Maxwell's Equations Applied Electromagnetic Waves 1

Notes 4 Maxwell’s Equations 1 Overview Here we present an overview of Maxwell’s equations. A much more thorough discussion of Maxwell’s equations may be found in the text and class notes for ECE 3318: http://courses.egr.uh.edu/ECE/ECE3318 Notes 10: Electric Gauss’s law Notes 18: Faraday’s law Notes 28: Ampere’s law Notes 28: Magnetic Gauss law Extra reference: D. Fleisch, A Student’s Guide to Maxwell’s Equations, Cambridge University Press, 2008. (This is on reserve in the Library.) 2 Electromagnetic Fields Four vector quantities E electric field [Volt/meter] D electric flux density [Coulomb/meter2] H magnetic field [Amp/meter] B magnetic flux density [Weber/meter2] or [Tesla] Each are functions of space and time e.g. E(x,y,z,t) Reminder: The Handscript SF font is used to denote time-varying vectors. J electric current density [Amp/meter2] ρv electric charge density [Coulomb/meter3] 3 MKS units Length – meter [m] Mass – kilogram [kg] Time – second [s] Some common prefixes and the power of ten each represent are listed below femto - f - 10-15 centi - c - 10-2 mega - M - 106 pico - p - 10-12 deci - d - 10-1 giga - G - 109 nano - n - 10-9 deka - da - 101 tera - T - 1012 micro - μ - 10-6 hecto - h - 102 peta - P - 1015 milli - m - 10-3 kilo - k - 103 4 Maxwell’s Equations (Time-varying, differential form) ∂B ∂t ∂D ∇× H= J + ∂t ∇⋅ B =0 ∇ ⋅D = ρv ∇ ×E = − 5 Maxwell James Clerk Maxwell (1831–1879) James Clerk Maxwell was a Scottish mathematician and theoretical physicist. His most significant achievement was the development of the classical electromagnetic theory, synthesizing all previous unrelated observations, experiments and equations of electricity, magnetism and even optics into a consistent theory. His set of equations—Maxwell's equations—demonstrated that electricity, magnetism and even light are all manifestations of the same phenomenon: the electromagnetic field. From that moment on, all other classical laws or equations of these disciplines became simplified cases of Maxwell's equations. Maxwell's work in electromagnetism has been called the "second great unification in physics", after the first one carried out by Isaac Newton. Maxwell demonstrated that electric and magnetic fields travel through space in the form of waves, and at the constant speed of light. Finally, in 1864 Maxwell wrote A Dynamical Theory of the Electromagnetic Field where he first proposed that light was in fact undulations in the same medium that is the cause of electric and magnetic phenomena. His work in producing a unified model of electromagnetism is considered to be one of the greatest advances in physics. (Wikipedia) 6 Maxwell’s Equations (cont.) ∂B ∇ ×E = − ∂t Faraday’s law ∂D Ampere’s law ∇× H= J + ∂t Magnetic Gauss law ∇⋅ B = 0 ∇ ⋅D = ρv Electric Gauss law Questions: When does a magnetic field produce an electric field? When does an electric field produce a magnetic field? When does a current flow produce a magnetic field? When does a charge density produce an electric field? 7 Charge Density ∆Q dQ x, y, z ) lim = ρv (= ∆V → 0 ∆V dV + + + + ++ + ++ + ++ ρ v ( x, y , z ) ( x, y , z ) dQ dV Non-uniform cloud of charge density Example: Protons are closer together as we move to the right. 8 Current Density Vector J = current density vector  A/m 2  + + + Medium ∆I J E σ ∆I= ∆S J ∆S Current flow is defined to be in the direction that positive charges move in. Note: If negative charges are moving, we can pretend that positive charges are moving in the opposite direction. 9 Current Density Vector (cont.) Ohm’s law J = σE J σ E Material σ [S/m] Silver 6.3×107 Copper 6.0×107 Copper (annealed) 5.8×107 Gold 4.1×107 Aluminum 3.5×107 Zinc 1.7×107 Brass 1.6×107 Nickel 1.4×107 Iron 1.0×107 Tin 9.2×106 Steel (carbon) 7.0×106 Steel (stainless) 1.5×106 http://en.wikipedia.org/wiki/Electrical_resistivity_and_conductivity 10 Current Density Vector (cont.) Current through a tilted surface: J + + + Medium E σ ∆I= n̂ ∆S ( J ⋅ nˆ ) ∆S 11 Current Density Vector (cont.) ∆I= = I ( J ⋅ nˆ ) ∆S ∫ J ⋅ nˆ dS Note: The direction of the unit normal vector determines whether the current is measured going up or down through the surface. S J n̂ S 12 Law of Conservation of Electric Charge (Continuity Equation) ∇ × H= J + (“zero identity”) ∂D ∂t  ∂D  ∇ ⋅ (∇ × H ) = ∇ ⋅ J + ∇ ⋅   ∂ t   ∂ 0 = ∇ ⋅ J + (∇ ⋅D ) ∂t Flow of electric current out of volume (per unit volume) ∂ρ v ∇⋅ J = − ∂t (Recall : ∇ ⋅ D = ρv ) Rate of decrease of electric charge (per unit volume) This is the continuity equation in point or differential form. 13