Maxwell's Equations Applied Electromagnetic Waves 2

Continuity Equation (cont.) ∂ρ v ∇⋅ J = − ∂t Integrate both sides over an arbitrary volume V: ∂ρ v ∫V ∇ ⋅ J dV = V∫ − ∂t dV Apply the divergence theorem: =  dS ∫ ∇ ⋅ J dV ∫ J ⋅ nˆ = V S iout S (current flowing out of V) Hence: ∂ρ v iout = ∫ − dV ∂t V V n̂ (outward normal) 14 Continuity Equation (cont.) ∂ρ v iout = ∫ − dV ∂t V S Physical interpretation: V n̂ Right-hand side: ∫− V ∂ρ v ∂ ∂Qencl dV = − ∫ ρ v dV = − ∂t ∂t V ∂t Hence iout ∂Qencl = − ∂t or (This assumes that the surface is stationary.) ∂Qencl iin = ∂t 15 Continuity Equation (cont.) ∂Qencl iin = ∂t J Qencl This implies that charge is never created or destroyed. It only moves from one place to another! 16 Maxwell’s Equations (cont.) Time - Dependent ∂B ∇ ×E = − ∂t ∂D ∇× H = J + ∂t ∇ ⋅B = 0 ∇ ⋅ D = ρv Time -Independent (Statics) ∇= ×E 0 ∇= ⋅ D ρv Statics decouples E and H ⇒ ∇× = H J ∇= ⋅B 0 E comes from ρ v and H comes from J Note: Regular (not script) font is used for statics, just as it is for phasors. 17 Maxwell’s Equations (cont.) Time-harmonic (phasor) domain ∂ → jω ∂t ∇ × E = − jω B ∇ × H= J + jω D ∇ ⋅ B =0 ∇ ⋅ D = ρv 18 Constitutive Relations The characteristics of the media relate D to E and H to B Free Space D = ε 0E (ε 0 = permittivity ) B = µ0 H (µ0 = permeability ) ε 0  8.8541878 ×10-12 [F/m] µ0 = 4π ×10-7 [H/m] (exact*) c= 1 µ 0ε 0 c ≡ 2.99792458 × 108 [m/s] *Prior to 2019 (exact value that is defined) (since 1983) 19 Constitutive Relations (cont.) Definition of the Amp*: *Prior to 2019 #1 I d Two infinite wires carrying DC currents #2 Fx2 I From ECE 3318: I 2 µ0 Fx 2 = 2π d x −7 µ= 4 π × 10 [H/m] 0 Definition of I =1 Amp: Fx 2 = 2 × 10−7 [ N/m ] when d = 1 [ m] 20 Constitutive Relations (cont.) Free space, in the phasor domain: D = ε0E (ε 0 = permittivity) B = µ0 H ( µ0 = permeability ) This follows from the fact that aV ( t ) ⇔ aV (where a is a real number) 21 Example Given the following electric field E in free space: 1 = E ( t ) θˆ ( E0 cos (ωt − k0 r + φ0 ) ) sin θ r Find the magnetic field H. k 0 = ω µ 0ε 0 ∇ × E = − jω B ∇ × H= J + jω D ∇ ⋅ B =0 ∇ ⋅ D = ρv In the phasor domain:  1  E = θˆ  E0 e − jk0 r e jφ0  sin θ  r  ∇ × E = − jω B Hence ∇ × E = − jωµ0 H = ∇ × E rˆ H = 1 − jωµ0 ∇× E [ V/m] = H 1 − jωµ0 1  ∂ ( Eφ sin θ ) ∂Eθ −  ∂θ ∂φ r sin θ  ∇× E  1  1 ∂Er ∂ ( rEφ )  ˆ 1  ∂ ( rEθ ) ∂Er  − −  + θ   +φ   θ ∂ φ ∂ ∂ ∂θ  r sin r r r     (no φ variation) 22 Example (cont.) 1 E = θˆ E0 e jφ0 e − jk0 r sin θ r ( ) = H 1 − jωµ0 ∇× E 1  ∂ ( rEθ )  ˆ ∇ × E =φ   r  ∂r  1  ˆ 1 jφ0 − jk0 r      jφ0 1 −= jk0 r  sin − H e E jk e φ θ ( ) 0 0   e  sin θ    ∂  r  E0 e − j r ωµ   0 r 1     ˆ   =φ  ∂r r      ∂ e − jk0 r    k0  1  − jk0r jφ0 1 ˆ jφ0 ˆ H φ E sin θ =  = φ e E0 sin θ   0 e e  r  ∂r    ωµ0  r  1 = φˆ e jφ0 E0 sin θ ( − jk0 ) e − jk0 r r ( )   k0   1 ˆ H φ  E0  =   sin θ cos (ωt − k0 r + φ0 )   ωµ0   r [ A/m ] 23 Example (cont.) Alternative approach (in the time domain directly): 1 = E ( t ) θˆ ( E0 cos (ωt − k0 r + φ0 ) ) sin θ r ∇ ×E = − ∂B ∂t ∂B = −∇ × E ∂t (no φ variation) ∇ ×E rˆ 1  ∂ ( Eφ sin θ ) ∂Eθ  −   ∂θ ∂φ  r sin θ  + θ 1  1 ∂Er ∂ ( r Eφ )  ˆ 1  ∂ ( rEθ ) ∂Er  − −  +φ  ∂r  ∂θ  r  sin θ ∂φ r  ∂r 1  ∂ ( rEθ )  ∇ × E = φˆ  r  ∂r  [ V/m] ∂B ∂t ∂D ∇× H= J + ∂t ∇⋅ B =0 ∇ ⋅D = ρv ∇ ×E = −    1   r E t k r ∂ − + ω φ θ cos sin ( )  0 0 0   1    r   ˆ =φ  r ∂r       ∂ ( cos (ωt − k0 r + φ0 ) )  1 ˆ = φ E0 sin θ   r r ∂   1 = φˆ E0 sin θ ( − k0 ) ( − sin (ωt − k0 r + φ0 ) ) r 24 Example (cont.) ∂B = −∇ × E ∂t 1 = ∇ × E φˆ E0 sin θ ( k0 ) ( sin (ωt − k0 r + φ0 ) ) r So All fields must be pure sinusoidal waves in the time-harmonic steady state. 1 ∂B = −φˆ k0 E0 sin θ ( sin (ωt − k0 r + φ0 ) ) r ∂t 1 1 B= −φˆ k0 E0 sin θ ( − cos (ωt − k0 r + φ0 ) ) + C ( r, θ , φ ) ω r B = µ0 H H  k0  1 ˆ φ  E0  sin θ ( cos (ωt − k0 r + φ0 ) )  ωµ0  r [ A/m] 25 Example (cont.) 1 = E ( t ) θˆ ( E0 cos (ωt − k0 r + φ0 ) ) sin θ r H  φˆ  E0  [ V/m] k0  1  sin θ ( cos (ωt − k0 r + φ0 ) ) ωµ0  r z [ A/m] This describes the far-field radiation from a small vertical dipole antenna. y x 26