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# tv - TimeVarying Fields EEL3472 EEL3472 TimeVaryingFields...

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EEL 3472 EEL 3472 Time - Varying  Time - Varying  Fields Fields

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EEL 3472 EEL 3472 2 Time-Varying Fields Time-Varying Fields Time-Varying Fields                Stationary charges            electrostatic fields                Steady currents            magnetostatic fields                Time-varying currents            electromagnetic fields Only in a non-time-varying case can electric and magnetic fields be considered as  independent of each other. In a time-varying (dynamic ) case the two fields are  interdependent. A changing magnetic field induces an electric field, and vice versa.
EEL 3472 EEL 3472 3 Time-Varying Fields Time-Varying Fields The Continuity Equation Electric charges may not be created or destroyed (the principle of conservation of charge).  Consider an arbitrary volume  V  bounded by surface  S . A net charge  Q  exists within this  region. If a net current  I  flows across the surface out  of this region, the charge in the volume  must decrease  at a rate that equals the current: Divergence theorem This equation must hold regardless of the choice of  V , therefore the integrands must be  equal: For steady currents that is, steady electric currents are divergences or solenoidal.  - = - = = S V V dv dt d dt dQ dS J I ρ - = V V V dv t dv J ρ t J V - = ρ 0 = J Kirchhoff’s current law follows from this ) / ( 3 m A the equation of continuity Partial derivative  because     may be a  function of both time  and space V ρ

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EEL 3472 EEL 3472 4 Time-Varying Fields Time-Varying Fields Displacement Current For magnetostatic field, we recall that Taking the divergence of this equation we have However the continuity equation requires that Thus we must modify the magnetostatic curl equation to agree with the continuity equation.  Let us add a term to the former so that it becomes where     is the conduction current density             , and     is to be determined and defined.  J H = × ( 29 J H = = × 0 0 - = t J V ρ d J J H + = × J E J E σ = d J
EEL 3472 EEL 3472 5 Time-Varying Fields Time-Varying Fields Displacement Current continued ( 29 d J J H + = = × 0 J J d -∇ = ( 29 t D D t t J J V d = = = -∇ = ρ t D J d = ∇× H = J + D t xH ds = S J + D t S ds H dl = I + D t S ds L H Taking the divergence we have In order for this equation to agree with the continuity equation, displacement current density Stokes’ theorem Gauss’ law

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EEL 3472 EEL 3472 6
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