Physics

 

 

 

Effects

 

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

 

 

 

Motion Along a Straight Line

Equations of Motion:

v = v0 + at

                x = x0 = v0t + at2/2

                v2 = v02 + 2a(x - x0)

                x - x0 = ((v0 + v)/2)t

                v = dx/dt

                a = dv/dt

 

Motion in Two or Three Dimensions

Projectile Motion in 2D

                x = (v0cos(a0)t

                y = (v0sin(a0))t - gt2/2

                vx = v0cos(a0)

                vy = v0sin(a0) - gt

Radial acceleration = v2/R ;   period = 2pR/v

Relative velocity: vP/A = vP/B + vB/A

 

Newton's Laws of Motion

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/s2

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.

 

Applications of Newton's Laws

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 v2/R.

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

 

Work and Kinetic Energy

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 = mv2/2

Unit of work is the Joule = 1 N*m = 1 kg*m2/s2

Work energy theorem: Wtotal = K2 - K1

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

Unit of power is the watt = 1 J/s = 1 kg*m2/s3

 

Potential Energy and Energy Conservation

Potential energy -DU = mgy1 - mgy2

Elastic work by compressed spring: W = kx12/2 - kx22/2

K1 + U1 +Wother = K2 + U2

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, Impulse, and Collisions

Momentum: p = mv

SF = dp/dt

Impulse: J = SF Dt = p2 - p1

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

Total momentum: P = pA + pB + ...

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: xcenter = (Smx)/Sm  ; ycenter = (Smx)/Sm

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

Fexternal = Macenter of mass

 

Rotation of Rigid Bodies

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 = q0 + w0t + at2/2

                q - q0 = (w0 + w)t/2

                w = w0 + at

                w2 = w02 + 2a(q - q0)

v = rw

atangential = ra

aradial (centripetal) = v2/r = w2r

Moment of inertia: I = Smr2

Kinetic energy: K = Iw2/2

Parallel axis theorem: Iparallel axis = Icm + Md2

Moments of inertia

                Slender rod, axis through center: ML2/12

                Slender rod, axis through one end: ML2/3

                Plate, axis through center: M(a2+b2)/12

                Plate, axis along edge b: Ma2/3  

                Hollow cylinder: M(R12 + R22)/2

                Solid cylinder: MR2/2

                Thin-walled hollow cylinder: MR2

                Solid sphere: 2MR2/5

                Thin-walled hollow sphere: 2MR2/3

 

Dynamics of Rotational Motion

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 = Mvcm2/2 + Icmw2/2

If the rigid body rolls without slipping, vcm = Rw

Work: W = tDq

Work-energy theorem for rotation of rigid body: W = Iw22/2 - Iw12/2

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.

 

Equilibrium and Elasticity

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 = (Ft/A)/(Dl/l0)

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/m2

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

B = -Dp/(DV/Vo)

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.

 

Gravitation

Newton's law of gravitation: F = Gm1m2/r2

Acceleration due to gravity near Earth's surface is GmE/RE2

Gravitational potential energy: U = - integral of F = -Gm1m2/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/c2 , the

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

 

Periodic Motion

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 = kA2/2 = 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 - b2/(4m2))

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

oscillator.

 

Fluid Mechanics

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: A1v1 = A2v2

Volume flow rate: Av = dV/dt

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

                p1 + pgy1 + pv12/2 = p2 + pgy2 + pv22/2

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 * (R4/n)((p1-p2)/L)

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

 

Temperature and Heat

TF = 9/5 TC + 32 ;  TC = 5/9 (TF - 32) = Tk - 273.15

The triple point of water is at 273.16 K

DL = aL0DT

DV = 3aV0DT 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(TH - TC)/L   (thermal conductivity k)

Heat current for radiation:  H = AesT4  (emissivity e, Stefan-Boltzmann constant s)

Net heat current: H = Aes(T4 - Ts4)

 

Thermal Properties of Matter

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: vrms = sqrt(3RT/M)

Mean free path: l = vt = V/(4pr2Nsqrt(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.

 

The First Law of Thermodynamics

W = p(V2 - V1)

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.

p1V1Cp/Cv = p2V2Cp/Cv

 

The Second Law of Thermodynamics

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

temperature QC .

Thermal efficiency: e = W/QH = 1 + QC/QH

Thermal efficiency of Otto cycle gasoline engine: e = 1 - 1/rCp/Cv-1

Coefficient of performance: K = | QC/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 - TC/TH
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/NA ,  number of possible microscopic states w)

 

Mechanical Waves

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 v2

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 = w2A2 sqrt(mF)/2    called intensity for longitudinal

 

Wave Interference and Normal Modes

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.

 

Sound and Hearing

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 = p2/(2rv)        (pressure amplitude p)

Sound intensity level (in decibels): b = (10 dB)log (I/I0)      I0 = 10-12 W/m2

Beats are two tones with slightly different frequencies sounded together;  f = f1 - f2

Doppler effect: fL = (v + vL)/(v + vS) fS

Shock wave: sin(a) = v/vS    (vS > v, the speed of sound)

 

Electric Charge and Electric Field

Charge is conserved; likes repel and opposites attract.

Coulomb's Law:  F = kq1q2/r2    

k = 1/(4pe0) = 8.99*109 Nm2/C2  ;  e0 is permittivity of free space = 8.85*10-12 C2/(N*m2)

Electric field is the force per unit charge exerted on a test charge at any point;  E = kq/r2 = 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)

 

Gauss's Law

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 = 4pkQenc    

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/r2

Inside solid insulating sphere:  E = kQr/R3

 

Electric Potential

Potential energy: U = kq1q2/r

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

V2 - V1 = 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.

Ex = -dV/dx

 

Capacitance and Dielectrics

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/C1 + 1/C2 + . . .

Capacitors in parallel: C = C1 + C2 + . . .

Energy of capacitor:  U = QV/2 = CV2/2 = Q2/(2C)

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

filled with a dielectric material;  C = KC0 ;  permittivity is Ke0

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

 

Current, Resistance, and Electromotive Force

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

Unit of current is ampere = C/s

I = n*|q|*vd*A = J*A  (concentration n, drift velocity vd, 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 = I2*R

 

Direct-Current Circuits

Resistors in series:  R = R1 + R2 + . . .

Resistors in parallel:  1/R = 1/R1 + 1/R2 + . . .

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 Field and Magnetic Forces

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*m2

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

 

Sources of Magnetic Field

B = (m0/(4p))*cross(q*v, r-hat)/r2

m0 is permeabilityh of free space, 4p*10-7 Wb/(A*m)

Law of Biot and Savart:  dB = (m0/(4p))*cross(I*dl, r-hat)/r2

Magnetic field at a distance from a long straight conductor:  B = m0I/(2pr)

F/L = m0II'/(2pr)

Magnetic field along axis from a circular loop: B = m0NIa2/(2*(x2+a2)3/2

Magnetic field in solenoid:  B = m0nI   (n turns per length)  ;  0 outside

Magnetic field in toroidial solenoid:  B = m0NI/(2pr)    (N turns) ;  0 outside

Ampere's Law states that the line integral of the magnetic field around any closed path equals m0 times the

net current through the area enclosed by the path;  integral of B times dl = u0Iencl

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;  ID = e*(dF/dt) and can be added into Ampere's

Law:  integral of B times dl = u0(ic + id)

 

Electromagnetic Induction

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 = -dFB/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.

 

Inductance

A changing current in one circuit causes a changing magnetic flux in another circuit; e1 = -M*(di2/dt) with

mutual inductance M = N1*FB1/i2  (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 = NFB/I

U = L*I2/2

Magnetic energy density: u = B2/(2*m0)

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) - R2/(4L2)), critically damped at

R2 = 4L/C

 

Alternating Current

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: Irav = 2*I/p

Root mean square current:  Irms = I/sqrt(2)

i = I*cos(wt)

v = V*cos(wt + f)   (phase angle f)

VR = IR, VL = IXL, VC = IXC

Inductive reactance of inductor, leading current by 90 degrees: XL = wL

Capacitive reactance of capacitor, lagging current by 90 degrees:  XC = 1/(wC)

V = I*Z,  Z is impedance

At resonance angular frequency w0 = 1/sqrt(LC) the current is maximum and impedance minimum;

impedance equals resistance and voltage and current are in phase.

 

Electromagnetic Waves

E = cB

c = 1/sqrt(e0m0)

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

E = Emaxsin(wt - kx);  B = Bmaxsin(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)/m0

The time-average value of the Poynting magnitude is the intensity;  I = e0cEmax2/2

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*Km)

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 / 108 Hz; Microwaves 10-2 m, 1010 Hz; Infrared 10-5 m, 1013 Hz; Visible 10-6 m, 1015 Hz (Violet 420

nm, Yellow 580 nm, Red 670 nm) ; Ultraviolet 10-7 m, 1016 Hz; X rays 10-10 m, 1018 Hz; Gamma rays 10-12 m,

1020 Hz

 

The Nature and Propagation of Light

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 = l0/n

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

Snell's law of refraction: nasin(qa) = nbsin(qb)

Total internal reflection if angle of incidence > critical angle:  sin(qcritical) = nb/na

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 = Imaxcos2(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) = nb/na

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.

 

Geometric Optics

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: na/s + nb/s = (nb-na)/R (=0 for plane surface);  m = -nas'/(nbs) (=1 for plane surface)

Lensmaker's Equation: 1/f = (n-1)(1/R1-1/R2)

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.

 

Interference

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 mth bright fringe ym = Rml/d

Amplitude of superimposing: Ep = 2*E*|cos(f/2)      (phase angle f)

Intensity of superimposing: I = I0cos2(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

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.

 

Relativity

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 = m0/sqrt(1-v2/c2)