ap physics reveiw

7/30/2019 ap physics reveiw

1/18

AP Physics B Review Sheet

UNITS OF MEASUEMENT

distance / displacement m

mass kg

time s

speed / velocity m /s

acceleration m /s2

force N=kgm

s2

pressure Pa=N

m2 =

kg

ms2

impulse Ns=kgm

s

momentumkgm

s

work / energy J=Nm=kgm

2

s2

power W=Js =kgm

2

s3

angular displacement rad

angular velocity rad / s

angular acceleration rad / s2

torque Nm

moment of inertia kgm2

angular momentum kgm2/s

rotational work/energy J

frequency Hz = cycles / s

mass density kg / m3

temperature KorC

heat J

entropy J / K

electric charge C

electric field N / C

electric flux Nm2/C

electric potential V = J / C

potential gradient V / m

capacitance F=C/ V

electric current A=C/s

electrical resistance =V/A

resistivity m

magnetic field T=Ns /Cm

magnetic flux Wb=Tm2

inductance H=Vs/A

To convert units, make sure you write downunits explicitly and treat them as algebraicquantities. You will multiply by conversionfactors in the form of fractions so units cancel.

TRIGONOMETRY

sin =opp

hyp =sin1 opphyp

cos =adjhyp

=cos1 adjhyp tan =

opp

adj = tan1oppadj

sin2

cos2

=1

KINEMATICS IN ONE DIMENSION

vavg= x t

vinstant=limt 0

x t

aavg= v t

a instant=limt0

v t

Variables

x0xv0va

t

Equations

v=v0at

x=x0v0t1

2a t2

v2=v0

22 a x

1. Make a drawing to represent the situation2. Decide which directions are positive

relative to a conveniently chosen coordinatesystem

3. Write down the values of the kinematicvariables. If there are multiple objects,remember that they may share comecommon variables.

4. Choose the equation which contains thegiven values and the unknown value.

5. When the motion is divided into segments,remember that the final velocity of onesegment is the initial velocity for the nextsegment.

6. Remember that there may be two possiblemathematical solutions, and you need toselect the appropriate answer or answerswhen this occurs.

Free Fall

The acceleration due to gravity at sea-level atthe Earth's equator is called one gee and isapproximately 9.80 m/s2.

KINEMATICS IN TWO DIMENSIONS

Separate vectors into components parallel to theaxes of the chosen coordinate system.

opp=hyp sin adj= hypcos Motion in the x-direction is independent of

motion in the y-direction.

Projectile Motion

ax=0m /s2

ay=9.80 m /s2

downvx=v0x

At the apex of the flight (vertex of parabola)vy=0m /s

At original height on way back downy=y0 vy=v0y

Relative Velocity

vAB is the velocity of A as seen by BvAB=vACvCB vAB=vBA

DYNAMICS (FORCES)

Inertia is the natural tendency of an object toremain at rest or in motion at a constantvelocity. Mass is the quantitative measurementof inertia.

An inertial reference frame is one in whichNewton's First Law of Motion is valid. The

acceleration of any inertial reference frame isalways zero.

An object is in equilibrium when it has zeroacceleration.

Newton's First Law of Motion

An object continues in a state of rest or a stateof motion at a constant velocity (speed anddirection) unless compelled to change that state

by a net external force.

Newton's Second Law of Motion

F=maThe direction of the acceleration is the same asthe direction of the net force.

Newton's Third Law of Motion

Whenever a body exerts a force on a secondbody, the second body will exert an oppositelydirected force of equal magnitude on the first

body.

This force pair doesn't cancel because they areacting on different objects.

Weight

The force of gravity,Fg, acting on an object isoften called the object's weight. If the local

value of the acceleration due to gravity isknown, this is Fg=mg

Normal Force

When an object is in contact with a surface, thenormal force,FN, is the component of the forcethat the surface exerts that is perpendicular tothe surface; it is the force that prevents theobject from passing through the surface.

Apparent Weight

The apparent weightof an object is the forcethat a scale exerts on the object resting on it,

FN. If the reference frame has an acceleration awhere up is positive, then FN= Fgma

Tension

Tension is a force applied to one end of a ropeor cable that is transmitted to an object attachedto the other end of the rope or cable.

If the rope is massless, the force applied to oneend would be completely transmitted to theobject at the other end. However, ropes dohave mass, so some of the force applied isneeded to accelerate the rope, which results is areduced force acting on the attached object.


2/18


Friction

When an object is in contact with a surface, thefriction,f, is the component of the force that thesurface exerts that is parallel to the surface; it is

present only when the object is moving orattempting to move along the surface due tosome other force acting on the object.

When the object is stationary, the magnitude ofthestatic frictional force isfs, which is onlylarge enough to prevent motion up to somemaximum amount fs

max= sFN s is

the coefficient of static friction, and measuresthe roughness of the surface and object.

When the object is moving, the magnitude ofthe kinetic frictional force isfk, which is given

by fk= kFN k is the coefficient ofkinetic friction, and measures the roughness ofthe surface and object. k s but eithermay be larger than 1 (normally not)

PressurePressure,P, is the force exerted per unit area.

P=F

A

IMPULSE AND MOMENTUM

The impulse of a force is given by F tAn object's linear momentum is p=mvMomentum is thought of as inertia in motion.

Impulse-Momentum Theorem

When a net force acts on an object, the impulseof the net force is equal to the change in themomentum of the object.

F t= p

If the mass remains constant while the net forceis acting, this becomes

Ft=mvfmvi

Conservation of Linear Momentum

The total linear momentum of an isolatedsystem remains constant. An isolated system isa system for which the vector sum of theexternal forces acting on it is zero.

Center of Gravity / Center of Mass

The center of gravity is the point that representsthe average location for the total weight of thesystem; it is the balance point for the object.The center of gravity of a thrown, rotatingobject is the point that moves along the

parabola.

The center of mass is the point that representsthe average location for the total mass of thesystem. The center of mass is generally thesame point as the center of gravity, unless theobject is tall enough thatgis smaller at its topthan at its base, in which case the center ofmass is slightly higher than the center ofgravity.

WORK AND ENERGY

The workdone on an object of mass m by aconstant forceFis W=Fcos s where is the angle between the force and thedisplacement. Work is a scalar.

Poweris the rate at which work is done.

P= W

t=Fv

The kinetic energy of an object with mass m

and speed v is KE=1

2mv2

Thegravitational potential energy of an objectwith mass m at a height h above a convenientlychosen zero point is PE= mgh

The mechanical energy is E=KEPE

A conservative force is a force where the workdone to move an object is independent of the

path taken between the starting and endingpoint. Alternatively, a force is conservativewhen the work done in moving an objectaround a closed path is zero.

Work-Energy Theorem

When a net external force does work on anobject, the work done is equal to the change inthe kinetic energy of the object.

W=KEThe net work done by all nonconservativeforces is equal to the change in the mechanicalenergy of the object.

Wnc=E=KEPE

Conservation of Mechanical EnergyThe total mechanical energy of an objectremains constant as the object moves, providedthat the net work done by any externalnonconservative forces is zero.

Conservation of Energy

Energy can neither be created nor destroyed,but can only be converted from one form toanother.

UNIFORM CIRCULAR MOTION

An object is in uniform circular motion when itis traveling at a constant speed on a circular

path.

An object spinning around an internal axis isrotating. An object spinning around an externalaxis is revolving.

Theperiod, T, is the time it takes to travel oncearound the circle.

The linear speed, v, around the circular path is

v=2 r

T

Centripetal Acceleration

Since an acceleration is the rate of change inthe velocity, which includes direction, an objectin U.C.M. is accelerating. This is called itscentripetal acceleration.

a c=v

2

r

Centripetal acceleration always points towardthe center of the circle, since that is thedirection the centripetal force points.

Centripetal Force

The force that keeps the object moving alongthe circular path is called the centripetal force.

Fc=ma c=mv

2

r

The centripetal force must point toward thecenter of the circle in order to force the objectalong the circular path. It is always

perpendicular to the direction of motion.

A very common trick to solving U.C.M.Problems is to equate the force providing thecentripetal force with the centripetal forceformula.

Maximum speed around an unbanked curve

v= sg r

Speed around a frictionless banked curve

tan =v2

r g

Orbital speed of satellites in circular orbits

v=

G ME

rNote that r=rEh is the orbital radius.

Artificial Gravity

v=r geffective


3/18


ROTATIONAL KINEMATICS

Rotational kinematics problems are solved likelinear motion problems, with angular variablessubstituted for linear variables.

Linear (m) Angular (radians)

Distance d =arclength

radius

Speed v= 2 r

T =

t

= 1T

Acceleration a =

t

Variables 0

0

t

Kinematics Equations = 0t

= 0 0 t12

t2

2= 0

22

As long as the angular variables are expressed

using radian measure, the followingconversions can be made:s=r vT=r aT=r

where the linear measure is tangential to thecircular path. (s is the arc length)

Centripetal Acceleration with Angular Speed

a c=vT

2

r=r

2

ROTATIONAL DYNAMICS

Torque

When a force F acts at a point adisplacement r from the axis of rotation, it

produces a torque =r F=Frsin =Fperpendicularr=Fl

Fperpendicular=Fsinl=rsin

The lever arm, l, is the distance between theline of action for the force and the axis ofrotation for the object.

If the object does not have an axis of rotationfixed by an external object, the axis of rotationwill be through the object's center of gravity.

Equilibrium

For an object to be in equilibrium, the net forceand the net torque acting on it must both be

zero.

Moment of Inertia

The equivalent of mass for a rotating object isthe moment of inertia, I. It is calculated by

I=i

miri2=

0

M

r2

dm

Common Moments of Inertia

Many more can be found on the Internet

Point mass I=m r2

Hollow cylinder, hoop I=m r2

Solid cylinder, disk I=12

mr2

Thin rod around center I= 112

m r2

Thin rod around end I=1

3m r2

Hollow sphere around center I=23

mr2

Solid sphere around center I=2

5m r

2

Newton's Second Law for Rotation

=I

Angular MomentumL=I Total angular momentum is conserved if the netexternal torque acting on the system is zero.

Conservation of angular momentum is why anice skater rotates faster when she pulls her armsand legs in during a spin.

Rotational Kinetic Energy

KErot=1

2I 2

Rotational kinetic energy is part of mechanicalenergy, and so must be part of the conservationof the total mechanical energy, though not

conserved itself.

Work-Energy Theorem for Rotation

net= 12I 2Power for Rotational Work

P=

UNIVERSAL GRAVITATION

Kepler's First Law of Planetary Motion

The paths of the planets are ellipses with theSun at one focus.

Kepler's Second Law of Planetary Motion

An imaginary line from the sun to a planetsweeps out equal areas in equal time intervals.

This implies that the planet moves faster whenclose to the sun (nearperihelion) and slowerwhen further from the sun (nearaphelion).

We often use the conservation of mechanicalenergy instead of this law to determine speedsat different points in an elliptical orbit.

Kepler's Third Law of Planetary Motion

For two objects orbiting the same body.

TATB2

= rArB 3

For one object orbiting a body with knownmassM

T2=4

2

G M

r3

Newton's Law of Universal Gravitation

Fg=G m1 m2

r2

G=6.674281011 m

3

kgs2

When using G, you must use meters fordistance, kilograms for mass, and seconds fortime.

Orbital speed of satellites in circular orbits

v=G MrNote that r=rEh is the orbital radius.

Escape speed at distance d

v=2G MdLocal Acceleration Due to Gravity and

Gravitational Field Strength

Outside a planet g is given by

g= G M

r2

where r=rplaneth is the orbital radius.

Inside a planet with uniform density it is

g= rcurrent

rplanetgsurfaceOcean Tides

Generally there are two high tides and two lowtides each day, caused by the difference in thegravitational pull by the Moon on oppositesides of the Earth.

Ftide =4G Mcause Rexperiencing

dcenter to center3

The Sun also causes tides, about half that of theMoon. When the Sun's tides and Moon's tidesalign (at the full moon and new moon) it is a

spring tide. When they are 90 out ofalignment (at the quarter moons) it is a neaptide.


4/18


SIMPLE HARMONIC MOTION

A vibration is a wiggle in time.A wave is a wiggle in space and time andcarries energy.

The equilibrium position is where the net forceon the object is zero.

In order for an object to be insimple harmonicmotion, the restoring force (the force that triesto return the object to its equilibrium position)must be proportional to the displacement fromequilibrium.

The amplitude,A, of the motion is themaximum distance that the object moves awayfrom equilibrium.

Theperiod, T, is the time needed for an objectto repeat one complete cycle of the motion.

Thefrequency of vibration,f, which is thenumber of cycles that repeat in one time period.

f= 1T

Springs - Hooke's Law

The restoring force provided by a spring withspring constantkis

Fspring=kx

whenx is the displacement from theequilibrium position.

The spring constant is referred to as thestiffness of the spring, and is inversely

proportional to the number of coils in the spring

Theperiodof the spring is given byT=2 mk

Potential Energy of Springs

PEspring=1

2k x

2

Pendulums

Theperiodof a pendulum is given by

T=2 LgResonance

A natural frequency of an object is one atwhich minimum energy is required to produceforced vibrations. It is also the frequency thatrequires the least amount of energy to continuethe vibration.

An object's natural frequencies depend onfactors such as its elasticity and the shape of theobject.

When the frequency of the application of aforce to an object matches the object's naturalfrequency, a dramatic increase in amplitudeoccurs. This increased amplitude of thevibrations is called resonance.

WAVE MOTION

Types of Waves

A transverse wave is created when the vibrationthat creates the wave is perpendicular to thedirection of the wave. (light)

A longitudinal wave is created when thevibration that creates the wave is parallel to thedirection of the wave. (sound)

Water waves are an example of waves thatinvolve a combination of both longitudinal andtransverse motions. As a wave travels throughthe water, the particles travel in circles. Theradius of the circles decreases as the depth intothe water increases.

Parts of Waves

The high points are called crests.

The low points are called troughs.

The solid dark center line represents the

midpointof the vibration, where there is norestoring force acting on the object.

The amplitude, A, is the displacement from themidpoint vibration.

The wavelength, , of the wave is the distancebetween successive identical parts of the wave.

Theperiod, T, of the wave is how long it takesfor one wavelength to pass a fixed location.

Thefrequency, f, of the wave is how manywaves pass occur in a given time.

Wave Speed

v=f

On a string: v=Ftension

m /L

REFLECTION

Wave Reflection in 1-Dimension

Waves encountering a hard boundary will flip(crest becomes trough). Waves encountering asoft boundary will reflect the way they come in(crest stays crest).

Waves moving from a less dense to a more

dense medium reflect as if encountering a hardboundary (less amplitude) and transmit a wavewith less amplitude and speed.

Waves moving from a more dense to a lessdense medium reflect as if encountering a soft

boundary (less amplitude) and transmit a wavewith greater amplitude and speed.

Wave Reflection in 2-Dimensions

A wave frontis a line that represents the crest ofa wave in two dimensions, and can be used toshow waves of any shape.

Rays are lines that are perpendicular to the

wave fronts and point in the direction of thevelocity of the wave.

When parallel wave fronts strike a solidboundary, they reflect.

r= i

Specular reflection occurs if all of the reflected

rays are parallel to each other which creates asharp image.

Diffuse reflection occurs if the reflected raysare not parallel to each other which creates afuzzy image or no image at all.

Images

A real image is formed when the rays of lightactually emanate from the image. A virtualimage is formed when the rays of light appearto come from the image, but do not.

An image is uprightif it is in the sameorientation as the object that formed it. Animage is invertedif it is in the oppositeorientation.

Plane Mirrors

A plane mirror forms an upright, virtual imagethe same size as the object that is located as far

behind the mirror as the object is in front of it.


5/18


Concave Mirrors

A concave mirror will have a focal point infront of the mirror. The mirror curves awayfrom the object.

Concave mirrors form an upright, enlarged,virtual image if the object is closer than thefocal point.

They will form an inverted, enlarged, realimage if the object is between the focal pointand the center of curvature.

They will form an upright, reduced, real imageif the object is further than the center ofcurvature.

Convex Mirrors

Convex mirrors curve toward the object, and

have their focal point behind the mirror.They will always form an upright, reduced,virtual image.

Ray Tracing

Ray 1 is initially parallel to the principal axis tothe mirror, then through the focal point.Ray 2 is through the focal point to mirror, then

parallel to the principal axis.Ray 3 can be drawn to confirm the point ofintersection, it is though the center of curvature.

Mirror Equations

fis the focal length

+ for a concave mirror for a concave mirrordois the object distance, diis the image distance

+ in front of mirror (real) behind mirror (virtual)

m is the magnification+ for upright for inverted

1

do

1

di=

1

f

m=hiho

=dido

REFRACTION

When parallel wave fronts strike a softboundary (like that caused by changing thedepth of water), the waves refract or changedirection.

Refraction is the bending of a wave resultingfrom a change in its velocity as its moves from

one medium to another. Since the frequency ofa wave cannot change, independent of thesource changing its frequency when itoriginally emits a wave, this change in wavevelocity must result from a change in itswavelength in the second medium.

Snell's Law

The index of refraction of a material is the ratioof the speed of light in a vacuum to the speedof light in the material. It must be at least 1.

Total Internal Reflection

When light encounters a boundary wheren1 > n2 then it is possible that all light reflectsand none refracts through the boundary. Thiswill happen at angles of incidence greater thanthe critical angle given by

sin c=n2

n1

Converging LensesConvex lenses are converging lenses in thatthey focus light rays closer together.

Outsidef

Insidef

Atfno image if formed because the rays areparallel.

Diverging Lenses

Concave lenses are diverging lenses in that theyspread the light rays further apart.

Object-Image Relations for Lenses

Ray Tracing

Ray 1 is initially parallel to the principal axis to

the lens, then through the focal point for thefirst lens surface.Ray 2 is through the focal point to secondsurface of the lens, then parallel to the principalaxis.Ray 3 can be drawn to confirm the point ofintersection, it is though the center of the lens.

Lens Equations

fis the focal length+ for a converging lens for a diverging lens

dois the object distance+ in front of lens (real) behind lens (virtual)

diis the image distance+ behind lens (real) in front of lens (virtual)

m is the magnification+ for upright for inverted

1

do

1

di=

1

f

m=hiho

=dido

n1 sin 1=n2 sin 2

n=c

v


6/18


INTERFERENCE AND DIFFRACTION

Wave Interference

Overlapping waves add together by adding theamplitudes of the waves at every point.

Constructive interference occurs where theamplitude of the combined wave is greater thanthe amplitudes of the component waves.

Destructive interference occurs where theamplitude of the combined wave is less than theamplitude of the component wave (because oneis above the midpoint and one is below themidpoint).

For two wave sources vibrating in phase, adifference in path length of zero, or an integralnumber of wavelengths leads to constructive

interference. A difference in path length that isa half-integer number (0.5, 1.5, 2.5, etc.) leadsto destructive interference.

Young's Double Slit Experiment

When a parallel wave with wavelength passesthrough two small slits separated by a distancedthen there are regions of maximum intensityfound at angles given by

sin =n

dand regions of minimum intensity at anglesgiven by

sin =n12dDiffractionThe bending of a wave around an obstacle orthe edges of an opening is called diffraction.

When a wave with wavelength passes througha slit of width dthen there are regions ofmaximum intensity found at angles given by

sin =n12dand regions of minimum intensity at anglesgiven by

sin =n

d

Resolving PowerTwo point objects are just resolved as separatewhen the first dark fringe in the diffraction

pattern of one falls directly on the central brightfringe in the diffraction pattern of the other.

min1.22

d

Beats

When two overlapping waves have frequenciesthat are only slightly different, they create acombined wave with a beat frequency equal tothe difference in the original frequencies.

Standing Waves

A traveling wave obviously advances, andmoves forward.

Astanding wave appears to vibrate in place.The parts of the standing wave that appearstationary are called nodes. The positions of thestanding wave with the greatest amplitude are

known as antinodes. Antinodes appear halfwaybetween nodes.

Standing waves are the result of interference.When two waves with equal amplitude passthrough each other in opposite directions, thewaves are always out-of-phase at the nodes,and in-phase at the anti-nodes.

A variety of standing waves can be produced byvarying the frequency of vibration. Differentstanding waves are called modes. In musicalinstruments, the different vibrational modesresult in different harmonics and overtones.

Natural FrequencyThe natural frequencies of an object are thefrequencies at which standing waves may becreated in the object.

For a string of mass m and lengthL fixed atboth ends with tensionFT, the naturalfrequencies are given by

fn=nvstring2L

where vstring=FT

m/L

For a tube of lengthL open at both ends,

fn=nvsound

2 L

For a tube of lengthL open at only one end,

fn=2n1vsound4L

f1 is the 1stharmonic or thefundamental freq.f2 is the 2stharmonic or the 1stovertone.f3 is the 3rdharmonic or the 2nd overtone.

Resonance

When an object vibrates near by or in contactwith a second object, and the frequency of

vibration is near one of the natural frequenciesof the second object, the second object willstart to vibrate at its natural frequency. This iscalled resonance.

Waves (including sound intensity or lightintensity) are amplified via resonance.

SOUND

Pitch is related to the frequency of the soundwave. Volume is related to the amplitude (halfof the pressure difference).

Speed of Sound

In a gas vsound= k T/m where =cP

cVkis Boltzmann's constant = 1.38065810-23 J/K

In a liquid vsound=Badiabatic/ where B isthe adiabatic bulk modulus.

In a solid vsound=Y/ where Y is Young'smodulus.

Sound Intensity

I=PowerArea

=10 dBlog II0 Note that an intensity level of 0 decibels is not0 W/m2, it is the threshold of human hearingwhich is 1.010-12 W/m2.

If the intensity level increases by 10 dB, thenew sound seems approximately twice as loud.

The Doppler Effect

TheDoppler effectis the change in frequencyor pitch of a sound detected by an observer dueto the relative motion of the source and theobserver relative to the medium of sound

propogation.

Source moving toward stationary observer:

fo=fs

1

1v s

v Source moving away from stationary observer:

fo=fs1

1vs

v

Observer moving toward stationary source:

fo=fs1vov Observer moving away from stationary source:

fo=fs1vov General:

fo=fs1vo

v

1vs

v


7/18


LIGHT

Electromagnetic radiation, including visiblelight, is produced by vibrating electric charges.This creates an oscillating electric field

perpendicular to the direction of propagation.The current formed by the moving electriccharges also creates an oscillating magneticfield perpendicular to both the direction of

propagation and the electric field.

Scientists now agree that light has a dualnature, part particle and part wave. Accordingto this theory, light also consists of massless

bundles of concentrated electromagnetic energycalledphotons.

Whether light appears to be a particle or a wavedepends on what is being measured and/or howthe experiment is designed.

The Electromagnetic Spectrum

Radio Waves - low frequency, long wavelengthMicrowaves

InfraredVisible Light (ROYGBIV)UltravioletX-raysGamma Rays high frequency, short

The Speed of Light

c=1

0 0=299,792,458m /s

Polarization

Inpolarized light, all of the oscillations of theelectric field are in the same plane. This planeis called the direction of polarization.

In unpolarized light, the direction ofpolarization is not fixed, but fluctuatesrandomly in time. The direction of oscillationof the electric field is different for different

photons.

Polarizing materials allow only the componentof the wave's electric field along one directionto pass through. This preferred transmissiondirection for the electric field is called thetransmission axis.

When unpolarized light is incident on a piece ofpolarizing material, the transmitted polarized

light has an intensity that is one-half that of theincident light.

When two pieces of polarizing material areused one after the other, the first is called the

polarizerand the second is called the analyzer.

If the average intensity of polarized light hittingthe analyzer is S0, then the average intensity oflight leaving the analyzer is

S=S0 cos2

where is the angle between the transmissionaxes of the polarizer and analyzer.

The angle of incidence at which the reflectedlight is completely polarized is parallel to thesurface is calledBrewster's angle.

tan B=n2

n1At this angle, the reflected and refracted raysare perpendicular to each other.

Shadows

A thin beam of light is often called a ray. Anybeam of light, no matter how wide, can bethought of as made of a bundle of light rays.

Ashadow forms where light rays cannot reach.Sharp shadows are produced by a small lightsource nearby or by a large light source furtheraway.Fuzzy-edged shadows are formed bylarger light sources close to an object, becauselight rays from one part of the light source may

be blocked while others reach that part of theshadow.

The darkest part of the shadow, where no lightreaches, is called the umbra. The lighter areaof partial shadow is called thepenumbra.

Opacity and Transparency

When an electromagnetic wave hits an atom,the electrons in the atom are forced into

vibration. The natural frequency of an electrondepends on how strongly it is held by a nearbynucleus.

When an electromagnetic wave encounters anelectron the electron may absorb the light as it

jumps up energy levels.

If the frequency of the light was the same as theelectron's natural frequency, the electron holdson to this energy for a longer time (about 100millionths of a second). During this time theatom collides with its neighbors many timesand the gives up this energy as heat.

If the frequency of the light is not similar to theelectron's natural frequency, it emits the energyquickly as light. The frequency of the light maychange depending on what energy level theelectron drops into.

If the light is emitted in the same direction itwas originally traveling in, the material istransparent. If the light is emitted randomly ina forward direction, but not exactly the samedirection, the material is translucent. If thelight is emitted backwards, the material isopaque.

COLOR

Coloris provoked by the frequencies of visiblelight emitted or reflected by things, but it is alsoin the eye of the beholder as whether or notthese frequencies of light are actually perceivedas colors depend on the eye-brain system. Forinstance, many organisms, including peoplewith red-green color blindness, will see no red

in a rose.

White lightis the combination of allfrequencies of visible light. Blackis theabsence of light.

Color by Reflection

When light hits an object light of somefrequencies is absorbed by the cells in theobject and some light is reflected. Thereflected frequencies create the color of theobject. Most objects do not have pure single-frequency colors, but are composed of a spreadof frequencies.

Note that only colors present is the originallight source could be reflected this way, whichis why objects look different colors underdifferent light sources.

Color by Transmission

The color of a transparent object depends onthe frequencies of the light it transmits. Thematerial in the object that selectively absorbscolored light is known as apigment, and thefrequencies absorbed by the pigment are nottransmitted.

From an atomic point of view, electrons in thepigment selectively absorb light of certainfrequencies, while other frequencies aretransmitted through the glass. The energy in theabsorbed light increases the kinetic energy ofthe atoms, and the object is warmed.

Additive Color (Lights)

All frequencies in the mix are seen.

The primary colors are red, green, and blue.

When mixing equal amounts of light,red + green = yellowred + blue = magentagreen + blue = cyan

red + green + blue = white

Subtractive Color (Pigments)

Only those frequencies not absorbed by any ofthe pigments are seen.

The primary colors are magenta, yellow, andcyan.

When mixing equal amounts of pigment,magenta + yellow = redmagenta + cyan = blueyellow + cyan = green

magenta + yellow + cyan = grey/black


8/18


FLUID MECHANICS

Static Fluids

Fluids are materials that can flow, and includeboth gases and liquids (and sometimesamorphous solids).

The mass density is the mass per unit volume.It is denoted by the Greek letter rho,.

Thespecific gravity of a substance is the ratioof its density compared to the density of acommon reference material. Specific gravityhas no units.

Usually the common reference material ischosen to be water at 4C with a density of1.000103 kg/m3.

In a static fluid, pressure is exertedperpendicularly to the surface of any object incontact with the fluid.

Pressure depends only on depth and density of

the fluid: Pbottom=Ptopg h

Average atmospheric pressure at sea level is1.013105 Pa.

The actual pressurePis known as the absolutepressure. The differenceP Patm= gh isknown as thegauge pressure.

Pascal's Principle

Any change in the pressure applied to acompletely enclosed fluid is transmittedundiminished to all parts of the fluid and theenclosing walls.

Hydraulics

F2

A2=

F1

A1g h

Buoyant Force

The upward force provided to an object whollyor partially immersed in a fluid is called thebuoyant force. The buoyant force exists

because fluid pressure is larger at greater

depths.FB= V g

Note that is the density of the liquid, not thedensity of the object, and Vis the volume ofdisplaced liquid (which will equal the volumeof the object if it is totally submerged, or may

be a fraction of the volume of the object if it isonly partially submerged).

Archimedes Principle

Any fluid applies a buoyant force to an objectthat is partially or completely immersed in it;the magnitude of the buoyant force equals theweight of the fluid that the object displaces.

Flowing Fluids

Insteady flow orlaminar flow the velocity ofthe fluid particles at any point is constant astime passes. Note that the velocity at different

points may be different from one another, but ateach point the velocity is constant.

Unsteady flow exists whenever the velocity at a

point in the fluid changes as time passes.

Turbulent flow is an extreme kind of unsteadyflow and occurs when there are sharp obstaclesor bends in the path of a fast moving fluid. Inturbulent flow the velocity at any particular

point changes erratically from moment tomoment, both in magnitude and direction.

A viscous fluiddoes not flow readily. Theviscosity hinders neighboring layers of fluidfrom sliding freely past one another. The flowof a viscous fluid is an energy-dissipating

process. A nonviscous fluidflows in anunhindered manner with no dissipation of

energy. No real fluid has zero viscosity atnormal temperatures, but some fluids havenegligibly small viscosities.

An incompressible, nonviscous fluid is calledan ideal fluid.

Flow Rates

The mass of fluid per second that flows througha pipe is called the mass flow rate.

mass flow rate= m t

=A v

The quantity Q=Av is the volume of fluidthat passes through the pipe each second and iscalled the volume flow rate. This remainsconstant as a pipe constricts or expands.

Bernoulli's Equation

P11

2 v1

2g y1=P21

2v2

2g y 2

Magnus Effect

When an object spins, air close to its surface isdragged around with it by surface irregularities.The air on the side rotating into its direction ofmotion is slowed down while the air on the siderotating away from the direction of motion issped up, resulting in pressure differences that

create a net force called theMagnus effect.

Torricelli's Law

Suppose you have a large tank, where thesurface is at atmospheric pressure, and there isa small hole or open pipe near the bottom of thetank. The speed at which the water exits thetank is called the efflux speed.

vefflux2

=vsurface2

2ghIf the tank is large, then the liquid level changesvery slowly and vefflux2gh Note that thisis the speed the water would have had it freelyfallen that height difference.

THERMODYNAMICS

Temperature

Temperature is a description of how hot or coldsomething is. It can be measured by observinga change in some thermometric property of anobject.

K= C273.15

F=9

5 C32

Absolute Zero

The phrase absolute zero means thattemperatures lower than -273.15 C cannot bereached by continually cooling a gas or anyother substance. If lower temperatures could bereached, then further extrapolation of thestraight line experimentally found on a P-Tgraph created with a constant volumethermometer would suggest that negativeabsolute gas pressures could be reached, whichis impossible as it has no meaning.

Thermal Expansion

is the coefficient of linear expansion.LL0

= T

Stress=F

A=Y

LL0

If a heated plate has a hole in it, the holeincreases in size in each dimension.

AA0

2 T

, is called the coefficient of volume expansion V

V0= T3 T

Heat

Internal energy is the sum of the molecularkinetic energy (due to the random motion ofmolecules), the molecular potential energy (dueto forces that act between the atoms of amolecule and forces that act betweenmolecules), and other kinds of molecularenergy.

Heat, Q, is energy that flows from a higher-temperature object to a lower-temperatureobject because of the difference intemperatures.

Q=mc TQ=mL

The proportionality constant, c, is thespecificheat capacity of the material. The latent heat,

L, is the heat per kilogram associated with aphase change.

Heat transfer will continue until a commontemperature, thermal equilibrium, is reached.

Mechanical Equivalent of Heat

1 cal = 4.186 J


9/18


Convection

When part of a fluid is warmed, the volume ofthe fluid expands and its density decreases.According to Archimedes' Principle, thesurrounding cooler and denser fluid exerts a

buoyant force on the warmer fluid and pushes itupward. As the warmer fluid rises, thesurrounding cooler fluid replaces it. This

cooler fluid, in turn, is warmed and pushedupward. This creates a continuous fluid flow,called a convection current, which carries alongheat. The transfer of heat is done byconvection.

Conduction

Conduction is the process whereby heat istransferred directly through a material, but any

bulk motion of the material plays no role in thetransfer.

Q=kA Tt

LThe proportionality constant, k, is called thethermal conductivity of the material.

Conduction happens best in metals because thefree electrons in the metallic bonds transferheat rapidly through the substance.

Radiation

The process of transferring energy viaelectromagnetic waves is called radiation; itdoes not require a material medium.

Q=eT4AtThe proportionality constant, , is called theStefan-Boltzmann constant.

=5.67 108J

sm2K4

The factore is the emissivity, which is a numberbetween 0 and 1 that represents the ratio of theenergy actually emitted by an object and what itwould emit if it were a perfect blackbody.

All bodies continuously radiate energy in theform of electromagnetic waves, though it may

be in a form other than visible light.

When a body has the same temperature as itssurroundings, the amount of radiant energy

being emitted must balance the amount ofradiant energy being absorbed.

Ideal Gas Law

An ideal gas is an idealized model for realgases that have sufficiently low densities. Thiscondition means that the molecules of the gasare so far apart that they do not interact otherthan via collisions that are effectively elastic.

PV= nRTPV=NkT

The proportionality constant is the universalgas constant,R, which has been experimentallydetermined to be 8.31 J/(mol K). Related tothis isBoltzmann's constant, k=R / which is1.3810-23 J/K.

Pressure of a Gas (Molecular Scale)

P= N3 m vrms2

L3 = N3 m vrms

2

V Kinetic Energy of a Gas (Molecular Scale)

KE=1

2m vrms

2 =3

2k T

Internal Energy of a Gas

U=N12 m vrms2 =N32 k T=32 n R TThermodynamics

Thesystem is the collection of objects uponwhich attention is being focused. Everythingelse in the environment is called the

surroundings.

The physical condition of the system is calledthestate of the system. It includes pressure,volume, temperature, and mass of the system.

The system and its surroundings must beseparated by walls of some kind. Walls that

permit the transfer of heat are called diathermalwalls. Perfectly insulating walls that do not

permit the flow of heat from the system to thesurroundings are called adiabatic walls.

The Zeroth Law of Thermodynamics

Two systems individually in thermalequilibrium with a third system are in thermalequilibrium with each other. Objects in thermalequilibrium will have the same temperature.

The First Law of Thermodynamics

U=U fUi=Q WQ is positive when the system gains heatQ is negative when the system loses heatWis positive when work is done by the systemWis negative when work is done on the system

Because internal energy depends only ontemperature,Uis determined once the initialand final temperatures are known.

Internal energy depends only on the state of asystem, not on the method by which the systemarrives at a given state.

Pressure-Volume Graphs

The area under the curve on a pressure-volumegraph is the work for any kind of process.

Isobaric Process

An isobaric process is one that occurs atconstant pressure.

W=P V

Isochoric Process

An isochoric process is one that occurs atconstant volume.

W=0

Isothermal Process

An isothermal process is one that occurs atconstant temperature.

W=nRTlnV fVi =Q

Adiabatic Process

An adiabatic process is one that occurs withoutany heat flow between the system and thesurroundings.

Q=0

W=32

nR TiTf

PiVi=Pf V f

=cPcV

monotomic ideal gas: =5/ 3diatomic ideal gas: =7/5


10/18


The Second Law of Thermodynamics

Heat flows spontaneously from a substance at ahigher temperature to a substance at a lowertemperature, and does not flow spontaneouslyin the reverse direction.

Heat Engines

A heat engine is any device that uses heat to

perform work.

The efficiency, e, of a heat engine is defined asthe ratio of the work done by the engine to theinput heat.

e=W

Qin=1

Q out

QinA reversible process is one in which both thesystem and its environment can be returned toexactly the states they were in before the

process occurred.

A process that involves an energy-dissipatingmechanism or a spontaneous process cannot bereversible because the energy wasted wouldalter the system, the environment, or both.

Carnot's Principle

No irreversible engine operating between tworeservoirs at constant temperatures can have agreater efficiency than a reversible engine

operating between the same temperatures.Furthermore, all reversible engines operating

between the same temperatures have the sameefficiency.

An important feature of a reversible engine,called a Carnot engine, is that all of the inputheat originates in a hot reservoir at a uniformtemperture ofTH and all the waste heat goesinto a cold reservoir at a uniform temperatureofTC .

QCQH

=TCTH

e=1TCTH

RefrigerationIf work is used, heat can be made to flow fromcold to hot, against its natural tendency. The

process of removing heat from the coldreservoir and adding it to the hot reservoir iscalled a refrigeration process.

Coefficient of performance=QCW

If the process occurs reversibly, we have anideal device called a Carnot refrigerator orCarnot air conditioner.

QCQH

=TCTH

Coefficientof performance

=TC

THTC

Heat Pumps

The term heat pump is reserved for a homeheating device which used workWto makeheat QCfrom the wintry outdoors flow up thetemperature hill into a warm house.

Coefficient of performance=QHW

If the process occurs reversibly, we have anideal device called a Carnot heat pump.

QCQH

= TCTH

Coefficientof performance

= THTHTC

Entropy

Irreversible processes lose some ability toperform work. This partial loss can beexpressed in terms ofentropy.

The quantity Q / Tis called the change inentropy and applies to any process in whichheat Q enters or leaves a system reversibly at aconstant temperature T.

S=QT REntropy, like internal energy, is a function ofthe state or condition of they system. Only thestate of a system determines the entropy S thata system has. Therefore, the change in entropy

Sis equal to the entropy of the final stateminus the entropy of the initial state.

The Second Law and Entropy

The Second Law of Thermodynamics statesthat if the physical process is irreversible, thecombined entropy of the system and theenvironment must increase. The final entropymust be greater than the initial entropy for anirreversible process:

Sf> Si (irreversible process)

When a reversible process occurs, thecombined entropy of the system and theenvironment does not change.

Sf = Si (reversible process)

When an irreversible process occurs, and theentropy of the universe increases, the energyfor doing work decreases.

Wunavailable=TC Suniverse

We associate an increase in entropy with anincrease in disorder, and a decrease in entropywith a decrease in disorder (or a greater degreeof order).

The Third Law of Thermodynamics

It is not possible to lower the temperature ofany system to absolute zero in a finite numberof steps.

ELECTROSTATICS

Conservation of Electric Charge

During any process, the net electric charge ofan isolated system remains constant.

Usually electrons are transferred rather thanprotons, because they take less energy to moveas they are on the outside of the atom.

Whenever two different materials rub againsteach other it is likely that one will leave withmore electrons than it started with...the otherwill leave with less. This is calledtriboelectricity.

When a rubber rod is rubbed with animalfur, some of the electrons in the fur transferto the rubber rod.

If a glass rod is rubbed with silk cloth,some of the electrons in the glass transfer tothe silk.

Items that allow the easy flow of electrons arecalled electrical conductors. Most metals areconductors because of the nature of metallic

bonds.

Items that inhibit the flow of electrons arecalled electrical insulators. Most nonmetalsare insulators because of the nature of covalent

bonds and ionic bonds in solids.

Rubbing two objects together to make anelectrical imbalance is called charging by

friction.

Transferring electrons from one material toanother by simply touching is called chargingby contact.

If we bring a charged object near a conductingsurface, even without physical contact,electrons will move in the conducting surface.This can be used to charge the object byinduction, if the object is grounded.

Charge polarization occurs when a charged rodis brought near an insulator. There is arearrangement of the position of charges withinthe atoms and molecules themselves.

Coulomb's LawThe electrical force between any two point-charges with charges q1 and q2, separated by adistance robeys an inverse-square law:

Fe=kq1 q2

r2

The constant kis often expressed in terms of amore fundamental constant called the

permittivity of free space,

0=8.8541878171012 C

2

Nm2

k=1

4 0=8.98755178810

9Nm2

C2


11/18


Electric Fields

The electric field at the location of a pointcharge q0 is

E=Feq0

The electric field due to a point charge q is

E=k q

r2 =

q

4 0r

2

Ifq is positive, thenEis directed away from qIfq is negative, thenEis directed toward q

Electric Field Inside Conductors

At equilibrium under electrostatic conditions,any excess charge will reside on the surface ofa conductor, and the electric field is zero at any

point within a conducting material.

Gauss' Law for a Point Charge

The product of the magnitude of the electricfield at any point on the Gaussian surface andthe area of the surface is called the electric flux,

E=E A=q

0

Gauss' Law

Suppose we have a charge distribution whosenet charge is Q. The charge distribution issurrounded by a Gaussian surface with anyarbitrary closed shape. The direction of theelectric field need not be perpendicular to thesurface, and the magnitude of the electric fieldmay vary on the surface.

E= EcosA=Q0

Or, using calculus

E= Ed A= Q 0

Electric Potential

The electric potential,V, at a given point is theelectric potential energy EPE of a small testcharge q0 at that point divided by the charge.

V=EPE

q0The electric potential difference between two

points is related to the work per unit chargeinvolved in moving a charge between those two

points.

V=EPE

q0=

WABq0

Potential Difference from a Point Charge

V= k qr

relative to a potential of 0 V at infinity.

When two or more charges are present, thepotential due to all the charges is obtained byadding together the individual potentials.

Equipotential Surfaces

An equipotential surface is a surface on whichthe electric potential is the same everywhere.

The net electric force does no work as a chargemoves on an equipotential surface.

Parallel Plate Capacitors

Aparallel plate capacitorconsists of two metalplates, each with areaA. A charge +q is spreaduniformly over one plate, while a charge -q isspread uniformly over the other plate.

The electric field between the plates is

E=k q

r2 =

q

0 A

=

0where is the charge per unit area (also calledthe charge density). Except near the edges, thefield has the same strength at all places betweenthe plates, and the field does not depend on thedistance from the edges.

The potential difference between the capacitorplaces is

V=E ss is the displacement along a lineperpendicular to the plates

E= Vs

V/s is called thepotential gradient.

The amount of charge on the plates isproportional to the potential difference betweenthe plates: q=CV The constant Cis thecapacitance of the capacitor.

It is common practice to fill the region betweenthe conductors or plates with an electricallyinsulating material called a dielectric. Adielectric reduces the electric field strength

between the plates, allowing for more charge tobe stored on them at the same potential.

=E0E

E0 is the field magnitude without the dielectricEis the field magnitude with the dielectric

A dielectric will increase the capacitance of acapacitor: C=C0C0 is the capacitance without the dielectricCis the capacitance with the dielectric

The energy stored is E=1

2C V

2

Energy Density of an Electric Field

Energy Density=EPE

V=

1

2 0E

2

ELECTRIC CURRENT

Motion of Charged Particles in a Potential

Positive charges will accelerate from a regionof high potential to a region or lower potential.

Negative charges will accelerate from a regionof low potential to a region of higher potential.

Electromotive ForceBecause of the positive and negative charges onthe battery terminals, and electric potentialdifference exists between them. The maximum

potential difference is called the electromotiveforce, emf, of the battery, for which the symbolE is used.

Note that generally the potential differencebetween the terminals of a battery is a bit lessthan the maximum value indicated by the emf.

Conventional Current vs. Electron Flow

Positive charges are repelled from the positiveterminal and travel through the wire toward the

negative terminal. This is conventional current.

We are now aware that it is electrons that move,not positive charges, but we continue to useconventional current.

Current

In a circuit the battery creates an electric fieldwithin and parallel to the wire, directed fromthe positive to the negative terminal. This fieldexerts a force on the free electrons in the wire,and they respond by moving from the negativeterminal to the positive terminal. This flow ofcharge is known as an electric current.

I= q t

If the charges move around the circuit in thesame direction at all times, the current is said to

be direct current.

If the charges move first one way, and then theopposite way, changing direction from momentto moment, the current is said to be alternatingcurrent.

Resistance

The resistance,R, is the ratio of the voltage, V,applied across a piece of material to the current,

I, through the material.R=

VI

Ohm's Law (does not apply universally)V=IR

Electrical Power

P=IV=I2R=

V2

R


12/18


ELECTRIC CIRCUITS

Symbols

Battery (DC) Generator (AC)

Resistor Capacitor

Inductor (Air Core) Inductor (Iron Core)

Transformer (Air Core)Transformer (Iron Core)

Voltmeter Ammeter

Resistors

To the extent that a wire or electrical deviceoffers resistance to the flow of charges, it iscalled a resistor.

Resistors play an important role in electriccircuits, where they are used to limit theamount of current and establish proper voltagelevels.

The electrical resistance of a piece of material

of lengthL and cross-sectional areaA, is

R= LAThe proportionality constant,, is the resistivityof the material.

The resistivity of a material depends ontemperature. In metals the resistivity increaseswith increasing temperature, while insemiconductors, the resistivity increases withdecreasing temperature.

=0 [1 TT0 ]The term has the unit of reciprocaltemperature and is the temperature coefficient

of resistivity.

Internal Resistance

In a battery, the internal resistance comes fromthe chemicals within the battery. In a generator,the internal resistance comes from theresistance of the wires and other componentswithin the generator.

The internal resistance causes the voltagebetween the terminals to drop below themaximum value specified by the battery's emf.This actual voltage is known as the terminalvoltage.

Series Wiring

Series wiring means that the devices have beenconnected so that all the current flows througheach device.

The current through each device in a seriescircuit is the same.

The voltage will drop through each device, tobe built up again by the battery or other emfsource.

Series resistors: RS=R1R2R3

Series capacitors:1

CS=

1

C1

1

C2

1

C3

Parallel Wiring

Parallelwiring means that the devices havebeen connected so that the same voltage isapplied across each device.

The current into a parallel circuit is splitbetween each device.

The voltage applied to each device in theparallel circuit is the same.

Parallel resistors:1

RP=

1

R1

1

R2

1

R3

Parallel capacitors: CP=C1C2C3

Circuits Wired Partially in Series and

Partially in Parallel

1. Draw a schematic diagram.2. Start with the most embedded portion of

the circuit and calculate a single equivalentresistance for those resistors. Draw a newschematic.

3. Repeat until you can reduce the circuit to asingle resistor. Find the total circuit current

and then go back through the circuits tofind the currents and voltages acrossindividual resistors.

Kirchhoff's Rules

There are many circuits in which no tworesistors are in series or in parallel. In thatcase, we need to use Kirchhoff's Rules.

Kirchhoff's Current Law (Junction Rule)

The sum of the magnitudes of the currentsdirected into a junction equals the sum of the

magnitudes of the currents directed out of thejunction.

Kirchhoff's Voltage Law (Loop Rule)Around any closed circuit loop, the sum of the

potential drops equals the sum of the potentialgains.

Using Kirchhoff's Rules1. Assume all voltage sources and resistances

are given. (If not label them V1, V2, ..., R1,R2, etc)

2. Label each branch with a branch current.(I1, I2, I3, etc)

3. Find Kirchoffs first law equations for each

node.4. Find Kirchoffs second law equations for

each of the independent loops of the circuit.5. Solve the simultaneous equations as

required to find the unknown currents.

RC Circuits

Many electric circuits contain both resistors andcapacitors.

When the switch is closed, the battery begins todeposit charge on the capacitor plates. Theresistor slows down this process.

Assuming that the capacitor is uncharged attime t= 0 s when the switch is closed, and it isconnected to a potential difference V0, it can beshown that the magnitude q of the charge on the

plates at time tisq=q0 1e

t/RC, where q0=C V0The voltage across the capacitor at time tis

V=V01et/RC

When a circuit containing a capacitor is

disconnected from the voltage source, thecapacitor will send charge through the circuitand power it, until the capacitor is fullydischarged.

q=q0 et/RC

V=V0 et/RC

The termRCin the exponent is called the timeconstant,, of the circuit.


13/18


MAGNETISM

Like poles on different magnets repel eachother; unlike poles attract.

Magnetic Field

Surrounding a magnet is a three-dimensionalmagnetic field. The direction of the magneticfield at any point in space is the direction

indicated by the north pole of a small compassneedle placed at that point.

The magnitudeB of the magnetic field at a

point in space is defined as

B=F

q0 v sinwhereFis the magnitude of the magnetic forceon a positive test charge q0 and v is the velocityof the charge and makes an angle with thedirection of the magnetic field.

The strength of the magnetic field near theEarth's surface is about 110-4 T, also known asa gauss, G.

Magnetic Force

When an electric charge is placed in a magnetic

field, it experiences a force provided certainconditions are met:1. The charge must be moving relative to the

magnetic field2. The velocity of the moving charge must

have a component that is perpendicular tothe direction of the magnetic field

FB=qvBFB=q v B sin

Drawing Magnetic Fields

Out of Page Into Page

Lorentz Force (Electric and Magnetic)F=q E qv B

F=q Eq v B sin

When an electric force is applied to a positivelycharged particle, the path of the particle bendsin the direction of the force. Because there is acomponent of the particle's displacement in the

direction of the electric force, the force doeswork on the particle.

When a magnetic force is applied to apositively or negatively charged particle, italways acts in a direction that is perpendicularto the motion of the charge. Consequently, themagnetic force cannot do work on the particleand change its kinetic energy, although it doesalter the direction of the motion by providing acentripetal force.

q v B sin90 =mv2

r

r=mvq B

Magnetic Force on a Long, Straight WireF=I L B

F=I L B sin

Torque on a Current Carrying Loop

If the wire is wrapped to form a coilcontainingNloops, each of areaA, thenet torque is

=N I A B sin The quantityN I A is known as the magneticmomentof the coil, and its units are Am2.

Magnetic Field Produced by a Wire

B= 0 I2 r0=4 10

7Tm/ A

The constant0 is known as thepermeability offree space.

Loop of Wire

If a current-carrying wire is bent into a circularloop withNturns, the magnetic field linesaround the loop are concentrated in the centerand loop radially around the loop.

At the center B=N 0 I2R

Along axis B=N0 2 R

2I

4 r2R

23/ 2

Solenoids

Asolenoidis a long coil of wire in the shape ofa helix.

If the wire is wound so that the turns are packedclose to each other and the solenoid is longcompared to the diameter, the magnetic fieldlines inside are are nearly constant in

magnitude and directed parallel to the axis.B=0 n I

The magnitude of the magnetic field outside thesolenoid is not constant and is much weakerthan the interior field. In fact, the magneticfield outside is nearly zero if the length of thesolenoid is much greater than its diameter.

Ampre's Law

The general law known as Ampre's Law givesthe magnetic field at any point around a wire ofany geometrical shape.

Consider any arbitrary closed path around a

current, and imagine it as being made up ofshort segments of lengthl. We take the

product of the length of each segment times thecomponent of the magnetic field parallel to thatsegment. If we sum all these terms, the result isthe product of0 and the net enclosed current

Ienc.B // l=0 Ienc

If you let the lengthlgo to zero, then this

becomes Bdl=0Ienc

Atomic Explanation for Magnetism

Electrons orbiting the nucleus behave likeatomic sized loops of current. Each electron

has a spin that also gives rise to a magneticfield.

In most substances the magnetism produced atthe atomic level tend to cancel out, with theresult that the substance is nonmagnetic overall.

Ferromagnetic materials are materials wherecancellation of the atomic magnetic fields doesnot occur for groups of approximately 1016 to1019 neighboring atoms, because they haveelectrons spins that are naturally aligned

parallel to each other.

This alignment results in a special type of

quantum mechanical interaction between spins.The result is a small but highly magnetizedregion of about 0.01 to 0.1 mm in size, called amagnetic domain.

The magnetic domains can be forced to alignby placing the object in an external magneticfield. The domains who magnetism is parallelor nearly parallel to the external field grow insize by absorbing unaligned domains, while themagnetic alignment of other domains mayrotate and become more oriented in thedirection of the external field.


14/18


ELECTROMAGNETIC INDUCTION

Induced EMF and Induced Current

When a magnet moves relative to a coil of wire,an ammeter connected to the coil will read a

positive or negative current, depending on thedirection of motion of the magnetic field.Since a source of electromotive force, emf, isalways needed to produce a current, the coil

itself behaves as if it were a source of emf.This emf is known as an induced emf. Thecurrent caused by the induced emf in the coil iscalled an induced current.

Motional EMF

When a conducting rod moves through aconstant magnetic field, an emf is induced inthe rod. The charged particles in the conductorare carried along with the moving conductor, sothey experience a force from the magnetic field

that causes positive charges to pile up on oneend of the conductor and negative charges to

pile up on the other end. This separation ofcharge creates an electric field inside theconductor. The charges that pile up create avoltage or emf across the length of the rod thatis constant. This induced emf is also called amotional emf.

qE=qvB sin 90

qEL =qvBE= vBL

Magnetic Flux

The magnetic fluxB for a uniform magneticfield through a loop of areaA is defined asB= BA=B A=B A cos

If the area is not a flat surface, or if themagnetic field is not uniform, then themagnetic flux is defined as

B= BdA

Faraday's Law of Electromagnetic Induction

Whenever there is a change in flux (over time)through a loop of wire, an emf is induced in theloop.

E=B / t

The minus sign reminds us that the induced emfwill oppose the change in the magnetic flux.

Faradays law states that an emf is generated ifthe magnetic flux changes for any reason. Since = BA cos , any change of B, A, orwillinduce an emf.

If the circuit containsNclosely wrapped loops,the emfs induced in each loop add together, so

E=N B

t

Induced Magnetic Fields

A changing magnetic field produces an inducedemf which drives a current around a circuit.This induced current produces an inducedmagnetic field which opposes the change in theoriginal magnetic field.

Lenz's Law

A current produced by an induced emf movesin a direction so that its magnetic field opposesthe original change in flux.

Determining the Polarity of the Induced EMF1. Determine whether the magnetic flux that

penetrates a coil is increasing ordecreasing.

2. Find what the direction of the inducedmagnetic field must be so that it canoppose the change in flux by adding to orsubtracting from the original field.

3. Use the RHR-2 to determine the directionof the induced current. The polarity of theinduced emf can be assigned because

conventional current is directed out of thepositive terminal, through the externalcircuit, and into the negative terminal.

Mutual Inducatance

When an ac current is passed through a primarycoil, it generates a magnetic field. If thischanging magnetic field penetrates nearbysecondary coil, the secondary coil experiencesa changing magnetic flux and an induced emfand induced current appears.

The effect in which a changing current in onecircuit induces an emf in another circuit iscalled mutual induction.

The induced emf in the secondary coil isproportional to the magnetic flux in thesecondary coil, which is proportional in turn tothe change in current in the primary coil. Weintroduce a proportionality constant,M, calledthe mutual inductance, which is usuallymeasured experimentally.

NSS=M IP

ES=NS S t

=M IP t

Self Inductance

An emf can be induced in a current-carryingcoil by a change in the magnetic field that thecurrent itself produces. This is referred to as

self-induction.

Suppose a coil is attached to an ac generator.The alternating current creates an alternating

magnetic field that creates a changing fluxthrough the coil. The change in flux induces anemf in the coil in accord with Faraday's Law.If is the magnetic flux through one turn of thecoil, thenN is the net flux through the coilwithNturns. Since is proportional to themagnetic field, and the magnetic field is

proportional to the currentI, we can stateN=L I whereL is the constant of

proportionality and is called theself inductanceof the coil.

E=N

t=L

I t

Inductors

Because of their self-inductance, coils areknown as inductors. Like capacitors, inductorscan store energy in a circuit. This stored energyarises because a generator does work toestablish a current in the inductor.

E=12

L I2

whereIis the final current through the indcutor.

For the special case of a long solenoid, the selfinductance is L=0 n

2A l where n is thenumber of turns per unit length,A is the cross-sectional area, and lis the length of thesolenoid. Thus the energy stored is

E= 12 0B2A l

Energy Density of a Magnetic Field

Energy Density=Energy

Volume=

1

20B

2

Transformers

A transformeris a device for increasing ordecreasing an ac voltage. It consists of aferromagnetic core on which two coils arewound. The primary coil is on the generatorside withNp turns, and a secondary coil on theappliance side withNs turns.

Vs

Vp =Ns

Np =Ip

IsIn astep-up transformer, the number ofsecondary coils is larger than the number of

primary coils, so the secondary voltage ishigher than the primary voltage.

In astep-down transformer, the number ofsecondary coils is smaller than the number of

primary coils, so the secondary voltage is lowerthan the primary voltage.

The ratioNs:Np is known as the turns ratio ofthe transformer.


15/18


QUANTUM MECHANICS

Black Body Radiation

A black-body is an object that completelyabsorbs all of the electromagnetic radiationfalling on it. Thus it also emits perfectly too.

When an object is heated it emits radiationconsisting of electromagnetic waves

(microwaves, infrared, visible light, etc.) with awide range of frequencies. This explains whyheated objects appear to glow dull red, cherryred, orange, yellow, or white as they get hotter.

The English physicists Lord Rayleigh and SirJames Jeans derived an equation that agreedwell with experiments, but only at the low-frequency, long-wavelength (infrared) end ofthe spectrum. The classical theory predicted aninfinite intensity for the ultraviolet region and

beyond. This was dubbed the ultravioletcatastrophe.

The German physicist Maxwell Planckassumed that there was some electric oscillatorin objects that vibrated at higher and higherfrequencies as the object was heated. With thisassumption, he found a formula that matchedthe experimental data, but lacked a physicalreality.

Using a trick from calculus, he broke theenergies up into small discrete bits proportionalto the oscillator frequencies, namelyE = h f.However, if he allowed the energy chunks to goto zero as the procedure demanded, theequation simplified to the incorrect Rayleigh-Jeans formula. BUT, if he did not require thatthe energies e or the constant h go to zero, butremained finite, he obtained his own radiationformula which matched experimental evidenceexactly!

Planck had stumbled across a theoretical basisfor his experimental radiation law, but only if

the energy is discontinuous. Even though hehad no reason to accept this notion (and hehated its implications), he accepted it

provisionally for he had nothing better.

The small, discrete bit of energy is called anenergy quanta. Planck's constant is

h = 6.626 069 9110-34 J.s

Photoelectric Effect

The variable voltage source turns the collectorplate into a cathode with a surplus of electronsand the emitter into an anode with a lack ofelectrons, creating a retarding voltage in thevacuum tube that tries to force electrons backtoward the emitter plate.

When a light source is turned on, some of theremaining electrons in the anode are ejected. Iftheir kinetic energy is enough to overcome theretarding voltage, they make it to the collector

plate, the circuit is completed, and the ammetermeasures a current.

The electrons that make the journey andcomplete the circuit must have had energygreater than q V0 where q is the charge of theelectron and V0 is the voltage value where thecurrent stops entirely.

There is a well defined minimum voltage, V0that stopped any electrons getting through; V0does not depend at all on the intensity of thelight!

Doubling the light intensity doubles the numberof electrons emitted, which doubles the current,

but did not affect the energies of the emittedelectrons.

He found that the maximum energy of theejected electrons did depend on the frequency(color); shorter-wavelength higher-frequencylight caused electrons to be ejected with moreenergy.

He also discovered that there is a certainthreshold frequencyft that depends on the typeof metal, below which no photoelectrons wereejected, no matter how bright the light beam.Einstein showed that the puzzling features of

the photoelectric effect are easily explainedonce the illuminating radiation is understood to

be a collection of particles, orphotons.

The photons have energy quanta of magnitudeh f. These energy packets penetrate the surfacelayer of the metal of the target electrode and hitan electron.

The photon's energy is transformed into thekinetic energy of the electron, and some areejected. Note that in order to be ejected, eachelectron must do an amount of work to climbout of the atom to get into free space.

KEphotoelectron=EphotonWq V0=hfphotonhfthreshold

Bright-line Emission Spectra

The spectrum of light from a hot gas whenpassed through a prism was completelydifferent from the well-known rainbow-like

pattern from a heated solid, and different gaseshave different patterns.

The bright-line emission spectra of eachelement is different, a chemical fingerprint.

Heated Solid

Heated Na vapor

When light from a heated solid is passedthrough a cool gas, the reverse pattern appears,called a dark-like absorption spectra.

The Swiss mathematics teacher Johann Jakob

Balmer published the results of months of workspent manipulating the numerical values of thefrequencies of the lines of the visible hydrogenspectrum.

1

=R 1n f2

1

ni2

n is an integer andR = 1.097107 m-1 is theRydberg constant.


16/18


Spectral Series predicted for HydrogenLyman series arein the ultraviolet.

Balmer seriesstart as visiblelight, some UV.

Paschen, Brackett,and Pfund seriesare in the infrared.

Bohr's Model (incomplete)

Each possible electron orbit in Bohr's modelhas a fixed energy called its energy level. Thefixed energy levels are like rungs in a ladder,

but the energy rungs are spaced closertogether the further you get from the nucleus.The radii rungs are further apart as you getfurther from the nucleus.

Electrons can jump from one energy level to

another, but they must gain or lose just the rightamount of energy; electrons can't be betweenenergy levels. Electrons can only jump to anorbit where the its angular momentum willincrease or decrease by a multiple of

= h/2 because angular momentum isquantized.

Ln=mvn rn=n h2 n is theprinciple quantum numberof theelectron.

A sudden transition of the electron between twostationary states will produce an emission (n

decreases) or absorption (n increases) ofradiation with a frequency given by thePlanck/Einstein relation

h f=EiEfwhereEi andEf are the energies of the atom inthe initial and final stationary states. Thisemission or absorption occurs in a single abruptstep called an electron transition.

If the angular momentum of an orbiting body isknown, it is a simple matter to compute theradius and the energy of the orbit.

rn= h2

42

me k e2n

2

Z

When n = 1 andZ= 1, the value is 5.3 nm. Atthis value, called theBohr radius, the energy ofthe hydrogen atom is a minimum and the atomis said to be in itsground state.

En=22m k

2e

4

h2 Z

2

n2

En=2.181018J Z

2

n2

En=13.6eV Z

2

n2

Quantum Mechanical Model

As in the Bohr model, theprinciple quantumnumber, n, determines the total energy of theatom and determines the size of the orbital.It can have only integer values, n = 1, 2, 3, 4,

The orbital quantum number, also called theangular quantum number, l, determines the

angular momentum of the electron due to itsorbital motion. It determines the differentshapes of the orbits. It can only have integervalues l= 0, 1, 2, , n 1.

s-shell: l= 0 d-shell: l= 2p-shell: l= 1 f-shell: l= 3

The magnetic quantum number, ml, determinesthe angular momentum of the electron due to itsorbital motion. It determines the orientation ofthe orbital. It can only have integer values,ml = -l, ..., -2, -1, 0, 1, 2, , l

p-shell:px, py, pz

Thespin quantum numberdescribes the spin of

the electron itself.spin up ms = +, clockwise rotation

spin down ms = -, counterclockwise

Pauli Exclusion Principle

Each quantum state, characterized by the fourquantum numbers n, l, ml, and ms, in the atom islimited to one electron. If a state is occupied,the next electron must to to an empty higherenergy state, filling up the empty states fromthe lowest energy to higher energy.This is what keeps the atom from alwayscollapsing to its lowest or ground state andgives each element its characteristic structure,and that gives the Periodic Table its form.

Wave-Particle Duality

Young's double-slit experiment proved lightbehaved as a wave. Einstein's solution to thephotoelectric effect proved light behaves as aparticle. Light can behave as either, dependingon how you measure it.

Prince Louis de Broglie explained this as pilotwaves which accompany particles throughspace and time. He called these wavespilot

De Broglie Wavelength

=h

p=

h

mvThis applies not only to light, but to particles aswell.

The Compton Effect

Arthur Compton used the photon model toexplain his research on the scattering of x-rays

by the electrons in graphite. The x-ray photonwill recoil from the collision in one directionwhile the electron recoils from the collision inanother.

Compton observed

that the frequency ofthe scattered photonis less than thefrequency of theincident photon,indicating that the

photon loses energy.

He also found that that the difference betweenthe two frequencies depends on the angle atwhich the scattered photon leaves the collision.

The difference between the wavelength ' of thescattered photon and the wavelength of theincident photon is related to the scattering angle

by '= hmc

1cos

Heisenberg's Uncertainty Principle

Heisenberg realized that quantum theoryimplied a fundamental limitation on howaccurately certain pairs of physical variablescan be measured simultaneously.

He showed from this that there is no way ofaccurately pinpointing the exact position of asubatomic particle unless you are willing to bequite uncertain about the particle's momentum.Conversely, there is no way of pinpointing theexact momentum of a subatomic particle unlessyou are willing to be quite uncertain about the

particle's position.

x p

2, where =

h

2

E t

2, where = h

2

Max Born's Matrix Mechanics

The basic idea is that the frequencies of theoptical spectrum can be represented as aninfinite square matrix, as can the momentum pand displacement q of the oscillators. ThenHeisenberg's formula becomes the matrixequation

pqqp= h2 i

I

whereIis the identity matrix.This leads to a system of equations which could

produce the values of the frequencies andrelative intensities of spectral lines of atoms.Heisenberg was able to use this matrixformulation to derive all the classical resultswith his new theory, showing Newtonianmechanics and Maxwell's electromagnetism to

be special cases, and deduce the spectra ofhydrogen and the additional lines in the

presence of magnetic fields.


17/18


Erwin Schrdinger's Wave Mechanics

Erwin Schrdinger developed another versionbased on de Broglie's concept of matter waves.He found an equation which can be applied toany physics system in which the mathematicalform of the potential energy Vis known.

hi

2

2

t2=

h2

8 2 m2 V

is the wave itself, and is a function of bothposition and time.The solution to Schrdinger's equation was awave that described the quantum aspects of thesystem. The quantum transitions are nowviewed as energy passing continuously fromone vibration pattern to another rather thanfrom jumping electrons.The wave determines the likelihood that theelectron will be in a particular position, but thewave has no physical reality of its own (unlikea sound wave, electromagnetic wave, or waterwave). Each point in space around the nucleushas a probability that the electron might bethere. The region where the electron is found90% of the time, according to the wavefunction solution to the Schrdinger equation,is often called an electron cloud.

Paul Dirac's Transformation Mechanics

At first puzzled by the non-commutingquantities in Schrdinger's wave mechanics,Dirac realized that this was the essence of thenew approach. He quickly found a link toclassical physics and used the new fundamentalidea of non-commutability to develop his ownversion of quantum mechanics.

Dirac showed that both of the otherformulations of quantum mechanics could beviewed as special cases of his own, moregeneral, formulation. In other words, all three,though appearing quite different, are allequivalent.

Dirac also showed that quantum theory had theanswer to the apparent paradox of light being

both a particle and a wave. The concept of acontinuous field was now broken up into bits inorder to interact with matter, transforming itinto a quantum field. This new approach couldtreat light as waves or particles, and give theright answers either way. Since this work of

Dirac, the dual nature of light as wave andparticle has been free of paradox for those whocan follow the mathematics.

Quantum Electrodynamics (QED)

After World War II, Dirac's pioneering workwas carried forward by Richard Feynman,Freeman Dyson, Julian Schwinger, and Sin-Itiro Tomonaga. Their quantumelectrodynamics theory describes theinteraction of light and matter with remarkableaccuracy.

NUCLEAR PHYSICS

The nucleus of an atom consists of protons andneutrons, collectively known as nucleons.

proton p+ H11

neutron n0 n01

electron e

e10

The proton charge is +1.602 177 2210-19 CThe electron charge is 1.602 177 2210 -19 C

The number of protons,Z, in the nucleusdetermines the type of element and is called theatomic numberof the element.

The number of neutrons,N, in the nucleus of anatom can differ between atoms of the sameelement. The total number of protons andneutrons is called the atomic mass numberorthe nucleon number, of that atom,A. Atoms ofthe same element with different atomic massnumbers are called isotopes.

Xcountcharge

Z

A

The atomic mass of an element is the weightedaverage of the atomic mass numbers of thedifferent isotopes of that element.

An atomic mass unit, amu or u, is defined asexactly one-twelth the mass of a carbon-12atom.

1 amu = 1.660510-27 kg = 931.5 MeV( Note: 1 eV = 1.602210-19 J )

A proton has a mass of 1.007 276 uor 1.672 623 110-27 kg

A neutron has a mass of 1.008 665 u

or 1.674 928 610-27 kgAn electron has a mass of 0.000 548 579 9 u

or 9.109 389 710-31 kg

Nuclear Size

r1.21015 mA1/ 3

Strong Nuclear Force

The force that holds the nucleus together iscalled thestrong nuclear force. It is about 100times stronger than the electrostatic force, butits range of action is very short, being verystrong at distances less than a femtometer(10-15 m) but essentially zero at larger distances.For comparison, the 1s energy level forhydrogen is about 52,918 fm.

The strong nuclear force is almost independentof electric charge; at a given separationdistance, nearly the same strong force exists

between two protons, between two neutrons, orbetween a proton and a neutron.

Nuclear Stability

As the numberZof protons in the nucleusincreases, the numberNof neutrons mustincrease even more in order to keep the nucleusstable.

As more protons are present in a nucleus, therecomes a point where adding neutrons still can't

hold the nucleus together. Bismuth-209 is thelargest stable atom, anything larger is unstableand must break down into smaller atoms viaradioactivity.

Binding Energy and Mass Defect

Energy is required to separate a stable nucleusinto its constituent protons and neutrons. Thisenergy is called the binding energy of thenucleus.

In Einstein's theory of special relativity, massand energy are equivalent by the famousequationE= mc2, where c is the speed of lightin a vacuum. Thus the binding energy used todisassemble the nucleus appears as extra massin the separated nucleons.

The difference in mass between the separatednucleons and the stable nucleus is called themass defectof the nucleus.

Nuclear Stability Again


18/18


Radioactivity

Alpha rays, rays, are the least penetrating,being blocked by sheets of lead approximately0.01 mm thick. They consist of positivelycharged particles which are He2

4nuclei.

Beta rays, rays, penetrate a lead sheet tentimes as far, approximately 0.1 mm. They

consist of negatively charged particles whichare electrons or positively charged + particleswhich are positrons. These come from thenucleus, not the electron cloud.

Gamma rays, rays, can pass through a greatamount of lead sheeting, approximately 100mm. They are photons with short wavelengths,high frequencies, and high energies.

Radioactive Decay

The original nucleus is called theparentnucleus. The new nucleus, after the removalof the particle or particle, is called thedaughter nucleus.

Conservation of mass/energy, conservation oflinear momentum, conservation of angularmomentum, conservation of electric charge,and conservation of nucleon number must beobeyed during radioactive decay.

Alpha Decay

When the nucleus is too big, or has too manyprotons, it disintegrates via decay.

PZA

DZ2A4

He24

When a nucleus releases an particle, energy isalso released, due to the change in mass. Theenergy released appears as the kinetic energy of

the daughter nucleus, the kinetic energy of the particle, and a ray.

vd=m

mdv

Beta Decay

When the nucleus contains too manyneutrons, it disintegrates via decay.

PZA

DZ1A

e10

eIn decay, the weak interaction converts aneutron n into a protonp while emitting anelectron e and an electron antineutrino.

When the nucleus has too many protons to be

stable, but not enough nucleons to throw out an particle, it disintegrates via + decay.

PZA

DZ1A

e10

eIn + decay, the weak interaction converts a

protonp into a neutron n while emitting anpositron e+and an electron neutrino.

A third kind of decay sometimes occurs whenthe nucleus pulls in or captures one of theorbital electrons from outside the nucleus. The

process is called electron capture, orK capture,since the electron normally comes from theinnermost or K shell (n = 1).

PZA

e10

DZ 1A

Gamma Decay

The nucleus, like the orbital electrons, existsonly in discrete energy states or levels. Whena nucleus changes from an excited energy state(den

ap physics reveiw

Documents