Download Physics 121 Lecture Summary

Physics 123 Concepts - Summary
Last updated October 3, 2010, 1:35 am
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
Please excuse the mess while this document is being updated.
The concept summary notes were originally created following the section numbers, outline, and
symbols of a different textbook than we are using this quarter (Fall 2010). The notes are good
and can be utilized right away. Just be careful of the section numbers listed until they are
updated. As the quarter progresses, I will be updating these notes to reflect the textbook we are
using for this class: Physics (Volume 2), 4th edition, by James S. Walker (Addison-Wesley, San
Francisco). I will also be adding notes for the chapters on modern physics.
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
k q1 q 2
o F
(magnitude only)
r2
 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)
k qQ
 F 2
(q is located outside sphere at distance r from center)
r
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
kq
o E 2
(magnitude only - direction depends on whether q is + or -)
r
 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
q


(magnitude only; only between plates away from edges)
0 A 0
 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
o


E
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
EPE
o V 
q0
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
EPEB EPE A  WAB
o V  VB  VB 


q0
q0
q0
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
V
o E
SI units = volts/meter = V/m
s
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)
kq
 V 
SI units = volt, V
r
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
 A
o Parallel plate capacitor without a dielectric, C  0
d
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
 A
o Parallel plate capacitor with a dielectric, C  0
d
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”
1
1   A 
2
o Energy  CV 2   0 Ed 
2
2 d 
 Where is energy of a capacitor stored? In the electric field between the plates
Energy 1
  0 E 2
o Energy density = uE=
Volume 2
o True for any electric field whether in capacitor or not
 κ=1 if no dielectric
uE = (½)ε0E2
The following has not yet been updated to reflect the current textbook (10/3/2010)
Electromotive “force” and current (20.1)
 Electromotive “force”, emf = maximum potential difference between the terminals of a
generator or battery in a circuit
q
 Electric current, I 
t
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
 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 (20.2-20.3)
 electrical resistance is voltage applied across a piece of material/current thru the material
V
o R
I
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)
L
o R
A
o Resistivity is an inherent property of the material and depends on the temperature
 ρ = ρ0 [1 + α(T-T0)]
 SI units = Ω·m = Ohm·meter
 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)
Electric Power (20.4)
 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 P  I 2R
V2
o P
R
Alternating Current (20.5) – only considering circuits with resistors (more in chapter 23)
 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)
V
o Voltage: Vrms  0
2
o Current: I rms 

I0
2
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
2
R
 Pave  I rms

Pave 
2
Vrms
R
Series and Parallel Wiring (20.6-20.8, 20.12)
 Series – devices connected one after the other so the same current passes through each
o Equivalent resistance, Rs = R1 + R2 + R3 + ···
 dissipates the same total power as the series combination
1
1
1
1



 
o reciprocal of the equivalent capacitance,
C s C1 C 2 C3
 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
1
1
1
1
o reciprocal of the equivalent resistance,



 
R p R1 R2 R3
 dissipates the same total power as the parallel combination
o 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 (20.9)
 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.
 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.
 Terminal voltage = emf – Ir
(r = internal resistance)
Kirchhoff’s Rules (20.10)
 Junction rule: sum of the current in = sum of the current out
o adding magnitudes only
 Loop rule: around any closed circuit loop, the sum of the potential drops = the sum of
potential rises
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 (20.11)
 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 (20.13)
 Circuit with resistors and capacitors
 Once circuit 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
-t/(RC)
o q = q0[e
]
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 (20.14)
 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!
Magnetic Fields (21.1)
 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
Magnetic force on a moving charge (21.2)
 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)
F
o B
(magnitude)
q 0 v sin  
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)
Motion of charged particle in magnetic field and mass spectrometer (21.3-4)
 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
mv
qB
 Radius of circle inversely proportional to magnitude of magnetic field
Mass spectrometer – used to determine abundance of ionized atoms or molecules with
different masses
 er 2  2
 B
o m  
 2V 
o

r
Force and torque on a current carrying wire (21.5-21.6)
 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
o magnetic moment = NIA
(SI units: A·m2)
Magnetic fields produced by currents (21.7)
 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 …
 I
o Infinitely long straight wire: B  0
2r
 two parallel current carrying wires
 attract if both have current going same direction
 repel if the wires have current going opposite directions
 I
o At the center of a circular loop (1 loop): B  0
2R
 I
o At the center of a circular loop (N loops, all same radius R): B  N 0
2R
 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: B  n 0 I
(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 (21.8)
- Skipped Summer 2006
 For static (unchanging through time) magnetic fields
  B|| l   0 I

Much easier to do with calculus
Magnetic materials (21.9)
 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
Induced emf and induced current (22.1-22.2)
 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
emf  vBL
 Magnitude of motional emf when v  B  L :
o Where does energy come from? From mechanical force keeping conductor
moving at a constant velocity.
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).
Magnetic flux and Faraday’s law of magnetic induction (22.3-22.4)
 All previously discussed ways to produce an induced emf can be described through the
concept of magnetic flux.
 Magnetic flux,   BA cos 
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
  0

 Faraday’s law: emf   N
 N
t
t  t0
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 (22.5)
 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.
Electric Generator (22.7)
 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
back emf or counter emf
o 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
o in an electric motor there are two sources of emf
 applied emf, V, to turn motor (from outlet plug)
 induced emf, emfb, from rotating coil (called back emf or counter emf)
o two sources have opposite polarities, so you get a net emf
 net emf = V - emfb

Mutual Inductance, Self Inductance, and Transformers (22.8-9)
 Mutual induction
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)
N 
 M  s s
SI units = Henry (H) = (V·s)/A
Ip
o Emf due to mutual induction, emf s   M




I p
t
Self induction
o 1 coil … changing current in circuit induces emf in same circuit
o Self inductance, or simply inductance of coil
N
 L
SI units = Henry (H) = (V·s)/A
I
I
o Emf due to self induction, emf   L
t
2
Energy stored in an inductor, energy = ½ LI
Energy density of magnetic field in vacuum, air, or non-magnetic material
1
B2
o Energy density 
2 0
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
emf s
N
V
 s  s
o Transformer equation,
(Vs, Vp = terminal voltages)
emf p N p V p
o If Ns > Np then Vs > Vp
(step-up transformer)
o If Ns < Np then Vs < Vp
(step-down transformer)
Resistors, capacitors and inductors individually in alternating current (AC) circuits (23.1-2)
 Review section 20.5 on alternating current
 Alternating voltage provided by AC generator
o Usually sinusoidal voltage: V = V0sin2πft
V
o Root mean square (a type of average) voltage: Vrms  0
2
 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
1
 Capacitive reactance: X C 
(SI units = ohm (Ω))
2fC
 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: X L  2fL
(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
Phasors – rotating arrows that rotate counterclockwise at a frequency f
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
Circuits with resistors, capacitors, and inductors in series (23.3)
 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: Z  R 2   X L  X C 
(SI units = ohm (Ω))
 Vrms = IrmsZ
Phase angle of the series RCL circuit
V  VC X L  X C
o tan   L

VR
R
o Angle between current phasor and voltage phasor across the series RCL circuit
On average, only resistance consumes power (not capacitors or inductors)
o Pave  I rmsVrms cos 
o Cosine of phi called power factor of circuit
2


Resonance (23.4)
 Only one natural frequency in RCL circuit
 Resonance when impedance is a minimum
o Z is minimum when Z = R because XC = XL
1
 Resonant frequency, f 0 
2 LC
Electromagnetic waves, the E-M spectrum and the speed of light (24.1-3)
 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


c
1
 0 0
Electromagnetic spectrum divided up into (from longest to shortest wavelength)
o Radio waves
o Infrared radiation
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
Energy carried by e-m waves (24.4)
1
1
B2
 energy density = u   0 E 2 
2
2 0
 in a vacuum (or air) the e-m wave carries equal amounts of energy/volume in electric
field and magnetic field
1 2
B
o u  0E2 =
0
o Since E and B fields oscillate, above formulas give u at a particular instant in time
E
B
o Or, can get a type of average … E rms  0 and Brms  0
2
2
1
1
cB 2
 Intensity of e-m wave, S = power/area = cu = u  c 0 E 2 
2
2 0