Important Stuff From Physics 442
Vector Identities:
Vector derivatives (gradient, divergence, and curl):
Laplacian Operator:
Vector Identities:
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
Second derivatives:
(9)
(10)
(11)
Chapter 7
Ohm's Law:
where 
is the conductivity, related to the resistivity
by 
.
Somtimes the symbol 
is used in place of for the
resistivity.
The resistivity has units of 
m.
Electromotive Force:
Faraday's Law:
Motional Emf:
Lenz's Law:
When a magnetic change is made, emf's are induced that oppose the change.
Self Inductance:
Mutual Inductance
Energy Stored in Inductive Circuits:
Magnetic Energy Density:
Magnetic Diffusion Coefficient:
Displacement Current:
Maxwell's Equations:
and 
:
Boundary Conditions:
where 
is the free surface charge per area and
where 
is the free surface current per length.
Charge and Energy Conservation:
where
and V:
Lorentz Gauge and Wave Equations:
Momentum Density:
Maxwell Stress Tensor (pressure and tension):
Time-Dependent Circuits
Loop Rules:
1. The voltage difference 
along a connecting wire is
taken to be zero.
2. When you traverse a battery, or some other source of emf,
in the direction that it tries to pump current, write down
If you traverse it opposite to its pumping direction write
3. If you traverse a capacitor from its negative side to
its positive side write
if you traverse it from plus to minus write
4. If you traverse a resistor in the direction of the current
flow through it write
if you traverse it opposite to the current write
5. If you traverse an inductor in the direction of the current
flow through it write
if you traverse it opposite to the current write
6. At a current junction, incoming current must equal outgoing current.
Circuit Energy Formulas:
Undriven LRC Circuit:
Steady State AC Circuits (ELI the ICEman):
The complex impedance Z of a circuit is obtained by treating
inductors and capacitors as if they were resistors in the network.
To interpret the meaning of Z it must be put in magnitude-phase
form and used as follows:
Useful Identities:
RMS Quantities and Average Power:
Chapter 8
Wave Equation in a Linear Medium:
Solutions of the 1-d Wave Equation:
Standard Sinusoidal Wave Functions:
Energy Density, Poynting Flux, Intensity, Momentum Density:
Disperson Relation and Index of Refraction n:
Reflection and Transmission at Normal Incidence:
Electromagnetic Waves in a Conductor:
Reflection and Transmission from a Conductor:
where
Phase and Group Velocities:
Deriving a Dispersion Relation:
Start with Maxwell's curl equations in plane wave form:
Then from somewhere else (Newton's second law, quantum mechanics, Ohm's law, etc.)
find a relation between 
and 
.
(It is sometimes helpful in this regard to recall that current
density and particle velocity are related by
where n is the number of particles of charge q per
unit volume.)
Use this relation to eliminate from the plane-wave
Maxwell's equations and re-express the right-hand-side of
the curl-
equation in terms of a frequency-dependent
dielectric constant 
.
The electromagnetic dispersion relation is then just our old
friend
Waveguide Guide: and V
Wave Equations:
Lorentz Condition:
and :
Boundary Conditions:
Dispersion Relation in Rectangular Coordinates:
It is the same for traveling, standing, or combined standing/traveling (waveguide) waves:
Rectangular Waveguide:
The wave guide is infinitely long in the x-direction and goes from 0 to a in the y-direction and from 0 to b in the z-direction.
Waveguide Waveform:
Two Mode Types
The rather complicated vector eigenvalue problem involved in
waveguides turns out to be simpler than might have
been thought.
There are two distinct types of solutions.
One type has 
, i.e., the mode electric field
only has components perpendicular (or transverse)
to the long axis of the wave guide.
These are the TE modes.
The second type has 
, i.e., the mode magnetic field
only has components perpendicular (or transverse)
to the long axis of the wave guide.
These are the TM modes.
If you work with and , this reduces
the number of vector components you need to find from
6 to 5, but if you work with and V, as
we are doing here, it reduces the number from 4 to 2.
TE
Modes (
and V=0):
The condition that turns out to require
that
and V=0.
Hence, we only have to find 
and 
.
Applying both the boundary conditions and the
Lorentz gauge condition leads to
and
with
This condition relates the two amplitudes, but does not determine the overall magnitude. This is determined by the person who shoots the energy into the waveguide.
TE
and TE
Special Cases:
If you set either m=0 or n=0 in the amplitude relation above you will see that only one of the vector field components survives.
TM Modes (
and 
):
The condition that turns out to require
that
and .
Hence, we only have to find 
and V(y,z).
Applying both the boundary conditions and the
Lorentz gauge condition leads to
TM and TM Special Cases:
Setting m=0 or n=0 in the formula for 
makes the fields vanish,
so 
and 
for TM modes.
TEM Mode:
In a completely hollow guide waves with both 
and 
parallel to the axis of the guide are impossible.
But with a conductor along the axis these waves are possible.
Their dispersion relation is simply
with 
parallel to the axis of the guide.
The electric field points outward from the central conductor
and terminates on the outer surface of the guide while
the magnetic field circulates around the central conductor
and runs parallel to the outer conductor.
Hence, they are just about like free space waves with ,
, and mutually perpendicular.
In a cylindrical guide (a coaxial cable) of radius a
these waves are
described in cylindrical coordinates by
where 
is the electric field amplitude at r=a.
Chapter 9
Retarded Potentials and Retarded Time:
Electric Dipole Radiation, Spherical Coordinates:
If there is an oscillating dipole at the origin of spherical
coordinates 
, then
the radiation fields and power it produces (far away from the dipole) are
where 
is the retarded time, 
and where
the double-dot means 
.
The average radiated power is
Note that
P is the instantaneous power through a distant sphere
while <P> is the average power through the same distant sphere.
If the radiation source is a non-relativistic particle,
in the power formulas above use 
, where
is the vector position of the particle and
a is the magnitude of the acceleration vector of the particle, e.g.,
Griffiths Chapter 10 (Relativity):
The Basics: There are two frames: the unprimed frame is stationary while the primed frame moves with speed v along the x-axis. If a variable is primed, it indicates that it was measured in the moving frame. Unprimed variables indicate quantities measured in the stationary frame.
Lorentz Transformation:
The inverse transformation is easy: just un-prime the
left side, prime the right side, and change v to -v.
Four-Vector Formulation:
Relativity is elegantly described by combining space and time
(and other physical quantities) into 4-vectors.
The space-time 4-vector is
In this formulation the Lorentz transformation involves the following rank-2
tensor (matrix):
Using this tensor and the space-time 4-vector the
Lorentz transformation is
Other useful 4-vectors, which all transform exactly the same as 
are:
where proper time 
is time as measured in the frame
of a moving particle and where 
is the charge density
the frame where locally the current density is zero.
In the cases of momentum and current, these are not just
formal definitions.
It is the 4-energy and the 4-momentum that are conserved
in collisions and are governed by Newton's second law;
and it is the 4-charge density and the 4-current that
go into Maxwell's equations to give and
in the inertial frame where the equations are being used.
Hence we usually forget about the proper velocity and just
write
Again: these quantities transform between moving inertial frames
just like the 4-vector.
Lorentz Invariants:
As transformations between different inertial frames
are made using the 4-vector formulation there is
a scalar quantity whose value never changes.
For any 4-vector 
this invariant quantity is
Relativistic Kinematics (Collisions):
In collisions and disintegrations the relativistic
momentum and energy are conserved.
The conservation of momentum and energy
coupled with the invariant relation
makes it possible to
find conditions after a collision in terms of
conditions before.
Relativistic Dynamics:
Newton's second law retains its validity in mechanics,
provided that it is the relativistic momentum that
is used for 
:
Note that 
in this equation is just the
force we are used to.
For instance, in the case of a particle in
an electromagnetic field
To solve problems with this equation, solve for the
momentum , then use the definition of the
components of in terms of 
to find the particle velocity.
Except in simple cases this last step is a nightmare.
The transformation law for forces is a little complicated
in general, but in one special case it is simple.
If the particle is momentarily at rest in the unprimed frame
and the force on it is observed from a moving
primed frame then
Relativistic Electrodynamics:
As pointed out above, charge density and current get mixed
together as we transform between frames, with 
transforming as a 4-vector.
The 4-vector formulation also makes
the continuity equation take a particularly simple form:
Just like and ,
the fields and also get mixed together, but
they don't transform like 4-vectors.
Instead and form a rank-2 tensor (matrix).
This formulation is hard to use in practical problems, so
the Lorentz transformations for the usual case of
a frame moving in the x-direction at speed v will be
given separately for each component.
The Electromagnetic Field Tensor:
The electric and magnetic fields fit into the 4-formulation
of relativity as components of two rank-2 4-tensors,
and 
:
Maxwell's Equations:
The 4-form of Maxwell's equations is particularly elegant.
In spite of their elegance, the equations are so cryptically packaged
that except for fundamental theoretical work they are not used much.
The Potentials V and in Relativity:
The scalar potential and the vector potential also (surprise)
form a 4-vector:
In terms of 
Maxwell's equations are even more compact than they are
in terms of
and .
Using the Lorentz guage the connections between the fields and the potentials
becomes symbolically very simple
The Lorentz gauge becomes symbolically simple:
And finally, if we define a special symbol for the wave-equation operator
(which also looks elegant in the 4-formulation)
then Maxwell's equations in terms of V and can
be written in the most compact form of all:
