Electromagnetic Fields
Review: Time–Dependent Maxwell’s Equations
G
G
∂B ( t )
∇ × E(t) = −
∂t
G
G
∂D ( t ) G
∇ × H(t) =
+J
∂t
G
∇ ⋅ D( t ) = ρ
G
∇ ⋅ B( t ) = 0
G
G
D( t ) = ε E ( t )
G
G
B( t ) = µ H ( t )
© Amanogawa, 2006 – Digital Maestro Series
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Electromagnetic Fields
Electromagnetic quantities:
Vector
quantities
in space
G
E
G
H
G
D
G
B
G
J
G
∂D
∂t
Electric Field
Magnetic Field
Electric Flux (Displacement) Density
Magnetic Flux (Induction) Density
Current Density
Displacement Current
ρ
Charge Density
ε
Dielectric Permittivity
µ
Magnetic Permeability
© Amanogawa, 2006 – Digital Maestro Series
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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]
In a 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 relative quantities are tensors:
 ε xx

ε r =  ε yx

 ε zx
ε xy ε xz 

ε yy ε yz 

ε zy ε zz 
© Amanogawa, 2006 – Digital Maestro Series
;
µ xx µ xy µ xz 


µ r = µ yx µ yy µ yz 


 µ zx µ zy µ zz 
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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 by adding an equation
that relates the fields to the motion of charged particles.
The electromagnetic fields exert a force
to the law (Lorentz force):
G
G
F(t) = q E(t) +
Electric Force
F on a charge q, according
G
G
G
G
G
q v ( t ) × B ( t ) = q  E ( t ) + v ( t ) × B ( t )
Magnetic Force
where v(t) is the velocity of the moving charge.
© Amanogawa, 2006 – Digital Maestro Series
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