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Electromagnetic Fields Review: Time – Varying Fields In the dynamics case, we can distinguish between two regimes: Low Frequency (Slowly-Varying Fields) – The displacement current is negligible in the Maxwell’s equations, since G G ∂D ( t ) << J ( t ) ∂t High Frequency (Fast-Varying Fields) – The general set of Maxwell’s equations must be considered, with no approximations. © Amanogawa, 2006 – Digital Maestro Series 24 Electromagnetic Fields In the low frequency regime we use the complete set of Maxwell’s equations, but the displacement current is omitted G G ∂B ∇× E= − dt G G ∇× H = J G ∇⋅ D= ρ G ∇⋅ B = 0 G G D=ε E G G B= µ H G G G G F = q ( E + v × B) © Amanogawa, 2006 – Digital Maestro Series 25 Electromagnetic Fields The concept of low frequency and slowly-varying phenomena is relative to the situation at hand. Any disturbance (time-variation) of the electromagnetic field propagates at the speed of light. If a length L is the maximum dimension of the system under study, the maximum propagation time for a disturbance is Maximum Propagation Time L td = vp Maximum Length Phase velocity of light We can assume slow-varying fields if the currents are practically constant during this time period. For sinusoidal currents, with a period of oscillation T , we have Wavelength Period Frequency 1 λ T= = >> t d f vp © Amanogawa, 2006 – Digital Maestro Series and L << λ 26 Electromagnetic Fields The electric potential is now by itself insufficient to completely describe the time-varying electric field, because there is also a direct dependence on the magnetic field variations. By recalling the definition of magnetic vector potential, we can derive a relationship between electric field and electric potential Time-Varying Fields G G G ∂ B( t ) ∂ ∇ × E(t) = − = − ∇ × A( t ) ∂t ∂t G ∂ A( t ) G ⇒ ∇ × E(t) + =0 ∂t G G ∂A ( t ) E(t) + = −∇φ ( t ) ∂t © Amanogawa, 2006 – Digital Maestro Series Statics G ∇× E=0 G E = −∇φ 27 Electromagnetic Fields We can also obtain an integral relation between electric field and magnetic flux, by integrating the curl of the electric field over a surface G G G G G ∂B ( t ) ∂ ∫ ∫S ∇ × E ( t ) ⋅ dS = ∫ ∫S − ∂ t dS = − ∂ t ∫ ∫S B ( t ) ⋅ dS Stoke’s Theorem Magnetic Flux Φ(t) G G ∂ Φ(t) v∫ E ⋅ d l = − ∂ t © Amanogawa, 2006 – Digital Maestro Series 28 Electromagnetic Fields In the electrostatic case, we do not need to distinguish between voltage and potential difference. The voltage between two points is always defined as G bG Vba = − E ⋅ d l = − e. m. f . a ∫ but in terms of potential φ we have Time-Varying Fields G b ∂A G Vba ( t ) = ∫ ∇φ + ⋅ d l a ∂t G ∂ bG = φ b − φ a + ∫ A( t ) ⋅ d l ∂t a © Amanogawa, 2006 – Digital Maestro Series Statics G bG Vba = − E ⋅ d l a ∫ = φb − φa 29 Electromagnetic Fields Note that for time-varying fields the line integral of the magnetic vector potential between two given points depends on the actual path of integration. In general: ∫ G G G bG A ( t ) ⋅ d l ≠ A( b, t ) − A ( a, t ) a Consider now the integral of the electric field along a closed path: Time-varying fields G G v∫ E ( t ) ⋅ d l ≠ 0 Statics G G v∫ E ( t ) ⋅ d l = 0 The closed path could be a metallic wire which confines the current due to moving electric charge. © Amanogawa, 2006 – Digital Maestro Series 30 Electromagnetic Fields The line integral of the electric field gives the work necessary to move a unit charge along the path of integration, under the influence of time-varying electric and magnetic fields. For a closed wire loop at rest, the work necessary to move a unit charge once around the loop is W = v∫ Force Charge G G G G G ⋅ d l = v∫ E ( t ) ⋅ d l = ∫ ∇ × ( E ( t ) ) ⋅ d S G G G ∂ B( t ) G ∂ =∫ − ⋅ d S = − ∫ B( t ) ⋅ d S S ∂t ∂t S ∂ =− Φ(t) ∂t Magnetic Flux © Amanogawa, 2006 – Digital Maestro Series 31 Electromagnetic Fields As a more general case, consider a wire loop in motion. The complete Lorentz force must be considered: W = e. m. f . = v∫ Force Charge G G G G G ⋅ d l = v∫ ( E ( t ) + v ( t ) × B ( t ) ) ⋅ d l G G G G = ∫ (∇ × ( E ( t ) + v ( t ) × B ( t ))) ⋅ d S G G G ∂ B( t ) G d G G = ∫− + ∇ × ( v ( t ) × B ( t )) ⋅ d S = − ∫ B ( t ) ⋅ d S ∂t dt G dB ( t ) dt Flux Φ(t) If the velocity of motion is constant, note that G G ∇ × ( v × B ( t )) = G G G G G G G G G G v∇ ⋅ B − B∇ ⋅ v + ( B ⋅ ∇ ) v − ( v ⋅ ∇ ) B = − ( v ⋅ ∇ ) B 0 0 © Amanogawa, 2006 – Digital Maestro Series 0 32