1 Time–Dependent Maxwell’s Equations

Electromagnetic Fields

Review: Time–Dependent Maxwell’s Equations

∂B ( t ) ∇ × E(t) = − ∂t ∂D ( t ) ∇ × H(t) = +J ∂t
∇ ⋅ D( t ) = ρ ∇ ⋅ B( t ) = 0 D( t ) = ε E ( t ) B(t ) = µ H ( t )

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Electromagnetic Fields

Electromagnetic quantities:

E H
Vector quantities in space

Electric Field Magnetic Field Electric Flux (Displacement)Density Magnetic Flux (Induction) Density Current Density Displacement Current Charge Density Dielectric Permittivity Magnetic Permeability
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D B J ∂D ∂t ρ ε µ

©Amanogawa, 2006 – Digital Maestro Series

Electromagnetic Fields

In free space:

ε = ε 0 = 8.854 × 10 −12 [ As/Vm] or [ F/m] µ = µ 0 = 4 π × 10 −7 [ Vs/Am] or [ Henry/m]
Ina material medium:

ε = ε r ε0 ; µ = µ r µ0 ε r = relative permittivity (dielectric constant) µ r = relative permeability
If the medium is anisotropic, the relativequantities are tensors:

 ε xx  ε r =  ε yx   ε zx

ε xy ε xz   ε yy ε yz   ε zy ε zz 

;

µ xx µ xy µ xz    µ r = µ yx µ yy µ yz     µ zx µ zy µ zz 
3Electromagnetic Fields

Electromagnetic fields are completely described by Maxwell’s equations. The formulation is quite general and is valid also in the relativistic limit(by contrast, Newton’s equations of motion of classical mechanics must be corrected when the relativistic limit is approached). The complete physical picture is obtained byadding an equation that relates the fields to the motion of charged particles. The electromagnetic fields exert a force to the law (Lorentz force):

F on a charge q, accordingF(t) = q E(t) +
Electric Force

q v ( t ) × B ( t ) = q  E ( t ) + v ( t ) × B ( t )  
Magnetic Force

where v(t) is the velocity of the moving charge.