WHAT IS PHYSICS ALL ABOUT? Physics seeks to understand the natural phenomena that occur in our universe; a description of a natural phenomenon uses many specific terms. definitions and mathematical equations Solving Problems in Physics In physics, we use the SI units (International System) for data and calculations
Base Quantity Length Mass Temperature Time Electric Current
.;;;;;;S=;;=;;;=~T~he
position ofa motion with position, velocity and acceleration as variables; mass is the measure of the amount of matter; the standard unit for mass is kg. I kg = 1000 g.; inertia is a property of matter, and as such, it occupies space I. Motion along a straight line is called rectilinear; the equation of motion describes the position of the particle and velocity for elapsed time. t a. Velocity (v): The mte of change of the displacement . h' () s WIt tIme ( t):v
cis = Tt Ll s = rlt
b. Acceleration (a): The rate of change of the
· .h . dv Ll v ve IoClty WIt tnne: a = dt = Tt a & v'are vectors , with magnitude and direction c. Speed is the absolute value of the velocity; scalar with the same units as velocity 2. Equations of Motion for One Dimension (I-D) Equations of motion describe the future position (x) and velocity (v) of a body in terms of the initial velocity (Vi), position (XII) and acceleration (a) a. For constant acceleration. the position is related to the time and acceleration by the following
x = vi, I ·1 ; a , t' y = vi, ( + ;a, (' 2. For a rotating body, use polar coordinates, an angle variable,
0 , and r. a radial distance from the rotational center C. 'lotion in TlJI'(~e Dimensions (3-D) I. Cartesian System: Equations of motion with x. y and z components 2. Spherical Coordinates: Equations of motion based on two angles ((} and 'P) and r. the radial distance from the origin.
m,M T t I
position and velocity: F = m a OR ~ F = m a 3. Newton's 3rd Law: Every action is countered by an opposing action F:. 1 ~ pe\ of Forcc\ I. A body force acts on the entire body, with the force acting at the center of mass a. A gravitational force. Fg. pulls an object toward the center of the Earth: Fg = mg b. Weight = Fg; gravitational force c. Mass is a measure of the quantity of material , independent of g and other forces 2. Surface forces act on the body's surface a. Friction. Fe. is proportional to the force normal to the part of the body in contact with a sUlface,
="
Fn·: Fr Fn i. Static friction resists the move-ment of a body ii. Dynamic friction slows the motion of a body For an object on a horizontal plane: F f = Il Fn = ll m g Net force = FI - F f
Polar: (r, 9) x = r cos9, . y = r sin9,
----r-
Circular Motion
....
(}
%,
Radian Radian/second
a
Radian/second 2
s
Meter
W
= F d cos (8) = F
F__
D
• r F_ _
r
D
Maximum work
r
p = LlWork = LlWork
Ll t
f P(t)dt
The Sl unit for power is the Watt (W):
I W = I Joule/second = I J/s
Work for a constant output of power:
W = P Lit Energ~
&
Enrrg~
Consenation
I. The total energy of a body, E, is the sum of kinetic. r
.
The angle between rand the (x) axis
te
angular velocity
The angular acceleration The circular motion arc
s = r8 (8 in rad)
3. Tangential acceleration & velocity:
v , = rw; a , = r a ; v and a along the path of the
K, & potential energy. U: E = K +:Eu 2. Potential energy arises from the interaction with a potential from an external force Potential energy is energy of position: U(r); the form of U depends on the force generating the potential: Gravitation: U(h) = mgh . q,q, Electrostatic: U (r,,) = '"'"F;; If there are no other forces acting on the system, E is constant and the system is called conservative I. Collisions & Linear 'loml'lItulll Collisions I. Types of Collisions a. Elastic: conserve energy b.lnelastic: energy is lost as heat or
deformation
2. Relative Motion & Frames of Reference: A body moves with vc:locity v in frame S; in frame S' the velocity is v' ; ifYs' is the velocity of frame S' relative to S, therefore: v = V,' + v' 3. Elastic Collision
Conserve Kinetic Energy: t m v ,' = t m vI
L:
motion arc
4. Centripetal acceleration: a ,. =
v'
r;
a is directed
toward the rotational center a. TIle centripetal force keeps the body in circular motion with a tangential acceleration and velocity
1
No work
4. Power (P) is energy expended per unit time:
II. Potrnthll
distance from the rotation center (center of mass)
z
Newton's Laws are the core x = r simp cos9, principles for describing the motion y = r sinep sin9, z = r coslj), of classical objects in response to r2=x2 + y2+z2 forces. The SI ullit of force is the Newton, N: IN=lkg m/s2 ; the cgs unit is the dyne: 1 dyne = I g cm/s2
~
1,- 0
I The
I Meter
r
J
Work =
2. Key Varia_b_le_s_: _ __ , _ _ _._ _ __ _---.
I
energy is the energy of motion ; mass. m and velocity. v: K = t mv' The SI energy unit is the Joule (J): \J = I kg m 2/s 2 2. Momentum, p: Momentum is a property of motion, defined as the product of mass and velocity: p = m v 3. Work (W): Work is a force acting on a body moving a distance; for a general force. F, and a body moving a path , s: W = F ds For a constant force. work is the scalar product of the two vectors: force. F. and path. r:
Ll time
Fn
polar coordinates: (r,8)
r2=x 2 +y2
x
Dynamk Friction
F. Circular 'lotion I. Motion along a circular path uses
~,
Spherical
Other physical quantities are derived from these basic units: Prefixes denote fractions or multiples of units; many variable symbols are Greek letters Math Skills: Many physical concepts are only understood with the use of algebra, statistics, trigonometry and calculus
Unit Meter - m Kilogram - kg Kelvin - K Second - s Ampere - A (C/s)
/, x
1. Newton's 1st Law: A body remains motion unless influenced by a force 2. Newton's 2nd Law: Force and acceleration determine the motion of a body and predict future
equation of motion: x (I) = X u + V i ( + t ar b. For constant acceleration. the velocity vs. time is given by the following: v r(t) = V i + at c.lf the acceleration is a function of time, the equation must be solved using a = aCt) R. 'Iotioll ill 1\\0 Dimcnsiolls (2-0) I. For bodies moving along a y Polar straight line. derive x- and y equations of motion
Symbol
Conserve Momentum:
Lm
Vi =
L:
L rn Vr
~
4. Impulse is a force acting over time
Impulse = F Ll t or
f F (t) dt
Impulse is also the momentum change:
z
Pfin - Pini!
.1. Rul:ltiulI 411 a Rigid Bud~ I. Center of Mass: The "average" position in the body, accounting for the object's mass distribution 2. Moment ofInertia, 1: The moment ofinertia is a measure of the distribution of the mass about the rotational axis: ~ rn, r,' rio is the radial distance from mj to the rotational axis Sample I for bodies of mass m: rotating cylinder (radius R): +rn R'
M.
O,cillatur~ Motion I. Simple Harmonic Motion a. Force: F = - k..1 x (Hooke's Law) b. Potential Energy: Uk = + k..1 x' c. Frequency of the oscillation:
T
Rotating Bodies
Law Spring
T
= 21l'jI
nr,
b. Frequency of oscillation: f=..L
Simple Pendulum
!K
21l'V T
\.
L = Iw = r • P =
f r • v dm
Torque is also the change in L with time:
T=r'F=~7 h:. Static E(llIilihrium &
Angular Momentum Elasticit~
t
1. Equilibrium is achieved when:
~f = O
, 0~
~T = O
~
The body has no linear or angular
acceleration
2. Deformation of a solid body a. Elasticity: A material returns to its original shape after the force acting on it is removed b. Stress & Strain i. Strain is the deformation of the body ii. Stress is the force per unit area on the body c. Hooke's Law: The stress IS linearly proportional to the strain; stress = elastic modulus x strain: i. Linear Stress: Young's Modulus, symbolized Y ii . Shape Stress: Shear Modulus, symbolized S iii. Volume Stress: Bulk Modulus, symbolized B L lIniH'rsal (;nl\ itatiull
...._......_._......... r ......_ . ...........................,
M 1 ...
Universal Gravitation
...
M2
1. Gravitational Force & Energy
a.
. . I energy: U,= -GM,M,
GravltatlOna -r-
GM,M, . . I rlorce: F, = ~ b. Gravltatlona Fg is a vector, along r, connecting M J and M2 c. Acceleration due to Gravity, g: For an object on the Earth's surface, Fg can be viewed as Fg
=m
g; g is the acceleration due to gravity on the Earth's surface: g = 9.8 m1s 2
Hooke's
2. Simple Pendulum a. Period of oscillation:
rotating sphere (radius R): trn r'
T = la = r • f (angular acceleration force) 5. Angular momentum is · the momentum associated with rotational motion:
u mll
f=..L Ik 21l'V m
twirling thin rod (length L): ,', rnL'
3. Rotational Kinetic Energy = +LQ' The rotational energy varies with the rotational velocity and moment of inertia. I 4. Angular force is defined as torque, T:
~.
Ful'l'~s in Solids & Fluids I. p , the density of a solid, gas or liquid:
p = mass/ volume = M/V
2. Pressure, P, is the force divided by the area of the forces acted upon: P = forcelarea The SI unit of pressure is the Pascal, Pa: I Pa = 1 N I m 2 a. Pascal's Law: For a Pascal's Law fluid enclosed in a vessel, the pressure is equal at all points in the vessel b. Pressure Variation with Depth Pf The pressure below the surface of a liquid: P, = PI + pgh h is the depth, beneath the surface p is the density of the water PI is the pressure at the surface
Pressure Variation
,-,.-.,.,..,.,.,.,.,,,.......==1.....,..,,......,..,.....,.'".,... Surface
Liquid
P Pl
h
c. Archimedes' Principle: An object of volume V immersed in liquid with density p, feels a buoyant force that tends to force the object out of the water: g, = p V g
Earth's
Archimedes' Principle
,,\.\I\i{fWV""hXXhhH omAA""""
Surface
Liquid
3. Examine Fluid Motion & Fluid Dynamics a. Properties of an Ideal Fluid i. Nonviscous - minimal interactions ii. Incompressible - the density is constant iii. Steady flow - no turbulence iv. At any point in the flow, the product of area and velocity is constant: AI VI = Al VI b. Variable Fluid Density If the density changes, the following equation described properties of the fluid: p,A,v, = p,A,v, Variable Fluid
Density
c. Bernoulli's Equation is a more general
b. Weight is the gravitational
description of fluid flow
force exerted on a body by the
i. For any point y in the fluid tlow: P + +p v' + p g y constant
Earth: Weight = Fg = mg
Weight is ill!! the same as mass
=
Gravitational
Potential
Energy
01 \\a,cs
•Transverse ·Traveling • Harmonic
• Longitudinal
• Standing
• Quantum mechanical 1. General fonn for a transverse OR traveling wave: y = fix - vt) (to the right) OR y = f(x + vt) (to the leli) 2. General form for a harmonic wave: y A sin (kx - w t) OR Standing '\
=
y = A cos (kx - w t)
3 Standing Wave: Multiples of ,1/2 fits
the length of the oscillating material
4. Superposition Principle: Overlapping
waves interact => constructive and
destructive interference
a. Constructive Interference: Thc
wave amplitudes add up to produce a
I ). wave with a larger amplitude than
either of the two waves
Harmonic ~ave b. Destructive Interference: The
wave amplitudes add up to
produce a wave with a smaller
amplitude than either of the two
waves
B. lIarnwnk \\:I'l' Propertil's Wavelength
A (m)
Period
T (sec)
Frequency
f(Hz)
Angular Frequency
w (rad/s)
Wave Amplitude
A
Speed
Distance between cycles
Cycles per ,eeond: f - IT
r::
=
21l'/ T =
2m
Height of wave
1
I v (m/s)
Linear velocity v = Af
C. Sound \\ aH', 1. Wave Nature of Sound: Sound is a compression wave that displaces the medium carrying the wave; sound cannot tra, el through a vacuum 2. General Speed of Sound: v =
~
b. p is the density 3. For a Gas: v =
gravitational
potential => Ug = mgh
T~fI~S
A . Esampfl's 01
a. B is the bulk modulus, the volume compressibilit. of the solid, liquid or gas
2. Gravitational Potential Energy, Ug a. The
WAVE MOTION
ii.For a fluid at rest (special case): P,-P,=pgh
2
J,r RT M
r = Cp/C,· (the ratio of heat capacities) 4. Loudness - Intensity & Relative Intensit~· Loudness (sound i11lensity) is the power carried by a sound wave a. Relative Loudness - Decibel Scale (dB): P(dB)
=
10 log
(f.)
i. The decibel scale is delined relative to the threshold of hearing, I,,: P(I,,) = 0 dB ii. A change in 10 dB, represents a lOx increase in sound intensity, I b. Doppler Effect The sound frequency shifts (f'/t) due to relative motion of the source of the sound and the observer or listener: Vo speed of the observer; vs Doppler Effect speed of the source;v speed of sound O => <=S i. Case I: If the source
of sound is
approaching the
observ.::r, the
frequency increases:
ii. Case #2: If the source
of sound is moving
away from the
observer, the
L _ (~) <=0 s => frequency decreases: f \ + \,
~j
'i~t'to.t~~l"".CiljJi.
THERMODYNAMICS
.
Thermodynamics
Thermodynamics is the study of the work. heat & energy of a process
Q
~
Heat: Q
+Q added to the system
Work: W
+W done by the system
Energy: E
System intemal E
Enthalpy: H
H
Entropy: S
Thermal disorder
-
----
E+ PV
=
I Measure of thermal E
Temperature: T
Force exerted by a gas
Pressure: P
Volume: V Space occupied I. Thermodynamic variables are variables of state and are independent of the process path; other variables are path-dependent 2. Types of Processes: Experimental conditions can be contro lied to allow for di fferent types of I'W"""'" r Thermodynamic Condition Constraints Result ~T
Isothermal
I ~E
Q=0
Adiabatic
No heat flow
I !
~P
= -w
~
against P ext; gas expands from V I to V 2 using an infinite number of steps; the system remains in
V= 0
~H =
Q
~E =
Q
w=
Fixed volume
0
O.
measures thermal energy
T(K) = T (OC)
+ 273.15
T is always in Kelvin, unless noted in the equation b. Z e roth Law of Thermodynamics: If two bodies. # I and #2. are separately in thermal equilibrium with a third body, #3, then # 1 and #2 are also in thermal equilibrium 2. Thermal E xpansion of Solid, Liquid or Gas a. Solid: tL
= a LIT = /3 LI T
c. Gas: LI V =
(T, - T,)nR , - p (Charles Law)
3. Heat C apacity, C C depends on LI l' and Q, the heat lost or gained:
C
= LI<;'
OR Q
b. Molar heat capacity is C per mole i. Heat capacity for constant pressure, C p : LI H is the key variable ii . Heat capacity for constant volume, C.: LI E is the key variable d. ldeal Gas:
5
3
C" = 2'R AND C, = 2'R
i. The ratio of these two heat capacities is called y ii.For Ideal Gas:
r=
g~
=
i = 1.667
2. Equations for Energy of an Ideal Gas:
3
E = M v' and E = 2' RT
+
a. Average Speed of a Gas Molecule:
~
b. Gas
+ PV
~H =
~
E
+P~V
=
ii .Exothermic: Negative ~ H; the system releases heat to the surroundings (EX: combustion of fuel, condensation of vapor to liquid) Transitions:
s olid
+-
liquid
+-
gas
ii. Enthalpy of fusion: ~ H,,,, c. Enthalpy & Variable Temperature:
= jCpdT
For constant C p : ~ H =
4. Examples of Work: W =
C. ~ T
j PdV
a. P opposes the ~ V for an expansion; P causes the ~
v "". ,
V for compression; W depends on the path
b. Single step isobaric expansion from V I to V 2 against an opposing pressure, Pext '
V{T; 1";'
!l:r
F.
F:1l'ro(l~
& 2nd I.a\\ of
rhenll(l(l~
Ita m ics
The 2nd Law of Thermodynamics is concerned with the driving force for a process
I . Entropy. S Entropy measures the thermal disorder of a system:
dS -- dQ T like E &
is a state variable,
~ S (universe)
=
~ S (system)
~
+
H:
S (thermal
reservoir)
2. 2nd Law of Thermodyna mics: spontaneous process. ~ s,,"" for a system at
equilibrium or for a reversible
process
3. Examples of Entropy Changes For ~
any
>0
S,,"h = 0
a. Natural Heat Flow: Heat flows from Thot to Tcold b. Entropy & Phase Changes:
LlS(changc)
LI H (change) =
Q
~ Heat Flow
T( h c a nge )
solid --. liquid
positive ~ S
~ S,...
liquid
positivI.' ~ S
~ S"p
-+
solid
c. Entropy & Temperature for an Ideal Gas:
SeT): LIS = nCpln( i : ) Increasing T increases the disorder d . Entropy & Volume for an Ideal Gas
A gas expands from V I to V~ :
S(V):
~S
= n
R In(~: )
The disorder of a gas increases if it expands
3
IS
change from TI to T2
e. For a Real Gas: Add heat capacity and energy terms for molecular vibrations and rotations
Entropy
a. ~ H Q for a process at constant pressure; the di fference between E and H is the work performed by the process
b. Phase
Temperature:
IT ; a
K - - lRT 2
= Q- W
determined by the difference between the heat gained (Q) by the system and the work performed (W) by the system on the mechanical surrounding 3. Enthalpy, H Enthalpy is a new state variable derived from the 1st Law of Thermodynamics at constant pressure:
H = E
&
Speed
d. Kinetic Energy for 1.00 Mole of an Ideal Gas:
E is independent of path ~E
/3~T
V n ", =
& I st Lan of Thumod~ l1amics I. Wand Q depend on path of the process; however,
~H
c. Two special experimental cases:
E. rhl' Kinetic rhe()r~ of Gases I . Gas particles of mass, M, are in constant motion, with velocity. v, exerting pressure on the container
El1lhalp~
i. Enthalpy of vaporization: ~ H ...
a. Specific heat capacity is C per gram
V
c. Gas Speed & Mass: v,,'" is proportional to
A phase change corresponds to a change in enthalpy:
= CLIT
p..
changes the speed by
i. Endothermic: Positive ~ H ; the system absorbs heat from the surroundings (EX: evaporation of liquid to gas; melting of a solid)
b. Liquid: LI:
PL'
Temperature(K)
2. lst Law of Thermodynamics:
a. The SI unit is Kelvin. absolute temperature:
SIa&Ie Step EspDIIoa
proportional to
a. The change in energy of the system (~ E ) is I.
process gives the maximum work
IL
Pressure (Pa)
W = P~V
= 0
equilibrium: W = n RT In (~: ) . This type of
Charlet' Law
~~
PV' = constant
Fixed pressure Isochoric
Carnot's Law is exact for monatomic gases; it
jl\ -,...
-Isobaric
c. Reversible, isothermal expansion of an Ideal Gas
-
must be modified for molecular gases C. Ideal Gas L:m: P\ = 11 R I I. The Ideal Gas Law a. Pressure, P: The standard unit is the Pascal (Pa), but the bar is more commonly used: I bar = 105 Pa b. Volume, V: The standard unit is the m 3, but the liter, L. is more common: I L = I dm 3 c. Temperature, T: The standard temperature unit is absolute temperature, the Kelvin scale: T(K) d. Amount of gas, n: # of moles of gas (mol) e. R is a proportionality constant, the gas constant, given the symbol R : R=O.083 L bar mol-I K-I 2. Applications of Gas Laws a. Boyle's Law (constant temperature, T): Pressure is proportional to I/volume b. Pressure is proportional to temperature, with volume fixed c. Charles' Law (constant pressure, P): Volume is proportional to temperature d. Avogadro's Law (constant P and T) : Volume is proportional to the # of moles, n e. General Ideal Gas Law Application i. Use PV = n RT to examine a gas sample under specific conditions ofP,V. nand T
I ~E = O, Q = w
= 0
C. = R
e. Carnot's Law: For ideal gas: C p
W
Thennocl namics
I. Thermal Engine: A heat engine transfers heat, Q,
Il(
from a hot to a cold
reservoir, to produce work. W a. The I st Law of Thermodynamic states that the work, W. must Q...w equal the di fference between the heat terms: W=
2. The efficiency of an engine. 7J, is defined as the ratio ofW divided by
Qhot: 7J
CanotCyde
p
= QW
"",
3. Idealized Heat Engine: The Carnot Cycle a. The Carnot Cycle consists of two isothermal steps and two adiabatic steps i. For overall cycle:
T.
t. T
n, t. H = 0 and t. S = 0
~
AT - O
i:
b. Camot Thermal Efficiency = 7J = 1
Qhot - QCOld
T.
~-----------------v
ELECTRICITY. MAGNETISM r----- r ------,
c. For a material with dielectric constant K:
Electric FIelds &; Eleetrle Charge
\. Electric Fi~lds & Electric Char:.:e Examine the nature of the field generated by an electric charge and the forces between charges I. Coulomb, given the symbol C, is a measure of the amount of charge: I Coulomb = I amp· I sec e is the charge of a single electron : e = 1.6022 x 10-19 C
2. Coulomb's Law for electrostatic force, Fcoul : F 1 q,q,
= 41r€o
rou'
--r;- r
3. Electric Field, E, is the potential generated by a charge that produces Fcoulon charge qo:
I. A capacitor consists of two separated electrical conducting plates carrying equal and opposite charge. A capacitor stores charge/electrical potential energy 2. Capacitance, C. is defined as the ratio of charge, Q, divided by the voltage, V, for a capacitor: C
V is the measured voltage; Q is the
charge a. Energy stored in a charged capacitor:
' Q'
c
= ,.
=
t QV
KV'
=
b. Parallel plate capacitor, with a vacuum, with area A, and spacing d :
E -- F...: qo
u'
4. Superposition Principle: The total F and E have contributions from each charge in the system:
. C ' C= apacltance:
I.
.0"
dA
ii. Energy Stored:
U
R. Sources flf Electric Fields: Gauss's La\\
I. Electric flux, (/J.. gives rise to electric fields and Coulombic forces 2. Gauss's Law: (1),.
=f E
• dA
=~
The electric flux, cp, • depends on the total charge in the closed region of interest C. Electric "otential & Coulflmhic Ener:.:~ I. Coulombic potential energy is derived from Coulombic force using the following equation:
f
U roo' = F"." dr a. Coulombic Potential Energy: 1 qq'
U e,.,ul = 4Jreor
b. Coulombic PotentiaWoltage i. The Coulomb potential, V(q), generated by q is obtained by di viding the Ucoul by the test charge, q':
U
=
V (q) - U - _1_3. - q' 47[.011 r
=
-
t CII AdE'
3. The Dielectric Effect a. Electrostatic forces and energies are diminished by placing material with dielectric constant K between the charges b. Voltage and electrostatic force (V & F) depend on the dielectric constant, K
V T
b. The SI resistance unit is the Ohm. ,Q 1 h 0
m
( n) = 0&
PanIIeI Plate Capacitor
E =.Y.=..JL d
c"A
c. ParaUei plate capacitor,
dielectric material with
dielectric constant K, with area A. spacing d:
C
=
Kc" A =
d
. f'
"'--II
1 = ,,1 -c L... C. 101
I
iii. Capacitors in Parallel: C ", = ~ C Two Capadton In Series
Two Capaciton In PanDel
~C' i C2 j .,
Capadtan In CIreuIIi
E. Current & Resistance: Ohm's L:m I. Current & Charge: The current, I, measures the charge passing through a conductor over a time; total charge, Q: Q = I • t
4
a:
0.
4. ReSistivity: The inverse of conductivity is resistivity, given the symbol p : p
=
1
5. Voltage for current I flowing through a conductor with resistance R: V = IR 6. Power Dissipation: Power is .Iost as 1 passes through the conductor with R:
Power =VR = 12 R
7. Resistors in Circuits: Certain groups of resistors in a circuit are found to behave as a single resistor
b. For resistors in parallel:
=
L
R
i,,,, - ~ A,
&
C, = vacuum capacitor i. Capacitors in Circuits: A group of capacitors in a circuit is found to behave like a single capacitor " . Capacltors "S ' II IRerles:
w u
Ivolt(V) 1 amp (A)
a. For resistors in series: Rio,
V(q)q '
q _1_fd 47[.00 r
by the current: R =
iii . Electric Field:
c. For an array of charges, qj' V",", = LV, 2. Potential for a Continuous Charge Distribution:
V-
conductivity: J = IJE 3. Resistance a. The resistance, R, accounts for the fact that
energy is lost by electron conduction ;
resistance is defined as the voltage divided
c.
= ~;
U
2. Ohm's Law: Current density, J, is in proportion to the fi eld; IJ is called the
F, Ilin'ct ( urn'nt ('il'cuit (1)( ) I. Goal: Examine a circuit containing battery. resistors and capacitors; determine voltage and current properties 2. Key Equations & Concepts EMF: The voltage of a circuit is called the electromotive force, denoted emf a. This voltage accounts for the battery, Vb' and the circuit voltage, denoted IR : emf=Vb + IR b. The battery has an internal resistance, r: Vb =1 r 3. Circuit Terminology a. Junction: Connection of three or more conductors b. Loop: A closed conductor path c. Replace resistors in series or parallel with RIOt
d. Replace capacitors in series or parallel with Ctot
4. Kirchoff's Circuit Rules a. Constraints on the Voltage i. For any loop in the circuit the voltage must
be the same:
LV = LIR
ii. The energy must be conserved in a circuit loop b. Constraints on the Current i. The current must balance at every node or junction ii.For any junction :~"
--<
LI
ii.For a closed current loop: F = 0 3. Magnetic Moment A magnetic moment, denoted M, is produced by a current loop a. A current loop, with current 1 and area A, generates a magnetic moment of strength M : M=IA b. Torque on a loop: A loop placed in a magnetic field will experience a torque,
6. Gauss's Law: The net magnctic flux through any closed surface is always zero:
f B • dA =
0
a. Gauss's Law is based on the fact that isolated magnetic poles (monopoles) do not exist I. I' arada~', I.a\\ - J' ll'ctrol1la~nctic Induction Faraday's Law: Passing a magnet through a current loop induces a current in the loop Faraday' Law
rotating the loop: r = M • B
= 0
iii. The total charge must be conserved in the circuit; the amount of charge entering and leaving any point in the circuit must be
equal ( •.
'Ja~lIl'1k
J kid . 1\
I. Magnetic Field: A moving electric charge or current generates a magnetic field, denoted by the symbol B; the vector 8 is also called the magnetic Induction or the magnetic nux density a. The SI unit for a magnetic field is the Tesla, T b. The SI unit for magnetic flux is the Weber, Wb
Wb N m N IT = Ill' = C . s = A • m
I. Faraday's Law of Induction
magnetic flux,
~ """LIBeI Z
.n'_
the
interaction
of
Band
M:
U (magnetic) = - M • B 5. Lorentz Force: A charge interacts with both E and 8, the force is given by the following expression: F = q E + q v • B a. Band E contribute to the force b. The particle must be moving to interact with the rna etic field
a. Given the current I and the conductor segment of length dl, the induced magnetic field contribution, dB, is described by the
III A.
passing through the circuit:
a. Special Case: Uniform field 8 over loop of area A; EMF =
B
is the angle formed by dA and 8:
d:
(BA cos 8)
b. Motional EMF: Moving a conductor of length I through a magnetic field 8 with a speed v induces an EMF (8 is perpendicular to the bar and to v): EMF = - B I v c. Lenz's Law: The direction of the induced current and EMF tends to maintain the original flux through the circuit; Lenz's Law is a consequence of energy cbnservation Eleetremapede Wave
p"ldI'r . followmg: dB = 41!' --r
a
b. The total magnetic field for the conductor is
~
e. For a current loop, the field is generated by the motion of the charged particles in the current.
pojdl'r . gIven by: B = 41!'1 ~ 2. The magnetic field strength varies as the inverse square of the distance from the conducting element 3. Special Case - Infinitely long straight wire: B (a) = : ;
i ;a is the distance from the wire;
I is the current; B, is inversely proportional to a I
BIot-savart Law
B
velocity,
v,
in
magnetic
field
0, minimum force) ii.For v perpendicular to
Z
a
4. Ampere's Law: For a circular path around wire,
=
B; F = q v B ( B = 7[/2, a "---->-..... A maximum force) RlPt-llaDd iii. The "right hand rule" Rule defines the force direction b. Force on a conducting segment: For a current I passing through a conductor of length I in a magnetic field B, the force is given by: F=II· B i. For a general current path s: F=ljds.B
.
E B
= c
b. The speed of light, c, correlates the magnetic constant, 11", and the electric constant, . _ _ 1_ Cu.c-
IlloE!)
speed oflight, c: c = fA d. X-rays have short wavelength, compared with radio waves e. Visible light is a very small part of the spectrum
F .... = q v·B = qvB sin 8
(B
.
followmg equatIOn:
wavelength, ..t ,and frequency, f, travels at the
B:
a. B is the angle between vectors v and 8 i. For v parallel to B; F = 0
I. Electromagnetic wa ves are formed by transverse 8 and E fields a. The relative field strengths arc defined by the
c. In a vacuum, an electromagnetic wave. with
2. Magnetic Force: F mag on charge, q, moving at
III A.
Q)""
EMF = fEds AND EMF = -~tl/)m 4. U (magnetic): Magnetic potential energy arises from
c. The CGS unit is the Gauss, G: 1 T = 104 G d. For a bar magnet, the field is generated from the ferromagnetic properties of the metal forming the magnet i. The poles of the magnet are denoted North/South. The field lines are show in the figure below
The EMF induced in a circuit is directly proportional to the time rate of change of the
Torque OD a Loop
the total of the magnetic flux, B . dS, must be consistent with the current, I:
f B • dS = Po I
5. Magnetic Flux, 1/)",
Summarize the general behavior of electrical and magnetic fields in free space I. Gauss's Law for Electrostatics:
fE. dA =
~
2. Gauss's Law for Magnetism:
a. The magnetic flux, 1/)"" associated with an area, dA, of an arbitrary surface is given by the following equation:
I/)m =
j
B • dA;
dA is vector perpendicular to the area dA b. Special Case - Planar area A and uniform B at angle I with dA: I/)m = B A cos B
5
fB'dA=O 3. Ampere-Maxwell Law:
f B • ds = p"I + p"e" ~~" 4. Faraday's Law: f E • dS =
-
~~..
I. Light exhibits a duality, having both wave and particle properties 2. Key Variables a. Speed of light in a vacuum, c b. Index of refraction, n: The index of refraction, symbolized n, is the ratio of the speed of light in a vacuum divided by the speed of light in the material: c (vacuum) n= c (material) c. View light as a wave--Iocus on wave properties: wavelength and frequency i. For light as an electromagnetic wave :
Af
=c
ii. Light is characterized by its wavelength ("color"), or by its frcquency, f. d. View light as a particle in order to
o
- .~--~'-
N
Images & Objects
frequcncy, f, with the proportionality constant
h,
Planck's
Constant:
E (photon) = h f 3. Reflection & Refraction of Light Renection of Light Incident Ray
\"2
2. Lenses and mirrors are characterized by a number of optical paramcters: u. The radius of curvature, R, defines the shape of the lens or milTor; R is two times the foca l lengt h, f: R = 2 f
i+ Sign
Parameters
::j '"'"g;", t.,
f foca l length I--
s obj ct distance
-
Y
diverging lens convex mirror
virtual image
erect
inverted
h' image size
erect
inverted
=
I
speed bends the
light ray as it
passes from n I
to 11 2 i. The angles of the incident and relracted rays are governed by Snell's Law: n, sin 8, n , s in 8,; n l, n2: indices of retraction of two materials . 0 n ·, c. Internal Reflectance: SID " n; ; Light
=
=
passing fromlllaterial of higher n to a lower nmay be trapped in the material if the angle of incidence is too large 4. Polarized Light: The E tield of th.: electromagnetic wave is not spherically symmetric (EX: plane (linear) polarized light, circularly polarized light) a. One way to generate a polarized wave is by retlecting a beam on a surface at a preci se angle , called B, b. The angle depends on the relative indices of refraction and is defined by Brewster's
n·, Law: tan B, =
n.
b. The optic axis: Line from base of object through center of lens or mi rror c. Magnification: The magnifying power of a
*
lens is given by M, the ratio of image si ze to object size: M =
d. Laws of Geometric O ptics i. The m irror equ ation: The focal length, image distance and object distance are described by the following relationship:
1
1
~
Q)::J.
C)
~ :::I a'
;:; w
a:0.
=
a, ...)
d sin B mA, (rn 0, 1, 2, 4. Single Slit Experiment: For a wave passing through a slit of width a, destructive interference is observed for:
=
=
sin () rnA / a, (m 0, ± 1, ± 2, ... ) 5. X-ray difi'raction from a crystal with atomic spacing d gives constructive interference lor:
2 d sin B = rnA,
(rn
= 0, I, 2, a....)
Fundamental Physical Con tanh
Mass of Electron Mass of Proton
ii. The object and image distances can also be
used to detennine the magnification:
h
"S'=-1l'=M C.
ill
lli
1
s+"S' =y s
ill :)i z« C)
c. Huygens' Principle: Each portion of wave front acts as a source of new waves
3. Diffraction of light from a grating with spacing d produces an interfer~nce pattern governed by the following equation:
-
+-
al
'Cc U Destructive Interference
h obj ect size
- OE gJU «$>~" ",0 :::1 ~~
--
=
+ y,
virtual object
s' image distance real image
a. Law of Reflection: For light rcfkcting from a mirrored surface, the incident and retlected beams must have the ame angle with the surface nOl1nal: 0, 0, b. Refraction: Li ght changes Refraction of Light speed as it pa sses through materials with
ditTcrent indices of refraction; this change in Glass
y = y,
)
J't
'bj,"
-
~x
Constructive Interference
- sign
lens concave mirror
o
b. Destructive interference occurs when wave amplitudes add up to produce a new wave with smaller amplitude than either of the component waves: the wave amplitudes cancel out
LeD II. Mirror Properties
i. Energy is quanti zed in packet s called ii. The energy of photon de pends on the
- ~ --"'-----:l
.. ~
understand the energetic properties of light
photons
2. Key Variables & Concepts a. Constructive interference occurs when wave amplitudes add up to produce a new wave with a larger amplitude than either of the component waves
I . Lenses and mirrors generate images of objects
A combination of two thin lenses gives a lens with properti es of the two lenses i. The focal length is given by the
1 follo\\ill1g equation' -f
-1f,
1
= + - f,
3. G ener al Guidelines for Ray Tracing a. Rays that parallel optic axis pass through 'T' b. Rays pass through center of the lens unchanged c. Image: Formed by convergence of ray tracings Ray Tracing d. Illustration of ray tracing for a C on verging Lens
l. Goal: Examine constructive and destructive interference of light waves
Avogadro
~C_o_sta_t n__n__ __ Elementary Charge Faraday Constant Speed of Light
I mr
1.67x10-27 kg
'hr-
.
A
I 6.022x I 0 23 mol-I 1.602x10- 19 C
e
5
96.4R5 C mol-I
1 ~ ms-I c
o
Jf\
, ~
lI"\-
0
rnoo= ~~lf)_ rn .... - rurn=
~DiiiiiiiiiiiiiiiO
, rn iiiiiiiiiiiiiiiO ___
r-=tnJ
~
~ru~tt"'I
~~_o
Molar Gas Constant
R
Boltzmann Constant
k
R.314 J mol-I K·I
.-N nlO_rt'I r-=!,...:!=N I
Gravitation Constant
11.3RX 10-23 JK -I I
G
I
22 CD CD
===:4' ===.
=co
~~-r--0
16.67XIO-llm3 kg·IS· 1
Permeability of Space
14;<
Permittivity of
Space
IR.R5 x 10. 12 F 111. 1
x 10-7 N A-l
