ECE 330
ELECTROMAGNETIC THEORY
Review List for Exam Number 2
Electric charges and fields:
~
• Definition of electric field in terms of force, F~ = q E.
• Coulomb’s Law
• Inverse r2 dependence of electric field from point charge Q, or charge element dQ
• Linearity of Coulomb’s Law leads to superposition, integration over charge densities to get field.
• Charge densities ρv (Coul/m3 ), ρs (Coul/m2 ) , ρl (Coul/m)
- dQ = ρl dl Coulombs
- dQ = ρs dA Coulombs
- dQ = ρv dv Coulombs
• Electric force (Newtons) and electric fields (Newtons/Coulomb) in terms of integrals over charge
distributions
• Electric field lines: originating on positive charges, terminating on negative charges (+ −→ −).
• Infinite line charge field varies as 1r , infinite surface charge field does not vary with r, and electric
dipole field varies as r13
• Utilizing symmetry (including proper choice of coordinate systems) in order to simplify problems
Gauss’s Law:
• Gauss’s Law takes two forms:
Integral form
ZZ
~ · dA
~ = Qenclosed
E
0
Differential form
~ = ρv
∇·E
0
• Integral form of Gauss’s Law indicates that the net total flux (total number of field lines) exiting
from a closed surface is proportional to the charge contained within the surface.
• Differential form of Gauss’s Law represents the strength of the sink (termination point) or source
(origination point) of electric field lines at a point in space is proportional to the volume charge
density at that point in space.
• The closed surface of integration for the integral form of Gauss’s Law is known as a “Gaussian
Surface”.
• The electric field inside a closed equipotential surface can be shown to be identically zero. This
result can be used for electrostatic shielding and is known as the “Faraday Cage” effect.
Work, Potential Energy, Energy and Voltage:
• Electric charge in an electric field has potential energy (Joules).
• Work (Joules) is equal to change in potential energy (Joules) of a charge in an electric field.
• Work (Joules) is energy required to assemble a system of charges. This is also the electrical
potential energy of the charges (Joules).
• Electric potential is the potential energy per unit charge (one Coulomb) in an electric field,
Joules/Coulomb = Volt.
• Voltage is difference in potential energy of 1 Coulomb between two points (Volt=Joules/Coulomb).
• Potential energy of charge Q in potential V is Q V (Joules).
• Potential between two points is voltage, equal to potential energy (Joules) of 1 Coulomb of charge
at one point relative to the second point,
is found by integrating the electric field along a path
R
~ · d~l.
from one point to the second V = − E
• A conservative field is a static (non time-varying) field. In a conservative field the integral of the
electric field between two points is independent of path. This leads to a uniquely defined voltage.
If a field is time-varying (non-conservative) then the path integral of the electric field between
two points is path dependent and voltage is not uniquely defined.
• Conservative Fields
~ =0
1. ∇ × E
H
~ · d~l = 0 −→ Kirchhoff’s Voltage Law
2. E
~ = −∇ V
3. E
~ is orthogonal (perpendicular, normal) to equipotential ( V =constant) surfaces.
• E
~ lines.
• Equipotential surfaces are normal to E
~ lines are normal to the surface of a good conductor (equipotential surface).
• E
• Energy in system of charges
WE =
1
1X
qk Vk =
2
2
ZZZ
ρ V dv
k
• Energy (Joules) in an electric field is proportional to |E|2 : WE =
0
2
RRR
E 2 dv
Dielectrics and Polarization:
• Electric dipole moment in Coulomb-meters
p~ = lim Qd~
|d|→0
• p~ points from − to +. In this way an electric dipole p~, when placed in an electric field, is parallel
~
to the electric field vector E.
• Potential of electric dipole varies as
1
r2 ;
~ = −∇V , field of electric dipole varies as
From E
1
r3 .
• Dielectric in electric field will become polarized.
• Polarization P~ is Dipole moment per unit volume in Coulombs per square meter
1 X
p~i
∆v→0 ∆v
i
P~ = lim
• Potential due to polarized dielectric
Vp =
1
4π0
I ~
ZZ
~
P · dA
1
∇ · P~
−
dv
r
4π0
r
• Equivalent polarization charge densities
ρpolarization, surf ace = P~ · ân
ρpolarization, volume = −∇ · P~
~ = 0 E
~ + P~ = E.
~
• Derivation of Electric Flux Density D
~ = ρv .
• Gauss’s Law ∇ · D
~ field at any point in dielectric.
• Polarization reduces E
~ where χE is the electric susceptibility of the material.
• P~ = 0 χE E
~ = 0 (1 + χE )E
~ = E.
~
• D
• Permittivity = r 0 where r is the relative permittivity (dielectric constant).
• r = 1 + χ E .
~ is a macroscopic field.
• D
~ is a microscopic field .
• E
• When the electric susceptibility χE is large (large dielectric constant r ), a small electric field
will cause a large polarization of the dielectric. Dielectric is susceptible to the field.
• Permittivity is a measure of how well strongly a material will respond to an electric field; describes
the electrical properties of the material.
• Definitions of homogeneous, linear and isotropic material.
Boundary Conditions:
• Boundary Conditions between two dielectrics (or dielectric and free space)
D1n − D2n = ρsurf ace
E1t = E2t
• Boundary Conditions between a dielectric and a good conductor
Dn = ρsurf ace
E1t = 0
• Electric field inside a perfect conductor is zero since charges are free to redistribute themselves
~ = −∇ V .
until potential is constant. If potential is constant then field is zero from E
• Charges are located on the surface of a perfectly conducting material (like charges repel so charges
will attempt to get as far apart as possible).
~ indicates that E
~ must be normal to surface of a perfect conductor.
• Boundary condition on E
Capacitance:
• Capacitance is a measure of the capability of a system to store electric energy.
• Capacitance (Farads) defined to be charge per unit voltage (Coulomb/volt), C =
Q
V .
• Capacitance of a parallel plate capacitor is
C=
A
A
= r 0
d
d
where A is the area of the plates, d is the plate separation, and r is the dielectric constant of the
dielectric. Note that this result neglects fringing, and only considers the electric field between
the plates, assuming this field is uniform and constant and normal to the plates.
• Energy stored in a capacitor is
WE =
1
1
CV 2 = QV
2
2
where V is the voltage across the capacitor (this is also the potential of one plate relative to the
second).
• Energy in the field
1
2
ZZZ
1
0
2
ZZZ
WE =
=
~ ·E
~ dv
D
|E|2 dv
• Stored electric energy −→ Capacitance; Capacitance −→ stored electric energy.
Poisson’s and Laplace’s Equation:
~ = −∇ V and ∇ · E
~ =
• Derived from E
ρv
0
• Poisson’s equation ∇2 V = − ρ0v
• When ρv is zero, Laplace’s equation results ∇2 V = 0.
• Fundamental equation of electrostatics. Solved subject to boundary conditions (potentials, voltage, etc.).
• More useful than either Gauss’s or Coulomb’s Law, since we generally have control over potentials,
not over charge distributions.
Important Equations and Relationships
(Note: These equations will NOT be provided for the exam)
Coulomb’s Law
q1 q2
âr
F~ =
4π0 r2
ZZZ
q1
ρv dv
F~ =
âr
4π0
r2
Electric Field
~ =
E
1
4π0
ZZZ
ρv dv
âr
r2
~
~ =F
E
q
Integral form of Gauss’s Law
ZZ
~ · dA
~ = Qenclosed
E
0
ZZ
~ · dA
~ = Qenclosed
D
Differential form of Gauss’s Law
~ =
∇·E
ρv
0
~ = ρv
∇·D
Potentials, Potential Energy, Work
ZZZ
1X
1
qk Vk =
ρv V dv
WE =
2
2
k
ZZZ
ZZZ
1
~ ·E
~ dv = 1
D
WE =
0 |E|2 dv
2
2
Electrostatic Potential
Z
V =−
~
~ · dl
E
~ = −∇ V
E
1
4π0
Flux Density and dielectrics, Capacitance
ZZZ
V =
ρv dv
r
~ = 0 E
~ + P~ = E
~
D
= r 0
Boundary Conditions
D1n − D2n = ρsurf ace
E1t = E2t