Physics

Antiferromagnetism
- magnetism in which adjacent ions that act as tiny magnets spontaneously
align

themselves at relatively low temperatures into
opposite (antiparallel) arrangements so that overall the

solid does not exhibit magnetism; occurs with
manganese oxide

Auger
Effect - outer shell electron replaces removed inner electron, releasing
energy as X-ray or ejecting

another electron

Cherenkov
Radiation - light produced by charged particles when they pass through an
optically transparent

medium at speeds greater than the speed of light in
that medium; named for 1958 Soviet Nobel winner

Compton
Effect - increase in wavelength of X-rays that have been elastically scattered
by electrons;

principal way in which radiant energy is absorbed in
matter; named for 1927 American Nobel

winner

Diamagnetism
- magnetism characteristic of materials that line up at right angles to a
nonuniform magnetic

field and that partially expel from their interior
the magnetic field in which they are placed

Doppler
Effect - the apparent difference between the frequency at which sound or light
waves leave a

source and that at which they reach an observer,
caused by the relative motion of the observer and the

wave source; red shift if receding, blue shift if
approaching

Dynamo
Effect - production of the Earth's main magnetic field by an electric field in
the core

Edison
Effect - thermionic emission; discharge of electrons from heated materials;
used as electron source

in electron tubes

Faraday
Effect - rotation of the plane of polarization of a light beam by a magnetic
field

Ferroelectricity
- spontaneous electric polarization occurs in crystals with perovskite
structure, such as

barium titanate

Ferromagnetism
- electrically uncharged materials strongly attract others

Hall
Effect - development of a transverse electric field in a solid material when
it carries an electric current

and is placed in a magnetic field that is
perpendicular to the current

Hysteresis
Effect - magnetization of ferromagnetic substances lags behind the magnetizing
field

Hawking
Radiation - when a particle pair is created near a black hole, one falls into
the hole and the other

escapes as radiation

Josephson
Effect - flow of electric current between two pieces of superconducting
material separated by a

thin layer of insulating material

Joule-Thompson
Effect - cooling of a gas as it undergoes adiabatic expansion

Lamb
Shift - separation of energy in nearly coincident electron levels of hydrogen;
named for 1955

American Nobel winner

Lorentz-Fitzgerald
Contraction - space contraction; the shortening of an object along the
direction of its

motion relative to an observer

Meissner
Effect - the expulsion of a magnetic field from the interior of a material
that is in the process of

becoming a superconductor

Mikheyev-Smirnov-Wolfenstein
Effect - neutrinos with mass oscillate among flavor states

Mossbauer
Effect - also called recoil-free gamma-ray resonance absorption; nuclear
process permitting the

resonance absorption of gamma rays; radioactive atoms
are imbedded in a crystal lattice, allowing

very precise measurements; named for 1961 German
Nobel winner

Paramagnetism
- magnetism characteristic of materials weakly attracted by a strong magnet

Paschen-Back
Effect - Zeeman spectral patterns are lost at magnetic separations for natural
doublets,

triplets, etc.

Peltier
Effect - the cooling of one junction and the heating of the other when
electric current is maintained

in a circuit of material consisting of two dissimilar
conductors

Photoelectric
Effect - charged particles are released from a material when it absorbs
radiant energy

Quantum
Tunneling - barrier penetration; particle travels through a barrier despite
the fact that the

particle's presence in the barrier is forbidden by
classical physics

Raleigh
Scattering - dispersion of electromagnetic radiation by particles that have a
radius less than one-

tenth the wavelength of the radiation; angle of
sunlight varies inversely as fourth power of

wavelength so blue is scattered most and sky appears
blue

Raman
Effect - change in wavelength of light that occurs when a light beam is
deflected by molecules;

named for 1930 Indian Nobel winner

Seebeck
Effect - production of an electromagnetic force and consequently an electric
current in a loop of

material consisting of at least two dissimilar
conductors when two junctions are maintained at

different temperatures; applicable in thermocouples

Stark
Effect - splitting of spectral lines in an electric field; named for 1919
German Nobel winner

Tachyons
- particles that travel faster than the speed of light

Tyndall
Effect - scattering of light by a colloid

Voigt
Effect - a constant magnetic field applied to a transparent gaseous medium
produces a birefringence

such that the vapor has different indices of
refraction in different directions

Zeeman
Effect - splitting of spectral lines in a magnetic field; named for 1902 Dutch
Nobel winner

Equations
of Motion:

v = v_{0 }+ at

x = x_{0} = v_{0}t + at^{2}/2

v^{2} = v^{0}2 + 2a(x - x_{0})

x - x_{0} = ((v_{0} + v)/2)t

v = dx/dt

a = dv/dt

Projectile
Motion in 2D

x = (v_{0}cos(a_{0})t

y = (v_{0}sin(a_{0}))t
- gt^{2}/2

v_{x} = v_{0}cos(a_{0})

v_{y} = v_{0}sin(a_{0})
- gt

Radial
acceleration = v^{2}/R ; period
= 2pR/v

Relative
velocity: v_{P/A} = v_{P/B} + v_{B/A}

Newton's
First Law states that when the vector sum of all forces acting on a body is
zero, the body is in

equilibrium; in inertial frames of reference, a body
in motion will stay in motion and a body at rest will stay

at rest.

Newton's
Second Law states that F = ma.

Unit
of force is Newton = 1 kg*m/s^{2}

Weight
is the gravitational force exerted on an object.

Newton's
Third Law states that two bodies exert forces on each other that are equal in
magnitude and

opposite in direction; action equals reaction.

Free-body
diagrams are used to show forces acting on a body.

The
coefficient of kinetic friction is directly proportional to the normal force
(perpendicular to interaction surface).

The
acceleration toward the center in uniform circular motion has magnitude v^{2}/R.

The
fundamental forces are gravitational, electromagnetic, strong, and weak.

Work
= F*s*cos(f)
= dot product of F and s vectors

Kinetic
energy is the amount of work required to accelerate a particle from rest to
speed v = mv^{2}/2

Unit
of work is the Joule = 1 N*m = 1 kg*m^{2}/s^{2}

Work
energy theorem: W_{total} = K_{2} - K_{1}

Power
is the time rate of doing work; P = dW/dt

Unit
of power is the watt = 1 J/s = 1 kg*m^{2}/s^{3}

Potential
energy -DU
= mgy_{1} - mgy_{2}

Elastic
work by compressed spring: W = kx_{1}^{2}/2 - kx_{2}^{2}/2

K_{1}
+ U_{1} +W_{other} = K_{2} + U_{2}

For
a conservative force, the work-kinetic energy relation is completely
reversible and can be represented by a

potential-energy
function; F(x) = -dU/dx

Momentum:
p = mv

SF
= dp/dt

Impulse:
J = SF Dt
= p_{2} - p_{1}

Momentum
equals the impulse that accelerated a particle from rest to its present speed.

Total
momentum: P = p_{A} + p_{B} + ...

Each
component of total momentum is conserved if the net external force is zero.

In
an elastic collision, the initial and final total kinetic energies are equal
and the initial and final relative velocities

have
the same magnitude.

In
an inelastic collision the final total kinetic energy is less than the initial
total kinetic energy (if same final velocity,

it
is completely inelastic).

Center
of mass: x_{center} = (Smx)/Sm ;
y_{center} = (Smx)/Sm

Total
momentum equals total mass times velocity of center of mass.

F_{external}
= Ma_{center of mass}

Angular
velocity: w
= dq/dt

Angular
acceleration: a
= dw/dt

Equations
of motion of rigid body about a fixed axis with constant angular acceleration

q
= q_{0}
+ w_{0}t
+ at^{2}/2

q
- q_{0}
= (w_{0}
+ w)t/2

w
= w_{0}
+ at

w^{2}
= w_{0}^{2}
+ 2a(q
- q_{0})

v
= rw

a_{tangential}
= ra

a_{radial
(centripetal)} = v^{2}/r = w^{2}r

Moment
of inertia: I = Smr^{2}

Kinetic
energy: K = Iw^{2}/2

Parallel
axis theorem: I_{parallel axis} = I_{cm} + Md^{2}

Moments
of inertia

Slender rod, axis through center: ML^{2}/12

Slender rod, axis through one end: ML^{2}/3

Plate, axis through center: M(a^{2}+b^{2})/12

Plate, axis along edge b: Ma^{2}/3

Hollow cylinder: M(R_{1}^{2} + R_{2}^{2})/2

Solid cylinder: MR^{2}/2

Thin-walled hollow cylinder: MR^{2}

Solid sphere: 2MR^{2}/5

Thin-walled hollow sphere: 2MR^{2}/3

Torque:
t = Fl
(l is lever arm) ; counterclockwise
positive; torque equals cross
product of r and F

St = Ia

Translational
and rotational kinetic energy: K = Mv_{cm}^{2}/2 + I_{cm}w^{2}/2

If
the rigid body rolls without slipping, v_{cm} = Rw

Work:
W = tDq

Work-energy
theorem for rotation of rigid body: W = Iw_{2}^{2}/2
- Iw_{1}^{2}/2

Power:
P = tw

Angular
momentum: L = Iw
= cross product of r and p, with p = mv

St = dL/dt

If
net torque of external forces is zero, the total angular momentum is zero.

For
a rigid body in equilibrium, the vector sum of forces is zero and the sum of
torques about any point is

zero.

The
weight can be assumed to be concentrated at the center of gravity, which is
the center of mass.

Hooke's
Law states that in elastic deformations, stress is proportional to strain;
elastic modulus =

stress/strain.

Tensile
stress is tensile force per unit area, and tensile strain is fractional change
in length.

Young's
modulus is the ratio of tensile stress to tensile strain; Y = (F_{t}/A)/(Dl/l_{0})

Compressive
stress and strain are defined the same way as tensile stress and strain.

Pressure
in fluid is force per unit area.

Unit
of pressure and stress is pascal = N/m^{2}

Bulk
modulus is the negative of the ration of pressure change (bulk stress) to
fractional volume change;

B = -Dp/(DV/V_{o})

Compressibility
is the reciprocal of the bulk modulus.

Shear
stress is force per unit area for a force applied parallel to a surface.

The
shear modulus is the ratio of shear stress to the shear strain angle; S =
(F/A)/f

The
proportional limit is the maximum stress for which stress and strain are
proportional; Hooke's law is

invalid beyond this.

The
elastic limit is the stress beyond which irreversible deformation occurs, and
the breaking (or ultimate)

strength is the stress at which the material breaks.

Newton's
law of gravitation: F = Gm_{1}m_{2}/r^{2}

Acceleration
due to gravity near Earth's surface is Gm_{E}/R_{E}^{2}

Gravitational
potential energy: U = - integral of F = -Gm_{1}m_{2}/r

Speed
of satellite: v = sqrt(Gm/r)

Period
of satellite: T = 2pr/v

Johannes
Kepler's Laws

Each planet moves in an elliptical orbit with the sun at one focus.

A line from the sun to a given planet sweeps out equal areas in equal
times.

The periods of the planets are proportional to the 3/2 powers of the
major axis lengths of their

orbits.

Weight
is 0.3% less at equator.

If
a nonrotating spherical mass distribution has radius less than the
Schwarzschild radius, 2GM/c^{2} , the

gravitational interaction prevents anything including
light from escaping; this is a black hole.

Periodic
motion occurs whenever a body has a stable equilibrium position and a
restoring force or torque.

Frequency
is inverse of period.

Angular
frequency: w
= 2pf

In
simple harmonic motion (SHM), the net force is a restoring force that is
directly proportional to the

displacement; F
= -kx ; a = F/m

The
projection on the horizontal axis of a rotating vector (using the circle of
reference construction) called

the phasor represents the actual motion of a body in
SHM.

w
= sqrt(k/m)

x
= A*cos(wt
+ f) (amplitude
A and phase angle f)

E
= kA^{2}/2 = constant

Angular
simple harmonic motion: w
= sqrt(K/I)
(torsion constant K)

Pendulum:
w = sqrt(g/L)

Physical
pendulum: w
= sqrt(mgd/I)

Damping
force: F = -bv

Damped
oscillation: critically damped if b = 2sqrt(km); w'
= sqrt(k/m - b^{2}/(4m^{2}))

Forced
oscillation occurs when a sinusoidally varying driving force is added to a
damped harmonic

oscillator.

Density:
p = m/V

Specific
gravity is ratio of density of a material to density of water.

Pascal's
law states that pressure applied to the surface of an enclosed fluid is
transmitted undiminished to

every portion of the fluid.

Gauge
pressure is the difference between absolute pressure and atmospheric pressure.

Pressure
at depth h of incompressible liquid is pressure at surface plus p*g*h.

Archimedes'
principle states that a fluid exerts an upward buoyant force on an immersed
body equal to the

weight of the fluid the body displaces.

Surface
tension is the force per unit length across a line on the surface.

An
ideal fluid is incompressible and has no viscosity.

A
flow line is the path of a fluid particle and a streamline is a curve tangent
at each point to the velocity

vector at that point.

A
flow tube is a tube bounded at its sides by flow lines.

In
laminar flow, layers of fluid slide smoothly past each other.

In
turbulent flow there is great disorder and a constantly changing flow pattern.

Equation
of continuity: A_{1}v_{1} = A_{2}v_{2}

Volume
flow rate: Av = dV/dt

Bernoulli's
equation relates pressure, flow speed, and elevation for steady flow in an
ideal fluid;

p_{1} + pgy_{1} + pv_{1}^{2}/2 = p_{2}
+ pgy_{2} + pv_{2}^{2}/2

The
viscosity of a fluid characterizes its resistance to shear strain; in a
Newtonian fluid the viscous force is

proportional to strain rate.

Poiseuille's
equation for total volume rate in a cylindrical pipe; dV/dt = p/8 * (R^{4}/n)((p_{1}-p_{2})/L)

Stokes'
law for a sphere moving through a fluid: F = 6pnrv

T_{F}
= 9/5 T_{C} + 32 ; T_{C}
= 5/9 (T_{F} - 32) = T_{k} - 273.15

The
triple point of water is at 273.16 K

DL
= aL_{0}DT

DV
= 3aV_{0}DT
for solids

Tensile
stress: F/A = -YaDT

Quantity
of heat required to raise the temperature of a material is mass times specific
heat capacity times

change in temperature.

Mass
= number of moles times molecular mass

The
Rule of Dulong and Petit states that the molar heat capacities of many solid
elements are

approximately 25 J/(mol*K).

Quantity
of heat required to change a material to a different state is mass times heat
of

fusion/vaporization/sublimation.

Conduction
is transfer of energy without bulk motion, convection involves mass motion,
and radiation is

energy transfer through electromagnetic radiation.

Heat
current for conduction: H = dQ/dt
= kA(T_{H} - T_{C})/L
(thermal conductivity k)

Heat
current for radiation: H = AesT^{4} (emissivity e, Stefan-Boltzmann constant s)

Net
heat current: H = Aes(T^{4}
- T_{s}^{4})

Ideal-gas
equation of state: pV = nRT

Isotherms
on pV diagram show pressure as a function for volume at constant temperature.

Average
translational kinetic energy of gas particles:
K = 3/2 nRT

Root
mean square speed of molecules in ideal gas: v_{rms} = sqrt(3RT/M)

Mean
free path: l
= vt = V/(4pr^{2}Nsqrt(2))

Molar
heat capacity for ideal: monatomic gas 3R/2, diatomic gas 5R/2, monatomic
solid 3R

Speeds
of molecules in an ideal gas are distributed according to the Maxwell-Boltzmann
distribution.

On
a phase diagram, the vaporization curve ends at the critical point, above
which there is no distinction

between the gas and liquid phases.

W
= p(V_{2} - V_{1})

Heat
and work in a thermodynamic process are not state functions.

First
Law: DU = Q - W

Internal
energy is a state function.

In
an adiabatic process, no heat is transferred in or out of the system;
isochoric is constant volume, isobaric

constant pressure, isothermal constant temperature.

The
internal temperature of an ideal gas depends only on its temperature; for all
else, it also depends on

pressure.

Molar
heat capacity at constant pressure equals molar heat capacity at constant
volume plus R.

p_{1}V_{1}^{Cp/Cv}
= p_{2}V_{2}^{Cp/Cv}

A
heat engine takes heat Q_{H} from a source, converts part of it to
work W, and discards the rest at lower

temperature
Q_{C} .

Thermal
efficiency: e = W/Q_{H} = 1 + Q_{C}/Q_{H}

Thermal
efficiency of Otto cycle gasoline engine: e = 1 - 1/r^{Cp/Cv-1}

Coefficient
of performance: K = | Q_{C}/W |

The
Second Law states that no cyclic process can convert heat completely into work
(engine statement) or

transfer
heat from a colder place to a hotter place with no work input (refrigerator
statement).

Thermal
efficiency of the Carnot cycle of reversible processes between two heat
reservoirs: e = 1 - T_{C}/T_{H}

All Carnot engines between the same two temperatures have the same efficiency.

A
backwards Carnot engine is a Carnot refrigerator.

Kelvin
scale is based on Carnot cycle efficiency.

Entropy:
DS = DQ/DT ; the entropy of an isolated system can
never decrease

S
= k*ln(w) (Boltzmann
constant k = R/N_{A} , number
of possible microscopic states w)

A
wave is any disturbance from an equilibrium condition that propagates from one
region to another; a mechanical

wave
travels within a medium material.

v
= lf

y
= A*sin(2*p*(t/T
- x/l)) = A*sin(wt
- kx) with k = 2p/l and w
= vk

Wave
equation: second partial of y with respect to x equals second partial of y
with respect to t, over v^{2}

Speed
of transverse wave on a string: v = sqrt(F/m)

Speed
of longitudinal wave in a fluid: v = sqrt(B/p) (bulk modulus B, density p)

Speed
of longitudinal wave in a solid rod: v = sqrt(Y/p) (Young's modulus Y)

Speed
of sound in ideal gas: v = sqrt((Cp/Cv)RT/M)

Average
power of sinusoidal wave on stretched string: P = w^{2}A^{2}
sqrt(mF)/2
called intensity for longitudinal

The
principle of superposition states that the total wave displacement at any
point where two or more

waves overlap is the sum of the displacements of the
individual waves.

When
a wave is reflected from a fixed or free end of a stretched string, the
incident and reflected waves

combine to form standing wave with nodes l/2 apart and antinodes between.

y
= (A*sin(k*x)*cos(w*t)
(angular frequency w = 2pf, wave number k = 2p/l)

If
both ends of a string are fixed, standing waves occur only when the string
length is an integer multiple of

l/2;
f = nv/(2L)

Each
frequency and its associated vibration pattern is called a normal mode.

The
lowest frequency is the fundamental frequency
f = sqrt(F/m)/(2L)

A
closed end of a pipe or tube is a displacement node and a pressure antinode.

Open
pipe: f = nv/(2L) for integral n;
Stopped pipe: f = nv/(4L) for odd n

Interference
(in-phase constructive or out-of-phase destructive) interference is the
overlap of waves.

Forced
oscillation is the vibration at the same frequency as an applied periodically
varying force, which

causes resonance if near a normal-mode frequency.

Pressure
amplitude of sound: p = BkA (bulk
modulus B)

Loudness
depends on amplitude and frequency, pitch on frequency, and tone quality
(timbre) on harmonic

content and the attack and decay characteristics.

Intensity
is the time average rate at which energy is transported by the wave per unit
area.;

I = p^{2}/(2rv)
(pressure amplitude p)

Sound
intensity level (in decibels): b
= (10 dB)log (I/I_{0})
I_{0} = 10^{-12} W/m^{2}

Beats
are two tones with slightly different frequencies sounded together;
f = f_{1} - f_{2}

Doppler
effect: f_{L} = (v + v_{L})/(v + v_{S}) f_{S}

Shock
wave: sin(a)
= v/v_{S} (v_{S}
> v, the speed of sound)

Charge
is conserved; likes repel and opposites attract.

Coulomb's
Law: F = kq_{1}q_{2}/r^{2}

k
= 1/(4pe_{0})
= 8.99*10^{9} Nm^{2}/C^{2}
; e_{0
}is
permittivity of free space = 8.85*10^{-12} C^{2}/(N*m^{2})

Electric
field is the force per unit charge exerted on a test charge at any point;
E = kq/r^{2} = F/q

Unit
of electric field is N/C or V/m

The
principle of superposition of forces or fields states that the total force or
field on a charge is the vector

sum of the forces or fields exerted by the individual
charges.

The
tangent to a field line is the direction of E.

Electric
dipole: magnitude p = q*d, torque
t = p*E*sin(f) (cross product of dipole moment and electric

field)

Potential
energy of an electric field: U =
-p*E (dipole moment
p)

Electric
flux is the flow of electric field through a surface; flux F
= integral of E*dA

Gauss's
Law states that the total electric flux through a closed surface equals 4pk times the enclosed

charge; F = 4pkQ_{enc}

Excess
charge on a conductor at rest resides entirely on the surface.

Line
charge: E = 2kl/r

Sheet
charge E = 2pks

Outside
solid insulating sphere: E = kQ/r^{2}

Inside
solid insulating sphere: E =
kQr/R^{3}

Potential
energy: U = kq_{1}q_{2}/r

Potential
is potential energy per unit charge; V
= kq/r

V_{2}
- V_{1} = integral of E times dl

1
eV = 1.602*10^{-19} J

An
equipotential surface is a surface on which the potential has the same value
at every point.; field lines

cross equipotentials perpendicular.

E_{x}
= -dV/dx

A
capacitor is any pair of conductors separated by an insulating material;
capacitance C = Q/V

Parallel
plate capacitor: C = A/(4pkd)

Unit
of capacitance is the farad = 1 C/V

Capacitors
in series: 1/C = 1/C_{1}
+ 1/C_{2} + . . .

Capacitors
in parallel: C = C_{1} + C_{2} + . . .

Energy
of capacitor: U = QV/2 = CV^{2}/2
= Q^{2}/(2C)

Capacitance
increases by a factor of K, the dielectric constant, when the space between
the conductors is

filled with a dielectric material;
C = KC_{0} ; permittivity
is Ke_{0}

In
dielectric breakdown, under electric field greater than dielectric strength,
dielectrics become conductors.

Current
is the amount of charge flowing through a specified area per unit time.

Unit
of current is ampere = C/s

I
= n*|q|*v_{d}*A = J*A (concentration
n, drift velocity v_{d}, current density J_{ })

Resistivity:
r = E/J

Unit
of resistivity is ohm-meter.

Resistivity
increases with temperature and is small in conductors.

Ohm's
Law: V = I*R

R
= rL/A

Unit
of electromotive force is volt.

V
= e - I*r

Power:
P = V*I = I^{2}*R

Resistors
in series: R = R_{1} + R_{2}
+ . . .

Resistors
in parallel: 1/R = 1/R_{1}
+ 1/R_{2} + . . .

Kirchhoff's
junction rule states that the sum of the currents into any junction must be
zero.

Kirchhoff's
loop rule states that the sum of potential differences around any loop must be
zero.

Deflection
is proportional to current in a d'Arsonval galvonometer.

A
galvanometer with a shunt resistor is an ammeter and should have low
resistance.

A
galvanometer with a resistor in series is a voltmeter and should have high
resistance.

Capacitor
charged by battery with resistor in series:
q = C*e*(1
- e^{-t/RC}), I = (e/R)*e^{-t/RC}

The
time constant t
= RC is time for charge to get within 1/e of its final value.

Capacitor
discharge: q = C*e*(e^{-t/RC}),
I = (e/R)*(1
- e^{-t/RC})

Household
wiring systems include hot, neutral, and ground wires.

Magnetic
force: F = cross product of q*v and B
(magnetic field B)

Unit
of magnetic field is tesla = Nm/A

Magnetic
flux: F
= integral of B times dA

Unit
of magnetic flux is weber = T*m^{2}

Gauss's
Law for magnetism states that the net magnetic flux through any closed surface
is zero; magnetic

field lines always close on themselves.

Magnetic
force is perpendicular to velocity; R = m*v/(|q|*B)

Electric
and magnetic forces cancel when v = E/B

Force
on a conductor: dF = cross product of I*ldl and B

t
= I*B*A*sin(f)

Magnetic
dipole moment: m = I*A, U
= -m*B*cos(f)

Motion
of a dc motor rotor throught the magnetic field causes an induced emf called
back emf.

The
Hall effect is a potential difference measured perpendicular to the direction
of current in a conductor

when the conductor is placed in a magnetic field;
electric force must just balance magnetic force on a moving

charge; E*n*q = -J*B

B
= (m_{0}/(4p))*cross(q*v, r-hat)/r^{2}

m_{0}
is permeabilityh of free space, 4p*10^{-7}
Wb/(A*m)

Law
of Biot and Savart: dB = (m_{0}/(4p))*cross(I*dl, r-hat)/r^{2}

Magnetic
field at a distance from a long straight conductor: B = m_{0}I/(2pr)

F/L
= m_{0}II'/(2pr)

Magnetic
field along axis from a circular loop: B = m_{0}NIa^{2}/(2*(x^{2}+a^{2})^{3/2}

Magnetic
field in solenoid: B = m_{0}nI (n turns per length)
; 0 outside

Magnetic
field in toroidial solenoid: B = m_{0}NI/(2pr)
(N turns) ; 0 outside

Ampere's
Law states that the line integral of the magnetic field around any closed path
equals m_{0}
times the

net current through the area enclosed by the path;
integral of B times dl = u_{0}I_{encl}

Magnetic
susceptibility is a material's relative permeability minus one; it is small +
for paramagnets, small

- for diamagnets, and large for ferromagnets

Displacement
current acts as a source of magnetic field like current;
I_{D} = e*(dF/dt)
and can be added into Ampere's

Law:
integral of B times dl = u_{0}(i_{c} + i_{d})

Faraday's
Law states that the induced electromotive force in a closed loop equals the
negative of the time

rate of change of magnetic flux through the loop;
line integral of E times dl = -dF_{B}/dt

Lenz's
Law states that an induced current or emf always tends to oppose or cancel out
the change that

caused it.

Induced
emf for moving conductor: e = v*B*L

Eddy
currents are induced in the volume of a bulk piece of conducting material in a
changing magnetic

field.

Maxwell's
equations include Gauss's law of electrostatics, Gauss's law for magnetic
fields, Ampere's law

with Maxwell's displacement current generalization,
and Faraday's law.

A
changing current in one circuit causes a changing magnetic flux in another
circuit; e_{1}
= -M*(di_{2}/dt) with

mutual inductance M = N_{1}*F_{B1}/i_{2}
(N wire coils)

Unit
of mutual inductance is the henry = Wb/A = V*s/A = ohm*s

Self-induced
emf: e = -L*(di/dt) with self-inductance L = NF_{B}/I

U
= L*I^{2}/2

Magnetic
energy density: u = B^{2}/(2*m_{0})

Time
constant of resistor - inductor circuit t
= L/R, the time for current to reach 1/e of its final value.

Angular
frequency of electrical oscillations for L-C circuit: w = sqrt(1/(LC))

Angular
frequency of damped oscillation in L-R-C series: w
= sqrt(1/(LC) - R^{2}/(4L^{2})), critically damped at

R^{2} = 4L/C

A
phasor vector rotates counterclockwise with constant angular velocity equal to
the angular frequency of

the sinusoidally varying emf produced by an
alternator or ac source.

Rectified
average current: I_{rav} = 2*I/p

Root
mean square current: I_{rms}
= I/sqrt(2)

i
= I*cos(wt)

v
= V*cos(wt
+ f) (phase
angle f)

V_{R}
= IR, V_{L} = IX_{L}, V_{C} = IX_{C}

Inductive
reactance of inductor, leading current by 90 degrees: X_{L} = wL

Capacitive
reactance of capacitor, lagging current by 90 degrees:
X_{C} = 1/(wC)

V
= I*Z, Z is impedance

At
resonance angular frequency w_{0}
= 1/sqrt(LC) the current is maximum and impedance minimum;

impedance equals resistance and voltage and current
are in phase.

E
= cB

c
= 1/sqrt(e_{0}m_{0})

E
and B are perpendicular and direction of wave is direction of E cross B.

E
= E_{max}sin(wt
- kx); B = B_{max}sin(wt - kx)

The
Poynting vector gives the energy flow rate (power per unit area) in an
electromagnetic wave in a

vacuum; S
= cross(E,B)/m_{0}

The
time-average value of the Poynting magnitude is the intensity;
I = e_{0}cE_{max}^{2}/2

For
an absorbing surface, radiation pressure is I/c; for a reflecting surface, it is 2I/c.

Wave
speed in dielectric v = c/sqrt(K*K_{m})

Nodal
planes for E in a standing wave reflected at x=0 are at multiples of p and nodal planes for B at odd

multiples of p/2.

An
oscillating electric dipole produces a wave in all directions, transverse far
from the source, zero

intensity along the axis.

Electromagnetic
Spectrum

Radio 10 m / 10^{8} Hz; Microwaves 10^{-2}
m, 10^{10} Hz; Infrared 10^{-5} m, 10^{13} Hz; Visible
10^{-6} m, 10^{15} Hz (Violet 420

nm, Yellow 580 nm, Red 670 nm) ; Ultraviolet 10^{-7}
m, 10^{16} Hz; X rays 10^{-10} m, 10^{18} Hz; Gamma
rays 10^{-12} m,

10^{20} Hz

A
wave front is a surface of constant phase, and a perpendicular line called a
ray is along the direction of

propagation.

Frequency
of light is the same in all materials.

Index
of refraction: n = c/v ; l = l_{0}/n

Incident,
reflected, and refracted rays and normal all lie in the plane of incidence.

Snell's
law of refraction: n_{a}sin(q_{a})
= n_{b}sin(q_{b})

Total
internal reflection if angle of incidence > critical angle:
sin(q_{critical})
= n_{b}/n_{a}

Dispersion
is the variation of the index of refraction with wavelength.

The
wave is polarized in the direction of the E field.

Malus'
law gives the intensity of polarized light transmitted through a polarizing
filter used as an analyzer:

I = I_{max}cos^{2}(f)

Brewster's
law states that the reflected unpolarized light striking an interface is
completely polarized

perpendicular to the plane of incidence if the angle
of incidence is tan(q)
= n_{b}/n_{a}

Superimposed
linearly polarized waves result in circular or elliptical polarized light.

A
birefringent material has different indexes of refraction for two
perpendicular directions of polarization.

A
dichroic material has preferential absorption for one polarization direction.

Huygen's
principle states that if the position of a wave front at one instant is known,
the position of the

front at a later time can be constructed by imagining
the front as a source of secondary wavelets.

Reflected
or refracted rays appear to diverge from an image point; if they actually
converge there it is a real

image; if not, it is a virtual image.

Lateral
magnification: m = y'/y (m>0
erect; m<0 inverted)

Paraxial
rays are close to and nearly parallel to the optic axis; nonparaxial rays
cause spherical aberration.

The
focal point of a mirror is the point to which rays parallel to the axis
converge after reflection form a

concave mirror or appear to diverge from a convex
mirror; focal length is distance to vertex.

Reflection:
1/s + 1/s' = 1/f (=0 for plane
mirror); m = -s'/s (=1 for plane mirror)

Refraction:
n_{a}/s + n_{b}/s = (n_{b}-n_{a})/R (=0 for
plane surface); m = -n_{a}s'/(n_{b}s)
(=1 for plane surface)

Lensmaker's
Equation: 1/f = (n-1)(1/R_{1}-1/R_{2})

Sign
rules: s>0 for real object, s' for real image, R>0 for center of
curvature on outgoing side, m>0 erect

image

**Optical Instruments**

Camera
forms real, inverted, reduced image.

Intensity
is inversely proportional to f-number = (focal length)/(aperture diameter)

Simple
magnifier: M = (25 cm)/f

Microscopes
and telescopes form an image with a lens on the barrel and then a virtual
image with the

eyepiece at infinity.

Coherence
is an unchanging phase relationship between two waves.

The
principle of superposition states that the total wave disturbance at any point
is the sum of the

disturbances.

Constructive
interference occurs at integral multiples of wavelength: dsin(q) = ml (integer m)

Destructive
interference occurs at odd multiples of l/2:
dsin(q)
= (m+1/2)l (integer
m)

Position
of m^{th} bright fringe y_{m} = Rml/d

Amplitude
of superimposing: E_{p} = 2*E*|cos(f/2)
(phase angle f)

Intensity
of superimposing: I = I_{0}cos^{2}(f/2)

Constructive
interference for reflection from film of thickness t: 2t = ml (integer
m)

The
Michelson interferometer, made to detect motion of earth relative to a
hypothetic ether, uses

monochromatic source and measures wavelengths.

Energy
is carried in photon bundles; E =
hf (Planck's constant h)

Diffraction
occurs when light passes through an aperture or around an edge.

In
Fraunhofer diffraction, the source and observer are far from the obstructing
surface.

In
Fresnel diffraction, the source or observer is relatively close to the
obstructing surface.

Destructive
interference occurs at sin(q)
= ml/a (slit
width a, integer m)

Maximum
interference intensity for a diffraction grating is at dsin(q) = ml (integer m)

The
Bragg condition for 3D x-ray diffraction grating states that constructive
interference occurs when

angles of incidence and scattering are equal and
2dsin(q)
= ml (integer
m)

The
Airy disk, of angular size sin(q)
= 1.22l/D is a central bright spot in the diffraction
pattern from a

circular aperture.

Rayleigh's
criterion states that two point objects are just barely resolved when their
angular separation is

given by the angular size of the Airy disk.

A
hologram is a photographic record of an interference pattern formed by light
scattered from an object and

direct light coming form the source.

In
time dilation, time intervals in frames of reference moving relative to each
other are compared.

In
length contraction, distances in frames of reference moving relative to each
other are compared.

The
Lorentz coordinate transformation relates the coordinates and time of an event
in an inertial coordinate

system to the coordinates and time of the same event
as observed in a second inertial system moving

relative to the first.

The
Doppler effect is the frequency shift from a source due to the relative motion
of source and observer.

Einstein's
theory of special relativity: m = m_{0}/sqrt(1-v^{2}/c^{2})