual vector potential. Poincare s lemma plus the Faraday s Law imply there is a scalar potential (sign chosen using traditional definition with E as the negative gradient) A A E E φ t = t = φ

22 Gauge Invariance Considerable flexibility/latitude in choosing potentials. If redefine them by A = A + dλ φ new new old Λ = φold t for an arbitrary space-time function, then the electric field and magnetic field will be identical when computed with the new potentials. In other words, the transformation from the old to new descriptions will leave the electromagnetic field invariant. One makes various choices on the gauge, for convenience of calculation.

23 Equations for the Potentials D ρ D D D x y z x y z = + + = D = ε ( ) E + + φ + A = x y z t and, by sorting the individual components D B H = J, H = t µ m ε ijk ε klm + ε µ φ + A j l i 2 i = µ Ji jlm,, = 1 x x t x t ρ A A ( A) φ = µ J x y z c t c t ρ ε

24 Lorenz Gauge Lorenz Gauge condition is 1 φ A + = c 2 t Applying this condition yields the following, very symmetrical version of the potential equations ρ + + φ = x y z c t ε A = µ J x y z c t

25 Energy/Power considerations Units of electric field, electric displacement, and current density Nt C C E = J D C = = sec m m J J E J = E D = 3 3 sec m m 2 2 Power delivered from electromagnetic field to a current element per unit volume involves E and J. Power delivered dto a current tin a small volume dv must be = E J dv ( ) Use Maxwell Equations to derive general result

26 Energy Conservation Energy/volume involves multiplication of E and D. The exact formula comes from the Maxwell Equations: 3 3 E Jd x = E H D / t d x V V 3 H E B / t d x V E D H B = E H d x d x t 2 V The RHS terms define the [1] Poynting (Energy Flux) Vector (J/sec m 2 ) S = E H V

27 Field Energy Evolution Equation For linear materials have [2] energy density (J/m 3 ) u E D H B ε E E µ H H = + = Field Energy Evolution u + S = J E t

28 Momentum Conservation Momentum is a vector quantity. Mechanical momentum delivered to a load by an electromagnetic field is dp mech dt = ρ E J B + dxdydx V Define the vector force density to be ρ E + J B Momentum Flux density must have nine components [ ] k l eˆ T ε dx dx = eˆ T dy dz + T dz dx + T dx dy i ij jkl i i1 i2 i3

29 In free space 1 E ρ = ε E J = B ε µ t then ( ) 2 E E + c B( B) ρ E + J B = ε ε E B ( ) ( ) 2 E E c B B t Momentum change equation dp mech dp + field = dt dt 2 2 ε E ( E ) c B ( B) E ( E ) c B ( B) + dxdydz V ( )

30 Momentum in field Maxwell Stress Tensor Stress Tensor 1 p field = E Hdxdydz 2 c V Tij = ε EiE j + c BiB j ( E E + c B B) δ ij = ExEx + c BxBx ExE y + c BxBy ExEz + c BxB z ε E E + ε E yex + c ByBx E yey + c ByBy E yez + c ByBz c B B EE z x + cbb z x EE z y + cbb z y EE z z + cbb z z 1