GAUS
Elεctrostatics
S’ LA
W
Conductors
- a conductor has a constant
potential
- the volume charge density is
zero
- surface charge density can be
non-zero
- a closed conductor shields
the inside from the outside
(immunity)
- a grounded closed conductor
shields the outside from the
inside (emission)
Poisson
Laplace
= only for linear & isotropic media
Linear medium
Homogenous medium
Isotropic medium
Dielectrics
Xe is independent of E
- Xe is independent of
space coordinates
P has the same direction
as E (Xe is a scalar)
Magnetic
materials
Xm is independent of H
- Xm is independent of
space coordinates
M has the same direction
as H (Xm is a scalar)
Determining capacitance
Assume +Q and -Q ⇒ Find E (Gauss’ law) ⇒ Get V by
integrating E from -Q to +Q ⇒ C=Q/V
A simple medium is linear,
isotropic and homogeneous.
Stεady cμrrents
Determining resistance
Assume V0 between terminals ⇒ Find E ⇒ Find J=σE ⇒
Find I=∮sJ·ds ⇒ R=V0/I
Determining mutual inductance
L12=L21
Assume current I1 ⇒ Find B1 ⇒ Find Φ12=∫s1B12·ds ⇒ Find
Λ12=NΦ12 ⇒ L12=Λ12/I12
Determining self inductance
Assume current I ⇒ Find B ⇒ Find Φ=∫ṣB·ds ⇒ Find Λ=NΦ
⇒ L=Λ/I
E: Electric field intensity (V/m=N/C) B: Magnetic flux density (T=Wb/m2)
D: Electric flux density (C/m2)
H: Magnetic field intensity (A/m)
P: Polarization vector (C/m2)
M: Magnetization vector (A/m)
V: Electric scalar potential (V)
A: Magnetic vector potential (Wb/m)
We: Stored electric energy (J)
Φ Magnetic flux (Wb)
we: Electric energy density (J/m3) Λ: Flux linkage (Wb)
ε: Permittivity (F/m)
Wm: Stored magnetic energy (J)
χe: Electric susceptibility (unitless) wm: Magnetic energy density (J/m3)
C: Capacitance (F)
µ: Permeability (H/m)
⍴: Volume charge density (C/m3) L: Self inductance (H)
⍴s: Surface charge density (C/m2) L12: Mutual inductance (H)
⍴l: Line charge density (C/m)
J: Volume current density (A/m2)
Q: Charge (C)
Js: Surface current density (A/m)
σ: Conductivity (S/m)
I: Current (A)
Magnεtostatics
Ampère’s circuital law
ds has the direction of the (outwards) surface normal vector
The magnetic field can only change
the direction of motion of a charged
particle and not its speed and thus
not its kinetic energy.
= only for linear & isotropic media
Differential
(general)
Integral
(general)
Differential
(simple,
lossless,
source-free)
Phasor
(general)
Phasor
(simple material)
Phasor
(simple,
lossless,
source-free)
Phasor
(simple,
lossy,
source-free)
Fawaday’s låw
Ampère-Maxwell’s låw
Gauss’ (electric) låw
Gauss’ (magnetic) låw
DJ
Ha
Uniform
plane waves
plane waves
v
wa
ne
(real)
e
at
rel
Pla
Poynting’s complex vector:
Average power flow density:
s
ion
Poynting’s vector
- the magnitude of the wave vector is the wavenumber!
- the direction of the wave vector is the direction of propagation!
- the direction of the electric field is perpendicular to the wave vector!
- the direction of the magnetic field is perpendicular to the wave vector!
Cartesian
Boundary Conditions
!!! Evaluate at the boundary (z=0) !!!
The tangential components of the electric field E and
the magnetic flux density B are continuous across any
boundary:
Cylindrical
The normal components of the electric flux density D and
magnetic field intensity H are discontinuous by an amount
equal to the surface charge density ⍴s and surface current
density Js respectively:
Spherical
e
xw
A boundary between two lossless dielectrics has zero
surface charge and surface current
A perfect conductor has zero electric field and zero
time-varying magnetic field
Polarization
ll