Electromagnetic Wave Propagation Lecture 1: Maxwell s equations

Electromagnetic Wave Propagation Lecture 1: Maxwell s equations Daniel Sjöberg Department of Electrical and Information Technology September 2, 2014

Outline 1 Maxwell s equations 2 Vector analysis 3 Boundary conditions 4 Conservation laws 5 Conclusions 2 / 30

Outline 1 Maxwell s equations 2 Vector analysis 3 Boundary conditions 4 Conservation laws 5 Conclusions 3 / 30

150th anniversary! A few of the equations back then: Nowadays: p = p + df dt, q = q + dg dt, r = r + dh J tot = J + D dt t dγ dy dβ dz = 4πp dα dz dγ dx = 4πq H = J tot dβ dx dα dy = 4πr e + df dx + dg dy + dh dz = 0 D = ρ de dt + dp dx + dq dy + dr dz = 0 J tot = 0 4 / 30

Historical notes Early 1800s: Electricity and magnetism separate phenomena. 1819: Hans Christian Ørsted discovers a linear current deflects a magnetized needle. 1831: Michael Faraday demonstrates that a changing magnetic field can induce electric voltage. 1830s: Electrical wire telegraphs come into use. 1864: James Clerk Maxwell presents his paper A Dynamical Theory of the Electromagnetic Field, joining electricity and magnetism and predicting the existence of waves. 1887: Heinrich Hertz proves experimentally the existence of electromagnetic waves. 1897: Marconi founds the Wireless Telegraph & Signal Company 1905: Albert Einstein publishes his special theory of relativity, emphasizing the role of the speed of light in vacuum. 5 / 30

Historical notes, continued 1930s: The first radio telescopes are built. Early 1940s: The MIT Radiation Laboratory substantially advances the knowledge of control of electromagnetic waves while developing radar technology. Late 1940s: The Quantum Electrodynamics theory (QED) is developed by Richard Feynman, Freeman Dyson, Julian Schwinger, and Sin-Itiro Tomonaga. 1957: Sputnik 1 transmits the first signal from a satellite to earth. 1970s: Low loss optical fibers are developed. 1981: The NMT system goes online. 2000: Mobile phones connect to internet. 2020: Initial observations by Square Kilometre Array(?) 6 / 30

Field equations and the electromagnetic fields B(r, t) Faraday s law: E(r, t) = t Ampère s law: H(r, t) = J(r, t) + Conservation of charge: J(r, t) + ρ(r, t) t Symbol Name Unit E(r, t) Electric field [V/m] H(r, t) Magnetic field [A/m] D(r, t) Electric flux density [As/m 2 ] B(r, t) Magnetic flux density [Vs/m 2 ] J(r, t) Current density [A/m 2 ] ρ(r, t) Charge density [As/m 3 ] D(r, t) t = 0 7 / 30

The divergence equations Often the equations D = ρ and B = 0 ( ) are considered as a part of Maxwell s equations, but they can be derived as follows. Since ( F ) = 0 for any vector function F, taking the divergence of Faraday s and Ampère s laws imply 0 = B ( t 0 = J + D ) t = ρ t + D t Thus B = f 1 and D ρ = f 2, where f 1 and f 2 are independent of t. Assuming the fields are zero at t =, we have f 1 = f 2 = 0 and the divergence equations ( ) follow. 8 / 30

Materials Maxwell s equations give 2 3 = 6 equations, but the fields E, B, H, D represent 4 3 = 12 unknowns. The remaining equations are given by the models of the material behavior. In vacuum we have D = ɛ 0 E, B = µ 0 H where the speed of light in vacuum is c 0 = 1/ ɛ 0 µ 0 = 299 792 458 m/s (exact) and ɛ 0 = 1 c 2 0 µ 8.854 10 12 As/Vm 0 µ 0 = 4π 10 7 Vs/Am (exact) In a material, there is in addition polarization and magnetization: P = D ɛ 0 E, M = 1 µ 0 B H The physics of the material in question determine the fields P and M as functions of the electromagnetic fields (next lecture!). 9 / 30

Outline 1 Maxwell s equations 2 Vector analysis 3 Boundary conditions 4 Conservation laws 5 Conclusions 10 / 30

Vector analysis The vectors have three components, one for each spatial direction: E = E x ˆx + E y ŷ + E z ẑ In particular, the position vector is r = xˆx + yŷ + zẑ. Vector addition, scalar product and vector product are r 1 r 2 r 1 r 2 r 2 r 2 ϕ r 1 ϕ r 1 + r 2 r 2 cos ϕ r 1 Addition: r 1 + r 2 = (x 1 + x 2 )ˆx + (y 1 + y 2 )ŷ + (z 1 + z 2 )ẑ Scalar product: r 1 r 2 = r 1 r 2 cos ϕ = x 1 x 2 + y 1 y 2 + z 1 z 2. Vector product: orthogonal to both vectors, with length r 1 r 2 = r 1 r 2 sin ϕ, and r 1 r 2 = r 2 r 1. ˆx ŷ = ẑ, ŷ ẑ = ˆx, ẑ ˆx = ŷ 11 / 30

The nabla operator is Vector analysis, continued = ˆx x + ŷ y + ẑ z The divergence and curl operations are E = x E x + y E y + z E z E = x ˆx E + y ŷ E + z ẑ E ) Cartesian representation: [E] = and [ E] = x 0 0 0 E x 0 0 1 E y 0 } 1 0 {{ E z } =[ˆx E] + y ( Ex E y E z 0 0 1 E x 0 0 0 E y 1 } 0 0 {{ E z } =[ŷ E] + z 0 1 0 E x 1 0 0 E y 0 } 0 0 {{ E z } =[ẑ E] 12 / 30

Integral theorems Generalizations of the fundamental theorem of calculus: Gauss theorem: V Stokes theorem: S b a F dv = f (x) dx = f(b) f(a) S ( F ) ˆn ds = ˆn F ds C F dr Typically: a higher order integral of a derivative can be turned into a lower order integral of the function. S S C V dr ˆn ˆn 13 / 30

Dyadic products The projection of a vector on the x-direction is defined by P x E = ˆxE x Since E x = ˆx E, we can write this P x E = ˆxE x = ˆx(ˆx E) = (ˆxˆx) E Thus P x = ˆxˆx, which is a dyadic product. In particular, the identity operator I E = E can be written I = ˆxˆx + ŷŷ + ẑẑ with the Cartesian representation 1 0 0 1 [I ] = 0 1 0 = 0 1 0 0 ) 0 + 1 0 1 0 ) 0 + 0 0 0 1 ) 0 0 1 0 } {{ } 0 } {{ } 1 } {{ } =[ˆxˆx ] =[ŷŷ ] =[ẑẑ ] 14 / 30

Outline 1 Maxwell s equations 2 Vector analysis 3 Boundary conditions 4 Conservation laws 5 Conclusions 15 / 30

Material interfaces Consider the interface between two media, 1 and 2: ˆn S 1 2 h What are the relations between field quantities at different sides of the interface? 16 / 30

Integral form of Maxwell s equations Integrate the field equations over the volume V (red and green): B E dv = V V t dv D H dv = J dv + V V V t dv B dv = 0 V D dv = ρ dv V V ˆn S 1 h 2 17 / 30

Integral form of Maxwell s equations, continued Use the integral theorems to find ˆn E dv = d B dv S dt V ˆn H dv = J dv + d S V dt ˆn B dv = 0 S ˆn D dv = ρ dv S V V D dv ˆn S 1 h 2 18 / 30

Pillbox volume Let the height of the pillbox volume V h become small, h 0. On metal surfaces, the current and charge densities can become very large. Introduce the surface current and surface charge as S lim J dv = J S ds ˆn h 0 V h S h lim ρ dv = ρ 1 S ds h 0 V 2 h S The limit of the flux integrals on the other hand is zero, lim D dv = lim B dv = 0 h 0 V h h 0 V h The limit of E is E 1 from material 1 and E 2 from material 2, implying lim ˆn E ds = (ˆn 1 E 1 + ˆn 2 E 2 ) ds h 0 S h S }{{} =ˆn 1 (E 1 E 2 ) 19 / 30

Boundary conditions We summarize as (when material 2 is a perfect electric conductor (PEC) the fields with index 2 are zero): In words, this means: ˆn 1 (E 1 E 2 ) = 0 ˆn 1 (H 1 H 2 ) = J S ˆn 1 (B 1 B 2 ) = 0 ˆn 1 (D 1 D 2 ) = ρ S The tangential electric field is continuous. The tangential magnetic field is discontinuous if J S 0. The normal component of the magnetic flux is continuous. The normal component of the electric flux is discontinuous if ρ S 0. 20 / 30

Example Fields ˆn D ρ S ˆn B Interface Distance to boundary 21 / 30

Outline 1 Maxwell s equations 2 Vector analysis 3 Boundary conditions 4 Conservation laws 5 Conclusions 22 / 30

Conservation of charge The conservation of charge is postulated, J + ρ t = 0 Alternatively, we could postulate Gauss law, D = ρ, and derive the conservation of charge from Ampère s law H = J + D t. 23 / 30

Energy conservation Take the scalar product of Faraday s law with H and the scalar product of Ampère s law with E: H ( E) = H B t E ( H) = E J + E D t Take the difference of the equations and use the identity (a b) = b ( a) a ( b) (E H) + H B t + E D t + E J = 0 This is Poynting s theorem on differential form. 24 / 30

Interpretation of Poynting s theorem Integrating over a volume V implies (where P = E H) S ˆn P ds = P dv V ( = H B t + E D ) t V dv E J dv V The first integral is the total power radiated out of the bounding surface S. The second integral is the rate of change of electromagnetic energy stored in the volume V (not obvious at this point). The last integral is the work per unit time (the power) that the field does on charges in V. This is conservation of power (or energy). 25 / 30

Momentum conservation Take the vector product of D with Faraday s law and B with Ampère s law, D ( E) = D t B B ( H) = B J + B t D = J B ( t D) B Adding the equations results in t (D B) + J B + D ( E) + B ( H) = 0 It can be shown that D ( E) = ( D)E + i=x,y,z implying (using D = ρ and B = 0) D i E i (DE) t (D B) + J B + ρe = (DE+BH) D }{{}}{{} i E i +B i H i momentum Lorentz force i=x,y,z 26 / 30

A simple material model Often materials can be described by using an energy potential φ: φ(e, H) φ(e, H) D = and B = E H which is interpreted D i = φ/ E i etc. For linear, isotropic media { φ(e, H) = 1 2 ɛ E 2 + 1 D = ɛe 2 µ H 2 B = µh Nonlinear materials have additional terms like E 4, E 6 etc. For a material described by potential φ, Poynting s theorem is (E H) + (D E + B H φ) = E J t and the conservation of momentum is t (D B) + ρe + J B = (DE + BH φi) 27 / 30

The EM momentum is still debated! 28 / 30

Outline 1 Maxwell s equations 2 Vector analysis 3 Boundary conditions 4 Conservation laws 5 Conclusions 29 / 30

Conclusions Maxwell s equations describe the dynamics of the electromagnetic fields. Boundary conditions relate the values of the fields on different sides of material boundaries to each other. Conservation laws can be derived to describe the conservation of physical entities such as power and momentum; these are usually products of two fields. Constitutive relations are necessary in order to fully solve Maxwell s equations. Topic of next lecture! 30 / 30