Physics 123 Concepts - Summary Last updated December 3, 2010, 2:15 pm

Includes Chapters 19, 20, 21, 22, 23, 24, 25, 29, 30, 31, 32 (Fall 2010)

Note: Although I have done my best to check for typos and list the formulas correctly, you should verify the formulas are correct before using them. Make sure you know what all the variables represent in any particular formula. Some letters are used in different formulas from different chapters and may represent different things. - Dr. Nazareth

Chapter 19 (Electric charges, forces and fields) – updated for current textbook – 9/22/10 Electric charge (19.1)

• Intrinsic property of matter • Two types: positive and negative • Magnitude of charge on an electron or a proton = e • SI units = coulomb (C) • e = 1.60 x 10-19 C • electric charge is quantized – it can only be an integer multiple of e

Electric conductors and insulators (19.2)

• insulator: material where charges are not free to move • conductor: materials that allow charges to move somewhat freely • semiconductor: material with properties in between conductors and insulators

Coulomb’s Law - Electrostatic Force (19.3)

• Law of conservation of electric charge • “Like charges repel and unlike charges attract each other.” • Electrostatic: all charges are at rest • Coulomb’s Law

o (magnitude only)

• note absolute value of point charges used • r = distance between the point charges • SI units = Newton (N) • direction – acts along line between the two point charges and is attractive

for oppositely charged point charges and repulsive for like charged point charges

• k = 8.99 x 109 N·m2/C2 o Similar in form to Newton’s law of gravitation, except, force may attract or repel,

depending on signs of the charges o If more than two charges, than total force on a charge is the vector sum of the

forces from each pair. Break problem into parts and calculate the net force (vector sum). See chapter 19, examples 19-2 and 19-3, pgs. 661-663

o If total charge of Q is distributed over surface of a sphere, then treat sphere as a “point” located at the center of the sphere.

• Q = σA (σ = surface charge density; A = surface area of sphere) • A = 4πR2 (R = radius of sphere)

• (q is located outside sphere at distance r from center)

Electric Field (19.4-19.5, some 19.6)

• Definition o E = F/q0 o SI units = Newton per coulomb (N/C) o Vector quantity – direction same as direction of electric force on positive test

charge o Test charge, q0

• Small enough not to disturb surrounding charges • Positive

• “the electric field is the force per unit charge at a given location” pg 665 o If you know E, then force felt by charge q is F = qE o Direction of force depends on sign of charge q

• If q = + then F same direction as E • If q = - then F opposite direction as E

• Point charge

o (magnitude only - direction depends on whether q is + or -)

• Points radial out for q = + and radial in for q = - o If have more than one point charge, then the electric field, E, is just the vector

sum of the electric fields due to each charge separately, at that particular location • See example 19-5, pg 668-669

• Electric field lines = lines of force o Point from + charges to – charges o Do not stop or start midspace

• Start at positive charges or infinity • End at negative charges or infinity

o Density is proportional to field strength (more lines per area when field is stronger)

• Parallel plate capacitor o See figure 19.17

o (magnitude only; only between plates away from edges)

• σ = q/A = charge per unit area = charge density • E points from positively charged plate to negatively charged plate

• Inside a conductor (19.6) o At equilibrium (electrostatic conditions)

• any excess charge is on surface of conductor • E = 0 at any point inside the material of the conductor (not a cavity within

the conductor) • E just outside conductor is perpendicular to surface

• Conductor shields inside from outside charges, but doesn’t shield outside world from charges enclosed within

• Sharp point in a conductor – charges more densely packed here so the electric field is more dense outside sharp point

Charging by Induction (19.6)

• Charge an object without making direct physical contact • How to:

o connect object to ground using a grounding wire o bring charged rod nearby – the like charge is repelled away down the grounding

wire (now object has net charge) o remove grounding wire while charged rod still in place o now remove rod and excess charge distributes itself about the object

• NOTE: induced charge is opposite charge of that on charged rod (object charged by touch has the same charge as the charged rod)

Gauss’s Law (19.7)

• Electric flux: Φ=EA cosθ o E = electric field magnitude o A = area of surface o θ = angle between direction of E and the perpendicular to the area A o think of electric field lines “flowing” through the surface of area A o SI unit: N m2/C o if surface A is closed (like a sphere - a rectangle would be open)

• flux is positive if E field lines are leaving the enclosed surface • flux is negative if E field lines are entering the enclosed surface

o Permittivity of free space, ε0 = 1/(4πk) = 8.85 x 10-12 C2/(N·m2) • Gauss’s law: if charge q is enclosed by any arbitrary surface, Φ = q/ε0

o shape of surface doesn’t have to be a sphere!!! • Use Gauss’s law to find the electric field in highly symmetric situations

Chapter 20 (Electric Potential and Electric Potential Energy) – updated for current textbook – 10/3/10 • NOTE: textbook uses U for electric potential energy and I use EPE Electric Potential Energy and the Electric potential (20.1) • As charge moves from A to B, work WAB is done by electric force:

o WAB = EPEA - EPEB o EPE = electric potential energy

SI units = joule (J) = N·m • For a positive test charge, q0, moving upward a distance, d, in a downward pointing uniform

electric field o W = -q0Ed o Since ΔPE = -W, then ΔPE = q0Ed o Electric force does negative work to move positive charge upward so the change

in potential energy is positive (it gets larger)

o Compare this to lifting a ball upward in the gravitational field … the potential energy gets larger as you lift the ball higher

• For a negative test charge, q0, moving upward a distance, d, in a downward pointing uniform electric field, the change in potential energy is negative (gets smaller) because the electric force does positive work to raise the negative charge upward

• CHANGE IN ELECTRICAL POTENTIAL ENERGY DEPENDS ON SIGN OF CHARGE AND ITS MAGNITUDE

• Electric potential, or simply, potential

o

o SI units = volt (V) = joule/coulomb (J/C) o Not a vector quantity, but can be positive or negative o Electric potential and electric potential energy are NOT the same thing

• Cannot determine V or EPE in the absolute sense because can only measure the differences, ΔV and ΔEPE, in terms of the work, WAB

o just like the gravitational potential energy is always relative to a reference level (e.g., ground level = 0 gravitational potential energy)

• Potential difference

o

o “a positive charge accelerates from a region of higher electric potential toward a region of lower electric potential”

• Electron volt: energy change an electron has when it moves through a potential difference of 1 V.

o 1 eV = e(1V) = (1.60 x 10-19 C) (1 V) = 1.60 x 10-19 J • Connecting electric field and rate of change of electric potential difference

o SI units = volts/meter = V/m

o “the electric field depends on the rate of change of the electric potential with position.” Pg. 693, Physics, 4th ed., J.S. Walker, 2010.

Can think of V like height of a hill and E as the slope of that hill o Electric potential decreases as you move in same direction as the electric field

Can think like going downhill … potential decreases o In general, only gives component of E along displacement, Δs

ΔV = -ExΔx (displacement in x-direction) ΔV = -EyΔy (displacement in y-direction)

Energy Conservation (20.2) • Total energy now Etotal = KEtranslational + KErotational + PEgravitational + PEspring + PEelectrical

o Etotal = ½ mv2 + ½ Iω2 + mgh + ½ kx2 + EPE • EPE = qV • If no work is done by non-conservative forces, then energy is conserved

o Initial Energy = Final Energy = E0 = Ef o Electrostatic (electric) force is conservative

• “Positive charges accelerate in the direction of decreasing electric potential.” pg 696, Walker o (can think: positive charges speed up rolling “downhill”)

• “Negative charges accelerate in the direction of increasing electric potential.” pg 696, Walker • For both positive and negative charges, as they accelerate, they move to a region of lower

electric potential energy Electric Potential Difference from point charges (20.3)

• SI units = volt, V

o V above not absolute, but rather how potential differs at a distance, r, as compared to a distance of infinity from the point charge.

o Assumes V = 0 at r = ∞ o So a positive q, puts potential everywhere above the zero reference value. o So a negative q, puts potential everywhere below the zero reference value. o Can add the potential from multiple point charges at a location

Its an algebraic sum (meaning signs matter), NOT a vector sum See chapter 20, examples 20-3 and 20-4

• Electric potential energy for point charges q and q0 separated by distance, r o EPE = q0V = kq0q/r SI units = Joule, J

• Note: r = distance NOT displacement so r is always positive Equipotential surfaces (20.4)

• Potential is same everywhere on an equipotential surface • “The net electric force does no work as a charge moves on an equipotential surface.” • Electric field is

o always perpendicular to an equipotential surface o points in direction of decreasing potential

• “Ideal conductors are equipotential surfaces; every point on or within such a conductor is at the same potential.” Pg. 703, Physics, 4th ed., J.S. Walker, 2010

o electric field lines meet the conductors surface at right angles Capacitors (20.5)

• Stores electric charge, thus it stores energy • Capacitance

o Q = CV o SI units of capacitance = farad (F) = coulomb/volt (C/V) o Depends on the geometry of the capacitor plate (or conductors) and the dielectric

constant of material between the plates

o Parallel plate capacitor without a dielectric,

• A dielectric can be inserted between the plates of a capacitor to increase the capacitance o Reduces electric field between plates in the dielectric. This increases the amount

of charge that can be stored for a given electric potential difference between the two capacitor plates.

o Dielectric constant, κ = E0/E (unitless) κ > 1

o Parallel plate capacitor with a dielectric,

o C = κC0 (applies to any capacitor, not just parallel plate) • Dielectric breakdown: when the electric field applied is large enough to force the

dielectric to conduct electricity o Dielectric strength: maximum e-field before breakdown

See table 20-2, pg 711, Physics, 4th, J.S. Walker, 2010 Electrical Energy Storage (20.6) • Energy stored – work done to charge up plates, increasing potential difference

o This work “stored as electric potential energy in the capacitor”

o

• Where is energy of a capacitor stored? In the electric field between the plates

o Energy density = uE=

o True for any electric field whether in capacitor or not κ=1 if no dielectric uE = (½)ε0E2

Chapter 21 (Electric Current and DC Circuits) – updated for current textbook – 10/08/10 Electromotive “force” and current (21.1)

• Electromotive “force”, emf = maximum potential difference between the terminals of a generator or battery in a circuit

o NOT a real force … comes from historic description

• Electric current,

o SI units = ampere (A) = coulomb/second (C/s) o Direct current (dc) – charge moves around circuit in same direction all the time

Batteries produce dc current o Alternating current (ac) – charges move first one way then the other, then back,

and so forth (Discussed in chapter 24) • Emf controls work that battery must do to move charge around a circuit

o W = ΔQε • Conventional current – hypothetical flow of positive charges in the circuit

o Flows from positive terminal of battery through circuit to negative terminal o Flows from higher potential to lower potential (hence the positive → negative) o In reality, negatively charged electrons flow in the circuit and go the opposite

direction of the conventional current Ohms’s Law, resistance, and resistivity (21.2)

• electrical resistance is voltage applied across a piece of material divided by the current thru the material

o

o SI units = ohm (Ω) = volts/ampere

• If V/I is constant (the same) for all values of voltage and current (at a given temperature) then the material follows Ohm’s Law (not really a “law” … it’s an observed relationship)

o Ohm’s Law: V/I = R = constant • Resistance of a material depends on geometry and resistivity (a material property)

o

o Resistivity is an inherent property of the material and depends on the temperature SI units = Ω·m = Ohm·meter ρ = ρ0 [1 + α(T-T0)] α = temperature coefficient of the resistivity (unit = 1/temperature)

• If α > 0, then ρ increases with temperature (e.g., metals) • If α < 0, then ρ decreases with temperature (e.g., semiconductors)

o R = R0[1 + α(T-T0)] (R, R0 are resistances at temperatures T, T0, respectively) Energy and Power in Electric Circuits (21.3)

• In a circuit with voltage, V, and current, I, electric power delivered to the circuit is o P = IV o SI units = Watt (W) = Joule/second (J/s)

• For a resistor, the power dissipated in the resistance is o

o

• energy usage: kilowatt-hour = (1000 W)(3600 s) = 3.6x106 J Series and Parallel Wiring (21.4, 21.6)

• Series – devices connected one after the other so the same current passes through each o Resistors: Equivalent resistance, Rs = R1 + R2 + R3 + ···

dissipates the same total power as the series combination

o Capacitors: reciprocal of the equivalent capacitance,

Carries the same amount of charge as any one of the capacitors in a combination

Stores the same total energy as the series combination • Parallel – devices connected so that the same voltage is applied across each device

o Resistors: reciprocal of the equivalent resistance,

dissipates the same total power as the parallel combination o Capacitors: Equivalent capacitance, Cp = C1 + C2 + C3 + ···

Each individual capacitor carries different amount of charge Equivalent capacitance carries same total charge as parallel combination Equivalent capacitance stores same total energy as parallel combination

• If circuit is wired partially in series and partially in parallel, often times the circuit can be analyzed part by part, each section following the rules of series or parallel wiring as applies.

• Internal Resistance

o Although we often assume perfect conductors, things like batteries and generators have resistance in the materials that make them up. We call this internal resistance.

o This internal resistance causes the voltage between the terminals of the battery (or generator) to be less than the emf when current is drawn from the battery.

o Terminal voltage = emf – Ir (r = internal resistance) Kirchhoff’s Rules (21.5)

• Junction rule: o Sum of the current in = sum of the current out (if adding absolute values only)

for example: I1 = I2 + I3 (absolute values) o OR all (positive) currents in plus all (negative) currents out = 0

For example: I1 - I2 - I3 = 0 Currents entering a junction are considered to be positive Currents leaving a junction are considered to be negative

• Loop rule: o Around any closed circuit loop, the sum of the potential drops = the sum of

potential rises o OR algebraic sum of all potential differences around any closed circuit loop is

zero Electric potential decreases as you cross a resistor in the direction of

current flow o Reasoning strategy

First, decide on the direction of the current on each segment of the circuit. Second, mark resistors with + and – signs. (Current flows from + to -). Third, choose a direction (clockwise or counterclockwise) and go around a

complete loop, adding up the potential drops and potential rises. • V = IR across a resistor.

If you solve for the current and you get a negative current, then you have chosen the current direction incorrectly. The current flows the opposite direction on that segment than what you originally chose.

Measurement of Current and Voltage (21.8)

• Ammeter – measures current o Must be inserted in the circuit (in series) o Analog version includes a galvanometer and a shunt resistor connected in parallel

Shunt resistor extends the range by providing a bypass for the current exceeding the galvanometer’s full scale limit

• Voltmeter – measures voltage between two points in a circuit o Must be connected across the part of the circuit to be measured (connected in

parallel) o Analog version includes a galvanometer and an external resistor connected in

series (so it is in series with the resistance in the galvanometer coil) External resistor extends the range of the galvanometer by splitting the

voltage between the coil resistance and the external resistance

RC Circuits (21.7) • Circuit with resistors and capacitors • Once circuit is complete (time, t = 0), charge starts flowing and capacitor charges up until

the charge on the plates reaches its equilibrium value, q0 = CV0 o (V0 = voltage from battery) o q = q0[1 – e-t/(RC)] capacitor charging

e = the “e” from the natural logarithm (i.e., e = 2.718…) τ = RC = time constant of the circuit (in seconds)

• τ = time for capacitor to gain ~63.2% of its total charge o q = q0[e-t/(RC)] capacitor discharging (no battery in the circuit)

τ = time for capacitor to lose ~63.2% of its total charge o Voltage across the capacitor at time t: V = q/C

Safety and the Physiological Effects of Current (Not covered in J.S. Walker textbook, but it it interesting)

• What causes electric shock to the body? … the current or the voltage? o The damage comes from the current passing through the body o But current flows through the body because there is a potential difference between

two parts of the body This is why there are so many warnings about high voltage.

o Safety feature – electrical grounding Third prong in a plug connected to a copper rod in the earth/ground Copper has lower resistance than the body so the current “prefers” to flow

through the copper rod into the ground and not your body → safe!

Chapter 22 (Magnetism) – updated for current textbook – 10/19/10

The Magnetic Field (22.1) • Magnetic force = force due to moving electrical charges • North and south magnetic poles • Like poles repel and unlike poles attract • Magnetic field surrounds the magnet

o A vector field has both a magnitude and a direction at every point surrounding the magnet

o Magnetic field lines (lines of force) point from north pole to south pole o Strength of field (magnitude) proportional to lines per unit area

Stronger where lines closer together Weaker where lines farther apart Strongest at the poles of the magnet

The Magnetic force on moving charges (22.2)

• A charge in a magnetic field experiences a magnetic force if … o Charge is moving AND o Velocity of charge must have some component perpendicular to the direction of

the magnetic field • Right Hand Rule Number 1 (find direction of magnetic force)

o Point fingers in direction of magnetic field o Point thumb in direction of the velocity of the charge o Palm faces direction of magnetic force on a positive particle o If the particle has a negative charge, the force points in the opposite direction.

• Magnetic force (magnitude) o F = q0Bvsinθ

0 ≤ θ ≤ 180° • Magnetic field (a vector field)

o (magnitude)

o Direction: determine using a small compass needle o SI units: telsa (T) = N·s/(C·m) = N/(A·m) o Earth’s magnetic field near the earth’s surface ~10-4 T = ~1 gauss (not a SI unit)

The Motion of Charged Particles in a Magnetic Field (22.3)

• Compare positively charged particle moving in a constant electric field (like between plates of parallel plate capacitor) and moving in a uniform magnetic field

o Electrostatic force direction is parallel (antiparallel) to the electric field direction Electric field does work on the charged particle

o Magnetic force direction is perpendicular to the magnetic field direction Magnetic field does NO work on the charged particle so does NOT change

the (kinetic) energy or the speed of the particle – only changes direction • A special case: velocity of charged particle is perpendicular to a uniform magnetic field

o Get circular trajectory of particle

o

Radius of circle inversely proportional to magnitude of magnetic field • Mass spectrometer – used to determine abundance of ionized atoms or molecules with

different masses

o

Force and torque on a current carrying wire (22.4-22.5)

• Current is moving charges, so a current carrying wires experiences a force due to the magnetic field if wire is oriented so that it has at least a small component perpendicular to the magnetic field.

• Use RHR #1 to find direction of the force on the wire. • F = ILBsinθ

o max when wire perpendicular to magnetic field (θ = 90°) o vanishes when wire parallel to magnetic field (θ = 0° or 180°)

• electric motor – changes electrical energy into mechanical energy o “When a current carrying loop is placed in a magnetic field, the loop tends to

rotate such that its normal becomes aligned with the magnetic field.” Physics, Cutnell and Johnson, 6th edition

• Torque on current carrying loop or coil

o τ = NIABsinφ • N = number of loops or turns • A = area of loop • I = current in the loop • B = magnitude of the magnetic field • φ = angle between the normal to the plane of the loop and the magnetic

field o magnetic moment = NIA (SI units: A·m2)

Magnetic fields produced by currents and Ampere’s Law (22.6-7)

• a current carrying wire produces a magnetic field of its own • Right hand rule #2 (RHR #2)

o Point thumb in direction of conventional current o Curl fingers in a half circle – they point in the direction on the magnetic field

• Magnetic field magnitude for …

o Infinitely long straight wire:

• two parallel current carrying wires • attract if both have current going same direction • repel if the wires have current going opposite directions

o At the center of a circular loop (1 loop):

o At the center of a circular loop (N loops, all same radius R):

• Current carrying circular loops like short bar magnets with “north” and “south” poles .. so two separate current loops can attract or repel depending on how the “poles” of the loops are aligned

o Solenoid: (n = # turns per unit length) • Long coil of wire wound tightly in helix shape; length is very long

compared to the diameter of the coil • Electromagnet • Magnetic field inside (away from the ends) is nearly constant in

magnitude and parallel to the axis of the coil • Ampere’s Law

• For static (unchanging through time) magnetic fields • enclosed

o ∆l = length of straight line segment of wire o µ0 = 4π x 10-7 T·m/A

• Much easier to do with calculus Magnetic materials (22.8)

• Where are moving charges in a solid magnet? o In the electrons in the atoms (not bulk motion of charges like current) o Two kinds electron motion contribute to make net magnetic field

Spinning motion (spin about own axis like a top)

Orbital motion (revolve around nucleus) • Most materials are not magnetic because the electrons spin in opposite directions and

cancel each other out • Ferromagnetic materials – you don’t get complete cancellation

o Iron has 4 electrons whose spin is not cancelled Each iron atom is a tiny “magnet”

o To lesser degree … nickel, cobalt, chromium and a few others o Get groups of neighboring atoms to align with each other and form a magnetic

domain (microscopic in size) o Common iron objects, like iron nail, not magnets because the domains are not

aligned o But when in external magnetic field, some of domains will align and magnet will

then pick up the iron nail o Remove external field and domains go back to random orientation and iron nail

no longer magnetized o If external field was very strong, some residual alignment may remain

Chapter 23 (all of it) – updated for current textbook – 11/19/10 11:55 pm Induced emf and induced current (23.1)

• Electricity can be produced from magnetic fields (just as magnetic fields are produced from moving charges)

• Ways to produce an induced emf (and an induced current on a complete circuit) o Relative motion between a magnet and a coil of wire o Changing the area of a coil in a uniform magnetic field o Moving a conductor in a uniform magnetic field (motional emf) o etc. … see Magnetic flux

Magnetic flux and Faraday’s law of magnetic induction (23.2-23.3)

• All previously discussed ways to produce an induced emf can be described through the concept of magnetic flux.

• Magnetic flux, o B = magnitude of magnetic field o A = surface area o φ = angle between direction of B and the NORMAL to the surface o SI units = Weber (Wb) = T·m2

• An induced emf is caused by a changing magnetic flux with time. o The magnetic flux can change because …

• The magnetic field changes (magnitude and or direction) • The area of the loop/coil changes • The angle between the normal to the surface of the loop/coil and the

magnetic field direction changes • Or more than one of these

• Faraday’s law:

o N = number of loops in the coil

o To get only the magnitude of the emf, take the absolute value of Faraday’s law. Faraday’s Law (23.4)

• Lenz’s law says that an induced current or emf always acts to oppose the change that caused it.

• If you get an induced current in a wire, you will get an induced magnetic field! So the net magnetic field is the original magnetic field that is related to the changing magnetic flux + the induced magnetic field. You have a net magnetic field!

• Reasoning strategy to determine polarity of induced emf (and induced current) o Determine whether the magnetic flux through your coil/loop is increasing or

decreasing. o Figure out which direction of induced magnetic field will oppose the change in

magnetic flux. o Once you know the direction of the induced magnetic field, use RHR #2 to

determine the direction of induced current. • Your induced magnetic field is not always pointing the opposite direction of the external

magnetic field, because Lenz’s law says it must oppose the change in magnetic flux.o Example: move a permanent magnet away from a loop of wire. The magnetic

flux is decreasing, so the induced magnetic field points the same direction as the permanent magnet.

Motional EMF (23.5)

• Magnitude of motional emf when : o Where does energy come from? From mechanical force keeping conductor

moving at a constant velocity. • Pmechanical = Fv

o Why do you need a mechanical force to keep something moving with constant velocity? There is a magnetic force trying to slow the conductor down … a magnetic force caused by the induced current in the conductor. This magnetic force points opposite the velocity (from RHR #1).

• FB = ilB • Fmechanical has same magnitude as FB, but points in opposite direction

Electric Generator (23.6)

• You input mechanical energy and the generator transforms it into electrical energy • Simple AC generator – planar coil of wire mechanically rotated in a uniform magnetic

field o emf = NBAωsinωt

ω = 2πf valid for any planar shape of coil with area A emf varies sinusoidally with time

Electric Motor (23.6) • electric motor similar to electric generator in that both consist of coil of wire that rotates

in a magnetic field and convert energy between mechanical energy and electrical energy • in an electric motor there are two sources of emf

o applied emf, V, to turn motor (from outlet plug)

o induced emf, emfb, from rotating coil (called back emf or counter emf) • two sources have opposite polarities, so you get a net emf

o net emf = V - emfb *** note: this textbook (and Fall 2010 class) only covers electric motors qualitatively, so don’t worry about net emf formula for electric motors

Mutual Inductance and Self Inductance (23.7)

• Mutual induction (covered in lecture, not in textbook) o Two coils … changing magnetic flux from primary coil causes an induced emf in

the secondary coil o Mutual inductance, M

Depends on geometry and ferromagnetic core material (in center of coil)

SI units = Henry (H) = (V·s)/A

o Emf due to mutual induction,

• Self induction (covered in lecture and textbook) o 1 coil … changing current in circuit induces emf in same circuit o Self inductance, or simply inductance of coil

€

L = N ΔΦΔI

SI units = Henry (H) = (V·s)/A

• common inductance values in mH (milliHenry) range

Solenoid inductance:

€

L = µ0N 2

l⎛

⎝ ⎜

⎞

⎠ ⎟ A = µ0n

2Al

• Inductor: coil of finite inductance o resists changes in electric current

o Emf due to self induction,

€

ε = L ΔIΔt

RL Circuits (23.8) • Resistor, R, and inductor, L, in series with a battery (emf) • Inductor resists changes in current … so takes time for current to reach full value in

circuit o exponential time dependence (like RC circuit had)

o

€

I =εR1− e− t τ( ) =

εR1− e−tR L( )

• Characteristic time: τ = L/R Energy Stored in Magnetic Field (23.9)

• Energy stored in an inductor, energy = ½ LI2 o Note: similar to energy = ½ CV2 for stored in capacitor

o Stored in solenoid’s magnetic field:

€

energy =12µ0

B2Al

• Energy density of magnetic field in vacuum, air, or non-magnetic material

o Energy density

• Transformers (23.10) • Transformers help reduce power loss in transmission lines by stepping up the voltage to

high levels while simultaneously reducing the current. o Small current, less power loss: P = I2R

o Transformer equation,

€

emfsemf p

=Ns

Np

=Vs

Vp

=IpIs

(Vs, Vp = terminal voltages)

o If Ns > Np then Vs > Vp (step-up transformer) o If Ns < Np then Vs < Vp (step-down transformer)

Chapter 24 (all of it) – updated for current textbook – 11/20/10 12:55 am Alternating Voltages and Current (24.1)

• Current that changes both magnitude and direction as a function on time o usually has a sinusoidal shape

• Caused by a voltage that is a sinusoidal function of time: V = V0sin2πft o V0 = peak value of the voltage o f = frequency at which voltage oscillates

• In a circuit containing only resistance: I = I0sin2πft o I0 = peak value of the current o I0 = V0/R

• Root mean square (a type of average)

o Voltage:

o Current:

• Power in circuit oscillates with time because current and voltage oscillate with time o Average power in the circuit, Pave = IrmsVrms o Average power dissipated in a resistor

• Phasors – rotating arrows that rotate counterclockwise at an angular frequency ω = 2πf o Represents magnitude and phase of voltage or current o Length of arrow represents the maximum (peak) voltage, V0, or maximum

(peak) current, I0 Instantaneous values of the voltage or current are equal to the vertical

components of the corresponding phasors (arrows) o The angle of the arrow, measured relative to the horizontal right pointing axis,

is the phase Resistors, capacitors and inductors individually in alternating current (AC) circuits (24.1-2, 24.4)

• Alternating voltage provided by AC generator o Usually sinusoidal voltage: V = V0sin2πft

o Root mean square (a type of average) voltage:

• Resistor only in AC circuit (with AC generator) o Resistance does NOT depend on frequency o Current in phase with voltage across resistor

In phase means voltage and current increase at the same time and decrease at the same time

I = I0sin2πft o Vrms = IrmsR

• Capacitor only in AC circuit (with AC generator) o Capacitor resists the flow of charge in the circuit

Capacitive reactance: (SI units = ohm (Ω))

Vrms = IrmsXC XC is proportional to 1/f and to 1/C

• Very large f → very small opposition to AC current flow • f = 0 → infinitely large opposition to the motion of charges so they

stop → no current o voltage and current NOT in phase (V and I do not reach peak (or trough) at the

same time) o voltage and current are π/2 radians or 90° out of phase o current “leads” the voltage by π/2 radians or 90°

if V = V0sin2πft, then I = I0sin(2πft+π/2) = I0cos(2πft) o because the voltage and current are out of phase by π/2 radians or 90°, a capacitor

consumes NO power on average • Inductor only in AC circuit (with AC generator)

o Inductor resists the flow of charge in the circuit AC current in coil sets up changing magnetic flux which causes an

induced voltage (Faraday’s Law) that opposes the change in the current (Lenz’s law)

Inductive reactance: (SI units = ohm (Ω)) Vrms = IrmsXL XL is proportional to f and to L

• As f increases, XL increases → very large opposition to AC current flow

• f = 0 (direct current), XL = 0 → doesn’t oppose current at all (DC) o voltage and current NOT in phase (V and I do not reach peak (or trough) at the

same time) o voltage and current are π/2 radians or 90° out of phase o voltage “leads” the current by π/2 radians or 90°

if V = V0sin2πft, then I = I0sin(2πft-π/2) = -I0cos(2πft) o because the voltage and current are out of phase by π/2 radians or 90°, an inductor

consumes NO power on average

Circuits with resistors, capacitors, and inductors in series (24.5)

• Although current is same through all circuit elements in a series RCL circuit, you cannot simply add up the voltages across the individual elements because they are not all in phase. It is like we have several velocity vectors to add but they point in different directions → you can’t ignore the directions, you have to take them into account when adding the vectors together.

• Taking the phase into account, we can determine the total opposition to the flow of charge in a series RCL circuit,

o impedance: (SI units = ohm (Ω)) Vrms = IrmsZ

• Phase angle of the series RCL circuit

o

o Angle between current phasor and voltage phasor across the series RCL circuit • On average, only resistance consumes power (not capacitors or inductors)

o o Cosφ called power factor of circuit

• If have only RC circuit or RL circuit, can use formulas for Z and φ above and just put in a zero for the variables corresponding to the absent circuit element

o e.g., if no capacitor, then C = 0 and XC = 0 so

€

Z = R2 + XL( )2 Resonance (24.6)

• Only one natural frequency in RCL circuit • Resonance when impedance is a minimum

o Z is minimum when Z = R because XC = XL o Z at a minimum gives maximum current in the circuit because Irms = Vrms/Z

• Resonant frequency,

Chapter 25 – updated for current textbook – 11/19/10 1:25 pm Electromagnetic waves, the E-M spectrum and the speed of light (25.1-3)

• Electromagnetic wave is made up of oscillating electric and magnetic fields that are oriented perpendicular to each other and to the direction of wave travel

• Transverse wave • Does NOT require a medium to travel through (unlike water waves, sound waves, etc) • All e-m waves travel through a vacuum at the same speed = speed of light

o c ≈ 3.0x108 m/s (in a vacuum or approximately air)

o

• Electromagnetic spectrum divided up into (from longest to shortest wavelength) o Radio waves o Microwaves

o Infrared radiation (Infrared waves) o Visible light o Ultraviolet (UV) radiation o X-rays o Gamma rays

• c = v = λf o low frequency means large wavelength o high frequency means small wavelength

Doppler shift of e-m radiation (25.2)

• speed of e-m waves is independent of motion of source and observer; only relative speed matters for what is observed

•

€

fo = f s 1±vrelc

⎛

⎝ ⎜

⎞

⎠ ⎟

o fo = frequency observed o fs = frequency output by source o vrel = relative speed between source and observer along line connecting source

and observer *** vrel << c *** o Use + when source and observer are moving toward each other o Use – when source and observer are moving away from each other

• “Blueshifted” when source and observer moving toward each other o inside parenthesis > 1 so fo > fs and λo < λs o wavelength shifted toward “blue” (shorter wavelength) end of the spectrum

• “Redshifted” when source and observer moving away from each other o inside parenthesis < 1 so fo < fs and λo > λs o wavelength shifted toward “red” (longer wavelength) end of the spectrum

Energy and momentum carried by e-m waves (25.4)

• energy density =

• in a vacuum (or air) the e-m wave carries equal amounts of energy/volume in electric field and magnetic field

o energy density, =

o Since E and B fields oscillate, above formulas give u at a particular instant in time

o Or, can get a type of average … and

o Electric and magnetic field energy densities are the same in an em wave

•

€

E =1ε0µ0

B = cB

• Intensity of e-m wave = amount of energy the wave delivers to unit area in unit time

o S = power/area = cu =

€

12cε0E

2 +12µ0

cB2 = cε0E2 =

cµ0B2

• Radiation pressure: pressure exerted by light

o Average pressure = average intensity/c = Save/c o Very small pressures compared to everyday life o Can be important for small particles like dust in solar system

Polarization (25.5)

• Direction of electric field in e-m wave defines polarization of that em wave • Unpoloraized light consists of waves with all different orientations for the electric field • Can polarize light by passing through a polarizer

o Intensity of unpolarized light cut in half when passes through a polarizer • S = ½ S0

o Intensity of polarized light cut when passes through polarizer at transmission angle θ relative to its polarization

• Malus’s Law: S = S0 cos2θ • Polarization of transmitted light now in direction of polarizer

• Can partially polarize light by reflection or scattering o Scattering: particles in atmosphere scatter sunlight; blue wavelengths (short)

affected most • Sunset is red because most of blue light is scattered away

o Reflection off smooth surface like smooth lake: reflected light is horizontally polarized

Chapter 29 – Added – 11/26/10 11:52 pm Special Relativity (29.1)

• Theory of Special Relativity (published 1905 by Albert Einstein • Altered our understanding of concepts of time. Length, mass, energy • Based on two simple postulates

o Equivalence of Physical Laws: “The laws of physics are the same in all inertial frames of reference.” Pg 1012, J.S. Walker, Physics, 4th edition

o Constancy of the Speed of Light: “The speed of light in a vacuum, c = 3.00 x 108 m/s, is the same in all inertial frames of reference, independent of the motion of the source or receiver.” Pg 1012, J.S. Walker, Physics, 4th edition

• Everything else that follows below is direct consequence of the above two postulates • [Remember: Inertial reference frames move with constant velocity (acceleration = 0) • Earth is almost an inertial reference frame; accelerations from orbit and rotation small

enough to ignore in most cases.] • [Light doesn’t need to move through a substance so it doesn’t move relative to that

substance. Experiments show same speed in all directions.] Relativity of Time and Time Dilation (29.2)

• Time is relative (it does not move forward at a constant rate) • Time (say as measured by identical clock on spaceship moving at v = 0.5c) moves slower

the closer to the speed of light that you travel time dilation

• Time dilation

€

Δt =Δt0

1− v 2 c 2 SI unit: second, s

o Δt0 is the proper time o Δt is time when object moves relative to observer with speed, v o Note: Δt = Δt0 for v = 0

• Event: “physical occurrence that happens at a specific location at a specified time” pg 1016, J.S. Walker, Physics, 4th edition

• Proper time: the amount of time that separates two events that occur at the same location

o Example: time between emission of light (event 1) and detection of light (event 2) in a light clock

• Example: spaceship moves at v = 0.800c relative to observer on earth. How long for spaceship’s clock to move one second (1.00 s) according to observer on earth?

€

Δt =Δt0

1− v 2 c 2=

1.00s1− (0.800c)2 c 2

=1.00s1− 0.640

=1.67s

• Actually tested time dilation in 1971, by flying highly accurate atomic clock on airplane at high speed for many hours. It ran “slow” compared to identical clock in laboratory (at rest).

• Time dilation occurs for all physical processes like aging, not just clocks. An astronaut traveling at high speed will age more slowly than on earth, but the astronaut will feel time progressing as usual.

• Can also see time dilation in decay of muons (unstable subatomic particles). Muons produced high in atmosphere by cosmic radiation. On average they decay after 2.20x10-6 s. Should decay after about 657 m. Shouldn’t make it to ground level. Many do. When they travel close to speed of light, the time the muons experience, slows down which allows them to travel a further distance through the atmosphere before decaying. This means that many more muons will last long enough to reach sea level.

Relativity of Length and Length Contraction (29.3)

• Distance is altered for observer moving close to speed of light, just like time is. Lengths and distances appear shorter when moving close to speed of light = length contraction

• Length Contraction

€

L = L0 1−v 2

c 2 SI unit: m

o L0 = length when object is at rest = proper length o L = contracted length when object moving at relative speed, v o Note: ΔL = ΔL0 for v = 0. o Another note: ONLY applies to lengths in the direction of relative motion.

Lengths at right angles to this are unaffected. • Proper length, L0: distance between two points as seen by an observer who is at rest

relative to both points. Relativistic addition of velocities (29.4)

• cannot simply add vectors. There is an additional factor. • 1-D Example: spaceship (#1) launches a probe (#2) at an asteroid (#3). Spaceship has a

relative velocity of v13 relative to the asteroid. Velocity of probe relative to ship is v21. What is velocity of probe relative to asteroid? v23 = v21 + v13 doesn’t work.

o

€

v23 =v21 + v131+

v21v13c 2

is correct answer.

o Note extra term on bottom that includes the speed of light Relativistic Momentum (29.5)

• p = mv from classical mechanics doesn’t work for all speeds. There is an additional factor.

• Relativistic momentum:

€

p =mv

1− v2

c 2

SI unit: kg m/s

o Note: if v << c then square root on bottom is ≈1 and you are left with p=mv • Some people view relativistic momentum equation as mass that increases with speed.

• Relativistic mass:

€

p =m0v

1− v2

c 2

=m0

1− v2

c 2

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

v where m =

€

m0

1− v2

c 2

⎛

⎝

⎜ ⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟ ⎟

and p = mv as

before. o m0 is called the rest mass, or the mass that the object has when it is at rest o As v approaches c, m approaches infinity, so a constant force acting on an object

makes less and less acceleration (a = F/m) as approach speed of light. Another way to see speed of light can’t be exceeded.

• Concept of relativistic mass works in many ways, but has its limitations as can be seen when considering energy.

Relativistic Energy and E = mc2 (29.6)

• From point of view of momentum, object’s mass increases as speed increases. So when work done, part goes to speeding up and part goes to increasing mass.

• So it follows that mass is another form of energy. o Completely unanticipated result before theory of relativity proposed.

• Relativistic Energy:

€

E =m0c

2

1− v2

c 2

= mc 2 SI unit: J

o m = relativistic mass o m0 = rest mass o Note: E does not go to zero when v = 0

• Rest Energy E0 = m0c2 SI unit: J • Examples of mass to energy

o fission reactions in nuclear power plants (heavy nuclei splits and releases difference in mass in form of energy)

o fusion reactions in the sun (two very small nuclei join to form larger one)

• Relativistic Kinetic Energy

€

KE =m0c

2

1− v2

c 2

−m0c2 SI unit: J

o Note: In limit of small speeds, v, binomial expansion gives for first couple of terms (rest of terms are very very tiny)

• KE = m0c2 + ½ m0v2 – m0c2 = ½ m0v2

General Relativity (29.8)

• Applies to accelerated frames of reference and to gravitation. • Principle of Equivalence: “All physical experiments conducted in a uniform

gravitational field and in an accelerated frame of reference give identical result.” Pg. 1034, J.S. Walker, Physics, 4th edition

• Leads to conclusion that gravity bends light • Can see this in gravitational lensing – gravity of sun bends light from distant stars around

it. • Black holes: collapsed star whose gravitational field is so strong that even light cannot

escape. (You can’t “see” them, you can only infer them from the effects seen like gravitational affects on nearby bodies and radiation from ionized matter falling into black hole)

• Schwarzchild radius: maximum radius on object of mass, M, can have to be a black hole (a little simplistic)

•

€

R =2GMc 2

SI unit: m o G = 6.67x10-11 Nm2/kg2

o Ex: Earth would have to be size of walnut to be a black hole (8.86 mm) Chapter 30 (Quantum Physics) – Added – 11/27/10 11:30 pm Blackbody radiation and quantized energy (30.1)

• All objects continuously radiate electromagnetic waves over a range of frequencies (wavelengths), no matter how hot or cold

• Ideal blackbody (Mathematical idealization) absorbs all radiation falling on it and readmits all of it

o Characteristic shape of curve of radiation intensity versus wavelength (or frequency)

o Higher temperature gives higher peak to curve and at shorter wavelengths Wien’s displacement law: fpeak = (5.88x1010 s-1 K-1) T

o Depends only on the temperature, not the material/substance o Max Planck came up with model to describe blackbody radiation curves by

assuming that blackbody is a large number of atomic oscillators, each of which emits and absorbs e-m waves. HAD TO ASSUME DISCRETE ENERGY VALUES FOR ATOMIC OSCILLATORS TO MATCH OBSERVED CURVES.

Energy is quantized • Energy: E = nhf

o n = 0, 1, 2, 3, … o f = frequency of vibration in Hz o h = Planck’s constant ≈ 6.63 x 10-34 J s

Photons and Photoelectric Effect (30.2)

• Einstein proposed light comes in bundles or packets on energy called photons o Energy of photon related to frequency

E = hf SI unit: J o Higher intensity beam (frequency, f) of light just has more photons per second

than lower intensity (dimmer) light beam o Used this idea to explain photoelectric effect

Photoelectric effect: a beam of light shines on a metal and ejects an electron

Einstein’s explanation: light shines on metal and photon gives energy to an electron in the metal; if photon has enough energy to do work to remove electron form metal than electron is ejected (how much work depends on how strongly electron is held … depends on the metal).

• Work function, W0: minimum value to eject least strongly held electrons

o Cutoff frequency, f0 = W0/h SI unit = Hz Frequency of photon with minimum energy

required to eject an electron o Light intensity is NOT deciding factor whether electron

ejected or not … frequency of light IS deciding factor • If a photon has excess energy beyond W0, then energy goes into

kinetic energy (KE) of ejected electron. o Least strongly held electrons ejected with maximum KE

KEmax = hf – W0 Mass and Momentum of a Photon (30.3)

• Photon is like a particle o Has energy but NO mass

Rest mass of a photon, m0 = 0 • Have finite momentum even though they have no rest mass

o Momentum of a photon, p = hf/c = h/λ Photon Scattering and the Compton Effect (30.4)

• Compton Effect: scattering of x-rays by collisions with stationary electrons o X-ray photon is scattered; direction and energy (frequency) changes

Frequency of scattered x-ray is less than frequency of incident x-ray Use conservation of energy and conservation of momentum to solve for

change in wavelength of x-ray

o Compton shift formula:

€

Δλ = λ'−λ =hmec

1− cosθ( ) SI unit: m

The de Broglie Hypothesis and Wave-Particle Duality (30.5) • Grad student, Louis de Broglie suggested since light waves could exhibit particle-like

behavior, particles of matter should exhibit wave-like behavior. • All matter has a wavelength associated with it

o Only particles with small mass, can we actually see this behavior Beam electrons at crystals of nickel and see diffraction pattern like that of

x-ray diffraction

Young’s double slit experiment with electrons instead of light (see interference patterns … fringes of light and dark)

etc. • De Broglie wavelength: λ = h/p

o h = Planck’s constant o p = relativistic momentum

• How to understand particle waves? … As waves of probability. o Magnitude of wave suggests probability that particle will actually be found at

particular point • Wave-Particle duality

o Electromagnetic waves can behave like particles (photons) o Particles (like electrons) can exhibit wave-like interference effects

Heisenberg Uncertainty Principle (30.6)

• You cannot simultaneously know both the position and the momentum of a particle precisely.

o Δpy Δy ≥ h/2π Δpy = uncertainty in y-component of linear momentum Δy = uncertainty in particle’s position in y-direction h = Planck’s constant

• You cannot simultaneously know both the energy of particle and the time precisely. o ΔE Δt ≥ h/2π

ΔE = uncertainty in energy of particle when particle is in specified state Δy = time interval during which particle is in specified state h = Planck’s constant

o ex: the shorter the half-life of an unstable particle, the greater the uncertainty in the energy of the particle

• Fundamental limit from nature (not “human error”) Chapter 31 (Atomic Physics) – Added – 12/02/10 10:20 pm Early Models of the Atom (31.1)

• Plum pudding model (early 20th century) o positive charge spread throughout (“pudding”) o negative charges suspended in “pudding” (the “plums”) o proved incorrect in 1911 Rutherford experiment

shoot alpha particles at gold foil and not all particles passed through the foil so positive charge is NOT spread out, but concentrated in a small region capable of deflecting alpha particles

• Rutherford miniature solar system model o Positive charge is in nucleus o Nucleus is like the sun o electrons (negative charge) are the planets

electrons are attracted to center by positive charge at center (GOOD) electrons moving in a circle …. Constantly accelerating …. Which means

they should be radiating e-m waves …. Which causes them to lose energy

… so they should spiral in to the nucleus … and the atom should collapse (BAD)

But matter is stable … so something else is going on The Spectrum of Atomic Hydrogen (31.2)

• Low pressure hydrogen gas emits e-m radiation when subjected to high voltage (produces an emission spectrum)

o Use diffraction grating to see colored lines o Particular colored lines and pattern is characteristic of hydrogen like a

“fingerprint” for the element Each element has its own particular set of colored lines (emission

spectrum) o Emission spectra not just in visible range (can also be infrared, ultraviolet, etc)

o

€

1λ

= R 1n f2 −

1ni2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ nf = 1, 2, 3, …. ni = nf+1, nf+2, nf+3, …

Rydberg constant, R = 1.097x107 m-1 nf = integer number of energy level where electron falls to (final) ni = integer number of energy level where electron falls from (final)

[More on energy levels for electrons in next section] o Common spectral series of hydrogen

Lyman series: nf = 1 (ultraviolet wavelengths) Balmer series: nf = 2 (visible wavelengths) Paschen series: nf = 3 (infrared wavelengths)

Bohr’s Model of the Hydrogen Atom (31.3) • Bohr’s model is a miniature solar system model, but with only certain orbital radii

allowed for the electrons • Four assumptions of Bohr’s model

o Electron moves in circular orbit about the nucleus o Only certain circular orbits are allowed and angular momentum of the electron in

this orbit is Ln = nh/2π n = 1, 2, 3, … Angular momentum is quantized

o Electrons do not give off e-m radiation when in allowed orbit, so orbits are stable o Electromagnetic radiation is only released or absorbed when electron changes

from one allowed orbit to another allowed orbit. Absolute value energy difference between orbits equal to energy of photon

released or absorbed |ΔE| = |Ef – Ei| = hf • Bohr orbits (for any atom with single electron orbiting the nucleus)

o Radius,

€

rn =h2

4π 2mkZe2⎛

⎝ ⎜

⎞

⎠ ⎟ n2 n = 1, 2, 3, …

m = mass of electron k = constant from Coulomb’s law Z = number of protons in nucleus (Z = 1 for hydrogen) e = electron charge (absolute value) h = Planck’s constant

o Radius for hydrogen (Z = 1) n = 1: r1 = 5.29x10-11 m ≈ ½ angstrom

• Called Bohr radius • Agrees with experimental observations size of hydrogen atoms

For n = 2, 3, … rn = (5.29x10-11 m) n2 o Speed (of electron in uniform circular motion)

€

vn =2πke2

nh n = 1, 2, 3, …

• speed gets smaller in orbits farther from nucleus • Energy of a Bohr orbit

o

€

En = −2π 2mk 2e4

h2⎛

⎝ ⎜

⎞

⎠ ⎟ Z 2

n2 = -(13.6 eV) (Z2/n2) n = 1, 2, 3, …

m = mass of electron k = constant from Coulomb’s law Z = number of protons in nucleus (Z = 1 for hydrogen) e = electron charge (absolute value) h = Planck’s constant

o Energy levels Ground state: n = 1 First excited state: n = 2 Second excited state: n = 3 etc.

o A photon is released when the electron falls from an excited state to a lower excited state or the ground state.

• Spectrum of hydrogen

o

€

1λ

= R 1n f2 −

1ni2

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

where

€

R =2π 2mk 2e4

h3c⎛

⎝ ⎜

⎞

⎠ ⎟ =1.097x107 m-1

o emission spectrum occurs because a particular frequency (energy, E = hf) photon is released when the electron falls from a higher energy level to a lower energy level

o absorption spectrum occurs because only particular energy (frequency) photons can be absorbed by the atom and move the electron up into a higher energy state

only photons whose energy matches a difference between two allowed energy levels can be absorbed. The rest just bounce off.

If the particular frequency is absorbed from a continuous spectrum (the full rainbow) say from a light bulb, then the frequencies (wavelengths) absorbed will be missing and show up as black lines in the rainbow. These absorption spectral lines are the exact same frequencies (wavelengths) as the emission spectral lines.

de Broglie Waves and the Bohr Model (31.4)

• de Broglie explained Bohr’s assumption that angular momentum of electrons is quantized (only particular values allowed)

• comes from wave nature of matter (talked about this in the previous chapter)

• Explanation: picture electron in orbit as a particle wave o Can get standing waves (resonance)

In Phy 122, you looked at standing waves on a string o Conditions for this total distance traveled = an integer # of wavelengths

Use same reasoning for the particle wave in orbit Total distance = circumference = 2πr = nλ n = 1, 2, 3, …

o de Broglie wavelength (from previous chapter) λ = h/p p = mv if v << c (if orbital speed is much less than speed of light) so λ = h/mv

o which gives us 2πr = nλ = nh/mv o Rearranging we get mvr = L = nh/2π

Which is what Bohr proposed 10 years earlier Quantum mechanical description of atoms and the periodic table (31.5-31.6))

• Bohr’s model helps us to understand atomic structure, but it’s too simple with only one quantum number (n)

• Need 4 different quantum numbers to describe state of hydrogen atom o Principal quantum number, n

Determines total energy, like Bohr’s model n = 1, 2, 3, …

o Orbital quantum number, l Determines angular momentum due to orbital motion l = 0, 1, 2, …., (n-1)

€

L = l l +1( ) h2π

• magnitude of angular momentum of electron due to orbital motion o Magnetic quantum number, ml

* When in external magnetic field * determines the component of angular momentum along the “z direction” ml = -l, …-2, -1, 0, +1, +2, … +l Lz = ml h/2π

• Magnitude of angular momentum in “z direction” o Spin quantum number, ms

Spin angular momentum • Intrinsic property of electron • Like earth spinning on axis, while also orbiting the sun

Two values • ms = +1/2 “spin up” • ms = -1/2 “spin down”

• Define the state of an atom by assigning values to each of the four quantum numbers o there are a limited number of states per energy level o Pauli Exclusion Principle: “Only one electron at a time may have a particular set

of quantum numbers, n, l, ml, and ms. Once a particular state is occupied, other electrons are excluded from that state.” J.S. Walker, Physics, 4th edition, pg. 1095, 2010.

o Energy levels (defined by n) are called shells in multielectron atoms

o Electrons with same orbital quantum number, l, for same energy level (n) are in the same subshell

o Electronic configuration – shorthand notation to describe arrangement of electrons in shells and subshells

o The periodic table was originally developed to group elements with similar chemical properties

Similar chemical properties can be explained on the basis of the configurations of the outer electrons in the group

o So, quantum mechanics and the Pauli exclusion principle can explain the chemical behavior of atoms!

SKIP (31.7) Atomic Radiation in Fall 2010 Chapter 32 (Atomic Physics) – Added – 12/03/10 2:15 pm Nuclear Structure (32.1)

• Nucleus contains neutrons and protons (collectively called nucleons) o Neutron: electrically neutral and slightly more mass than a proton o Number of protons defines the element

• Electrons “orbit” the nucleus • Atomic number, Z: number of protons in nucleus • Atomic mass number, A: number of protons + number of neutrons in nucleus

o Also called nucleon number o A = Z + number of neutrons

• Shorthand notation:

€

ZA X

o example:

€

1327Al aluminum: 27 nucleons, 13 protons, (14 neutrons)

• isotopes: atoms with same number of protons, but different number of neutrons in the nucleus

• nucleus size – protons and neutrons clustered together in approximately spherical shape o experiments show size, r ≈ (1.2x10-15 m) A1/3 o example: Al rAl ≈ (1.2x10-15 m) 271/3 ≈ 3.6 x10-15 m

• If protons and neutrons packed so closely together, why doesn’t the nucleus fly apart because of the repulsion of the positively charged protons by the electrostatic force (Ch 19)?

o Gravitational force is too weak at the tiny mass of the protons o Strong nuclear force

One of three fundamental forces in the universe Almost independent of electric charge Nearest neighbors attract (range of action for force is short) Doesn’t matter if nearest neighbor is a proton or a neutron

o Electrostatic repulsion of the protons (against each other) is balanced by the strong nuclear force

As the number of protons increases the number of neutrons increases even faster to maintain stability of the nucleus

At some point, more neutrons and the strong nuclear force can’t balance out long range repulsion of so many protons and the elements are no longer stable

• Bismuth

€

82209Bi is last stable nuclei (has 126 neutrons)

• All nuclei with more than 83 protons are unstable and spontaneously break apart or rearrange internal structures as time passes ⇒ called radioactivity

Radioactivity (32.2) • When an (unstable) radioactive nucleus disintegrates spontaneously (= decay of the

nucleus) it releases particles or high energy photons collectively known as radioactivity o From naturally occurring radioactivity – three types

Alpha rays (α) (least penetrating) Beta rays (β) Gamma rays (γ) (most penetrating)

• When a radioactive nucleus decays the number of nucleons is conserved o Number of protons + neutrons before decay = number of protons + neutrons after

decay • Alpha decay (α)

o Alpha particles:

€

24He nucleus (only) so +2 charge

o General form for alpha decay

€

ZAP →

€

Z −2A −4D +

€

24He

• P = parent nucleus • D = daughter nucleus • Energy also released

o Process is a transmutation: parent and daughter have different number of protons so the element changes

• Beta negative (β-) decay o β- particles are negatively charged (-e) particles (electrons)

not called electrons because came from nucleus not from orbiting electrons

o General form for beta negative decay

€

ZAP →

€

Z +1AD +

€

−10e + antimatter neutrino

• P = parent nucleus • D = daughter nucleus • Neutrino has no electric charge and a tiny fraction of the mass of

an electron • Energy also released

A neutron from nucleus decays to become a proton and an electron • So electron comes from nucleus not from orbiting electrons • Total number of nucleons remains the same

o Process is a transmutation: parent and daughter have different number of protons so the element changes

• Beta positive (β+) decay o β+ particles are positively charged (+e) particles (positrons)

same mass as an electron, but positively charged

didn’t exist in nucleus, it was formed when a nuclear proton is transformed into a neutron

o General form for beta positive decay

€

ZAP →

€

Z −1AD +

€

+10e + neutrino

• P = parent nucleus • D = daughter nucleus • Neutrino has no electric charge and a tiny fraction of the mass of

an electron • Energy also released • A nuclear proton is transformed into a neutron

o Total number of nucleons remains the same o Process is a transmutation: parent and daughter have different number of

protons so the element changes • Gamma (γ) decay

o The nucleus can only exist in discrete energy states or levels (like we had for orbital electrons with Bohrs model of the atom)

o A nucleus can change from an excited state to lower energy state and release a high energy photon (gamma ray)

Much higher energy photon released than with energy level transitions of orbital electrons

o General form for gamma decay

€

ZAP* →

€

ZAP + γ

• P* = excited energy state • P = lower energy state

o Does NOT cause transmutation from one element to another Radioactive activity, decay and dating (32.2-32.3)

• Individual disintegrations of unstable/radioactive nuclei occur randomly, but if look at a number of the same kind of radioactive nuclei, a there is a pattern in the decay over time

• Exponential decay: N =

€

N0e−λt

o N = number of radioactive nuclei remaining at time t o N0 = number of radioactive nuclei at time t = 0 o e = is “e” from natural logarithm (ln) and is �pproximately equal to 2.718… o t = time o λ = decay constant λ = (ln 2)/ T½

• half-life, T½ : time for the one-half the number of radioactive nuclei present to disintegrate (decay)

o ex: at t = 0 N = N0 at t = T½ N = ½ N0 at t = 2T½ N = ½(½ N0) = ¼ N0 at t = 3T½ N = ½ (¼ N0) = 1/8 N0 etc

o this is a constant value for each particular type of radioactive nuclei o ranges from fraction of a second to billions of years o T½ = (ln 2)/λ ≈ (0.693)/λ

• activity: number of disintegrations per specified time interval that occurs o 1 curie = 1 Ci = 3.7x1010 decays/s

o 1 becquerel = 1 Bq = 1 decay/s (SI unit)

o Activity:

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A =ΔNΔt

= −λN

Minus sign because decay reduces the number of nuclei present N = number of radioactive nuclei Also called decay rate, R A = R = λ

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N0e−λt =

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R0e−λt

• R0 =λN0 initial value of the activity (at t = 0) is based on initial number of radioactive nuclei

o More radioactive nuclei present means more disintegrations are likely to occur

• Radioactive dating: if object contains radioactive nuclei when formed, then the decay of the nuclei contained within is like a clock (½ radioactive nuclei will decay each half-life)

o Two methods Compare the original and remaining number of nuclei Compare the original and remaining activity

o The age range of dating is limited by the half-life of the radioactive nuclei you are looking at

• Carbon-14 dating

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614C isotope

o Undergoes beta negative decay o Half-life = 5730 years o Isotope constantly produced in atmosphere by cosmic rays so there is a constant

proportion between carbon-12 and carbon-14 (1 carbon-14 atom for every 8.3x1011 carbon-12 atoms)

o Living organisms continuously take in carbon so ratio is constant o Once organism dies, carbon-14 starts to decay and no new carbon-14 is taken in

Can use to see how long ago organism died o Activity of living organism = 0.231 Bq = A0 o Example: A living organism has an activity of A0 = 0.231 Bq; remains of Stone

Age man found trapped in ice in glacier has activity of A = 0.121 Bq. How old are the remains?

Decay constant, λ = (ln 2)/(5370 yr) = 1.21x10-4 yr-1 A = R =

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R0e−λt R = 0.121 Bq & R0 = 0.231 Bq

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RR0

= e−λt ⇒

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ln RR0

⎛

⎝ ⎜

⎞

⎠ ⎟ = ln e−λt( ) = −λt

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t = −1λln R

R0

⎛

⎝ ⎜

⎞

⎠ ⎟ =

1λln R0

R⎛

⎝ ⎜

⎞

⎠ ⎟ =

1(1.21x10−4 yr−1)

ln 0.231Bq0.121Bq⎛

⎝ ⎜

⎞

⎠ ⎟ = 5344 years

SKIP 32.4-32.9 for Fall 2010 32.4 Nuclear Binding Energy 32.5 Nuclear Fission 32.6 Nuclear Fusion 32.7 Practical Applications of Nuclear Physics 32.8 Elementary Particles 32.9 Unified Forces and Cosmology