Quick Study Academic Physics 600dpi

WHAT IS PHYSICS ALL ABOUT?

Physics seeks to understand the natural phenomena that occur in our universe; a description of a natural

Base Quantity Symbol Unit

Length /, x Meter - m

Other physical quantities are derived from these basic units: Prefixes denote fractions or multiples of units; many variable

.;;;;;;S=;;=;;;=~

T

~

he

position ofa

motion with position,

1. Newton's 1st Law: A body remains

motion unless influenced by a force energy is the energy of

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

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' ( cis Ll s

s WIt tIme t):v

=

rlt =

Tt

b. Acceleration (a): The rate of change of the · . h . dv Ll v

I

ve oClty 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 equation of motion: x (I) = Xu

+

Vi (

+

t ar b. For constant acceleration. the velocity vs. time is

given by the following: vr(t) = Vi

+

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 straight line. derive x- and y­

equations of motion

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.

Newton's Laws are the core principles for describing the motion of classical objects in response to 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 y Polar

~

_{Polar: (r, 9)}

,

x = r cos9, . y = r sin9, r2=x2 +y2 Spherical z

%,

x x = r simp cos9, y = r sinep sin9, z = r coslj), r2=x2+ y2+z2 Mass m,M Kilogram - kg Temperature T Kelvin - K Time t Second -s

Electric Current I Ampere - A (C/s)

2. Newton's 2nd Law: Force and acceleration determine the motion of a body and predict future 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 _{Dynamk Friction} i. Static friction resists the

Fn

move-ment of a body

ii. Dynamic friction slows

~

the motion of a body

1

,

-

0

For an object on a

horizontal plane: Circular Motion

Ff = Il Fn= ll m g

_....

Net force = FI - Ff

F. Circular 'lotion

I. Motion along a circular path uses r

polar coordinates: (r,8)

_..

2. Key Varia_b_le_s_: _ __ , _ _ _._ _ __ _---.

I

----r-

I

The distance from the

r

IMeter

rotation center (center of

mass)

The angle between rand (} Radian

the (x) axis

Radian/second

t

e

angular velocity

a

Radian/second2 The angular acceleration

The circular motion arc s Meter

s = r8 (8 in rad)

3. Tangential acceleration & velocity:

v, = rw; a, = ra; v and a along the path of the motion arc

v'

4. Centripetal acceleration: a,. =

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

symbols are Greek letters

Math Skills: Many physical concepts are only understood with the use of algebra, statistics, trigonometry and calculus

motion; mass. m and velocity. v: K = t mv' The SI energy unit is the Joule (J): \J = I kg m 2/s2

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 =

J

F ds

For a constant force. work is the scalar product of the two vectors: force. F. and path. r:

W

=

F d cos (8)

=

F • r

F _ _

D

_r F_ _

D

Maximum work

r No work 4. Power (P) is energy expended per unit time:

p = LlWork = LlWork Ll time Ll t Work =

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

II. Potrnthll Energ~ & Enrrg~ Consenation I. The total energy of a body, E, is the sum of kinetic.

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, Eis 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:

L:

t

m

v,' =

L:

t

m v

I

Conserve Momentum:

L

m Vi =

L rn V

r

z

4. Impulse is a force acting over time

~

Impulse = F Ll t or

f

F (t) dt

.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' twirling thin rod (length L): ,', rnL' rotating sphere (radius R): trn r'

Rotating Bodies

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: T = la = r • f (angular acceleration force) 5. Angular momentum is · the momentum

associated with rotational motion: L = Iw = r • P =

f

r • v dm

Torque is also the change in L with time:

T=r'F=~7

Angular

Momentum

h:. Static E(llIilihrium & Elasticit~

t

1. Equilibrium is achieved when:

~f=O ~T=O

,

0

~

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

. . I

U

GM,M

,

a.

G

ravltatlOna energy:

,=

- - r - ­

. .

I

r

F

GM

,

M

,

b G

_{. ravltatlona lorce:}

_,

₌

~

Fg is a vector, along r, connecting MJ 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 m1s2

2. Gravitational Potential Energy, Ug a. The Earth's gravitational

potential => Ug = mgh

b. Weight is the gravitational force exerted on a body by the Earth: Weight = Fg = mg Weight is ill!! the same as mass

Gravitational Potential

Energy

M. O,cillatur~ Motion

I. Simple Harmonic Motion a. Force: F = - k..1 x (Hooke's

Law)

~

.

b. Potential Energy: Uk = +k..1 x' _umll

c. Frequency of the oscillation:

Hooke's f=..L Ik Law 21l'V m _Spring 2. Simple Pendulum a. Period of oscillation: T

=

21l'jI

T

b. Frequency of oscillation: nr, Simple f=..L

!K

21l'V

T

Pendulum

\. 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 m2

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 PPl 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

Archimedes' ,,\.\I\i{fWV""hXXhhH omAA"""" Surface

Principle

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 description of fluid flow

i. For any point y in the fluid tlow:

P

+

+p v'

+

p g y

=

constant ii.For a fluid at rest (special case):

P,-P,=pgh

WAVE MOTION

A. Esampfl's 01 T~fI~S 01 \\a,cs

•Transverse • Longitudinal ·Traveling • Standing

• Harmonic • 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 '\<ave

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 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) Distance between cycles Period T (sec)

Frequency f(Hz) Cycles per ,eeond: f- IT Angular

r::

w (rad/s) =

21l'

/

T

=

2m

Frequency Wave A Height of wave Amplitude

Speed

I

v (m/s)

1

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 =

~

a. B is the bulk modulus, the volume compressibilit. of the solid, liquid or gas

b. p is the density

,

RT

3. For a Gas: v =

J

r

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

frequency decreases:

L

f

_

(

~

\ + \,

)

<=0 s=> I ).

-

-V

~j 'i~t'to.t~~l"".CiljJi.

THERMODYNAMICS

.

Thermodynamics is the study Thermodynamics e. Carnot's Law: For ideal gas: Cp - C. = R c. Reversible, isothermal expansion ofan Ideal Gas of the work. heat & energy of a

Q

W

_{Carnot's Law is exact for monatomic gases; it} against P ext; gas expands from V I to V 2 using an process

~

-Heat: Q +Q added to the system

I

Work: W +W done by the system _!

Energy: E System intemal E

Enthalpy: H H = E+ PV

Entropy: S Thermal disorder Temperature: T

I

Measure of thermal E Pressure: P Force exerted by a gas

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

controlied to allow for di fferent types of

I'W"""'"

r Thermodynamic Condition Constraints Result Isothermal ~T = 0 I

~E

= O,Q = w I ~E

= -w

Q =0 Adiabatic

No heat flow _{PV' = constant}

Isobaric _{~P = 0 W = P~V}

Fixed pressure ~H =

Q

~ V = 0 ~E =

Q

Isochoric

Fixed volume

w

= 0

I. measures thermal energy

a. The SI unit is Kelvin. absolute temperature:

T(K) = T (OC)

+

273.15

T is always in Kelvin, unless noted in the equation b. Zeroth 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 Expansion of Solid, Liquid or Gas a. Solid: tL

=

a

LIT

b. Liquid: LI:

=

/3

LI T

(T, -

T,)nR

,

c. Gas: LI V = - p (Charles Law) 3. Heat Capacity, C

C depends on LI l' and Q, the heat lost or gained: C

=

LI<;' OR

Q

=

CLIT

a. Specific heat capacity is C per gram b. Molar heat capacity is C per mole

c. Two special experimental cases:

i. Heat capacity for constant pressure, Cp :

LI H is the key variable

ii.Heat capacity for constant volume, C.:

LI E is the key variable

5 3

d.ldeal Gas:

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

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 m3, but the

liter, L. is more common: I L = I dm3

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

Charlet' Law

jl

\

-,...

~

~

I

L

Pressure (Pa) Temperature(K) O. El1lhalp~ & I st Lan of Thumod~l1amics

I. Wand Q depend on path of the process; however,

~

E

is independent of path

2. lst Law of Thermodynamics: ~E =

Q -

W a. The change in energy of the system (~E ) is

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

+

PV ~H = ~ E

+

P ~V

a. ~H

=

Q

for a process at constant pressure; the difference between E and H is the work performed by the process

i. Endothermic: Positive ~H ; the system absorbs heat from the surroundings (EX: evaporation of liquid to gas; melting of a solid)

ii.Exothermic: Negative ~H; the system releases heat to the surroundings (EX: combustion of fuel, condensation of vapor to liquid)

b. Phase Transitions: solid +- liquid +- gas A phase change corresponds to a change in enthalpy:

i. Enthalpy of vaporization: ~H ... ii. Enthalpy of fusion: ~

H

,,,,

c. Enthalpy & Variable Temperature:

~H

=

jCpdT

For constant Cp : ~

H

=

C.

~

T

4. Examples of Work: W =

j

PdV

a. P opposes the ~V for an expansion; P causes the

~V for compression; W depends on the path b. Single step isobaric expansion from V I to V 2

against an opposing pressure, Pext'

infinite number of steps; the system remains in equilibrium: W = n RT In

(~

:

)

.

This type of process gives the maximum work

SIa&Ie

Step EspDIIoa

P L '

p..

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 2. Equations for Energy of an Ideal Gas:

3

E

=

+

M

v' and

E

=

2' RT

a. Average Speed of a Gas Molecule: V n", =

/3~T

b. Gas Speed & Temperature: v"". , IS

proportional to

IT

;

a change from TI to T2 {T;

changes the speed by

V1";'

c. Gas Speed & Mass: v,,'" is proportional to

!l:r

d. Kinetic Energy for 1.00 Mole of an Ideal Gas:

K

-

_-

lRT

₂

e. For a Real Gas: Add heat capacity and energy terms for molecular vibrations and rotations 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

Entropy is a state variable, like E & H:

~

S

(universe)

=

~

S

(system)

+

~

S

(thermal reservoir)

2. 2nd Law of Thermodynamics:

For any spontaneous process. _~s,,""

_>

₀ ~S,,"h = 0 for a system at

equilibrium or for a reversible process

3. Examples of Entropy Changes

a. Natural Heat Flow: Heat flows Q

from Thot to Tcold

b. Entropy & Phase Changes:

~

LI H (change) Heat Flow

LlS(changc) = T( h )

c ange solid --. liquid positive ~

S

~

S

,

...

liquid -+ solid positivI.' ~

S

~

S

"p

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

Thennocl namics

I. Thermal Engine: A heat engine transfers heat, Q,

from a hot to a cold

Il(

reservoir, to produce work. W

a. The I st Law of Thermodynamic states

that the work, W. must

Q...w

T.

equal the di fference between the heat terms:

W = Qhot - QCOld

ELECTRICITY. MAGNETISM

r - - - - -

r

- - - ,

Electric FIelds &; Eleetrle Charge

\. Electric Fi~lds & Electric Char:.:e

2. The efficiency of an engine. 7J, is defined as the

ratio ofW divided by Qhot: 7J

=

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

~

n,

t.

H

=

0

and

t.

S

=

0

b. Camot Thermal Efficiency = 7J = 1 ­

i

:

c. For a material with dielectric constant K:

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, F coul :

F 1 q,q,­

rou'

=

41r€o

--r;-

r

3. Electric Field, E, is the potential generated by a charge that produces F coulon charge qo:

E - F...:

u '

- qo

4. Superposition Principle: The total F and E have contributions from each charge in the system:

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:

U

roo'

=

f

F"." dr

a. Coulombic Potential Energy:

1 qq'

U e,.,ul = 4Jreo

r

b. Coulombic PotentiaWoltage

i. The Coulomb potential, V(q), generated by q is obtained by di viding the Ucoul by the

test charge, q': V (q) - U - _1_3.

- q' 47[.011 r

U = V(q)q'

c. For an array of charges, qj' V",", = LV, 2. Potential for a Continuous Charge Distribution:

V -

_1_fd

q

- 47[.00 r

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

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'

U

= ,.

c

=

t

QV

=

KV

'

b. Parallel plate capacitor, with a vacuum, with area A, and spacing d:

. C ' C A

I. apacltance: = .0"

d

ii. Energy Stored:

U =

t CII AdE'

iii . Electric Field:

PanIIeI Plate Capacitor

E

=.Y.=..JL

d c"A

c. ParaUei plate capacitor, dielectric material with dielectric constant K,

with area A. spacing d:

&

= Kc" A = .f ' C _{d "'--II}

C

,

= vacuum capacitor

i. Capacitors in Circuits: A group of capacitors in a circuit is found to behave like a single capacitor

"C " S e ' 1 ,,1

II . apacltors IR rles:

-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

p CanotCyde

AT -O

T. ~---v

2. Ohm's Law: Current density, J, is in proportion to the field; IJ is called the

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 V

by the current: R =

T

b. The SI resistance unit is the Ohm. ,Q

w

u

1 h ( n) = Ivolt(V)

a:

c. 0 m 0& 1 amp (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

a. For resistors in series: Rio,

=

L

R

b. For resistors in parallel:

i,

,,

,

-

~

A,

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

= 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 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

~

"""LIBeI

Z

.n

'

_

III

A.

a

~

e. For a current loop, the field is generated by

the motion of the charged particles in the current.

2. Magnetic Force: Fmag on charge, q, moving at

velocity, v, in magnetic field B:

F .... = q v·B = qvB sin

8

a. B is the angle between

vectors v and 8 i. For v parallel to B; F

=

0 (B = 0, minimum force) ii.For v perpendicular to B; F

=

q v B (

B

= 7[/2,

_a

"---->-...

A

maximum force) RlPt-llaDd

Z

iii. The "right hand rule" Rule

defines the force direction

b. Force on a conducting segment: For a current

A.

I passing through a conductor of length I in a

magnetic field B, the force is given by:

a

F=II· B

i. For a general current path s:

F = l j d s . B

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, rotating the loop: r = M • B

Torque OD a Loop

4. U (magnetic): Magnetic potential energy arises

from 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

. p"ldI'r

followmg: dB = 41!'

--r­

b. The total magnetic field for the conductor is

. p o j d l ' 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

4. Ampere's Law: For a circular path around wire, the total of the magnetic flux, B . dS, must be consistent with the current, I:

f

B • dS = Po I 5. Magnetic Flux, 1/)",

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

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

I. Faraday's Law of Induction

The EMF induced in a circuit is directly proportional to the time rate of change of the

magnetic flux, Q)"" passing through the circuit:

EMF

=

fEds

AND EMF

=

-~tl/)m

a. Special Case: Uniform field 8 over loop of area A;

B

is the angle formed by dA and 8:

EMF =

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

I. Electromagnetic waves are formed by

transverse 8 and E fields

a. The relative field strengths arc defined by the

. . E

followmg equatIOn:

B

= c

b. The speed of light, c, correlates the magnetic constant, 11", and the electric constant,

. _ _ 1_

C u . c - IlloE!)

c. In a vacuum, an electromagnetic wave. with wavelength, ..t ,and frequency, f, travels at the 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 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:

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

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)

2. Key Variables & Concepts

I. Lenses and mirrors generate images of objects a. Constructive interference occurs when

wave amplitudes add up to produce a new

-.~--~'- _- wave with a larger amplitude than either of

wave

~--"'---:l

o ..~ _{the component waves}

N

Images & Objects

o

n=

_{c (material)} 2. Lenses and mirrors are characterized by a

number of optical paramcters:

\"2

~

x

c. View light as a wave--Iocus on wave

u. The radius of curvature, R, defines the

properties: wavelength and frequency

shape of the lens or milTor; R is two times Constructive Interference i. For light as an electromagnetic wave:

the focal length, f: R = 2 f

b. Destructive interference occurs when

Af

=

c

wave amplitudes add up to produce a new ii. Light is characterized by its wavelength _{LeD II. Mirror Properties}

wave with smaller amplitude than either of ("color"), or by its frcquency, f.

the component waves: the wave amplitudes

d. View light as a particle in order to _{Parameters i+ Sign -sign cancel out}

understand the energetic properties of light

- y = y,

+

y,

-

al

E

_O

i. Energy is quantized in packets called J't ) ill

::

j

'"'"g;",

diverging

Y

_gJU

:)i

photons lens lens

_«$

_>-

_«

z

f focal length C)

ii. The energy of photon depends on the concave convex

_~

_"

",0 :::1 ill

mirror mirror

_lli

frequcncy, f, with the proportionality

_~

_~

<Ii

I-- ­

~

;:;

constant h, Planck's Constant:

s obj ct virtual 'CcU _C) w

E (photon) = h f Q)::J.

distance

t.,

'bj,"

object _~:::I

a:

0.

3. Reflection & Refraction of Light _{Destructive Interference} a'

Renection of Light

Incident s' image distance real image virtual c. Huygens' Principle: Each portion of wave

Ray image front acts as a source of new waves

-

_+-

-

3. Diffraction of light from a grating with

spacing d produces an interfer~nce pattern h object size erect inverted

governed by the following equation:

d sin B

=

mA, (rn

=

0, 1, 2,

a, ...

)

h' image size erect inverted _{4. Single Slit Experiment:}

I

For a wave passing through a slit of width a,

a. Law of Reflection: For light rcfkcting

destructive interference is observed for:

b. The optic axis: Line from base of object

through center of lens or mirror sin ()

=

rnA/a, (m

=

0,

±

1,

±

2, ... )

from a mirrored surface, the incident and

retlected beams must have the ame angle

5. X-ray difi'raction from a crystal with atomic

with the surface nOl1nal:

0,

=

0

,

c. Magnification: The magnifying power of a

spacing d gives constructive interference lor:

b. Refraction: lens is given by M, the ratio of image size

2 d sin

B

=

rnA, (rn

=

0, I, 2,

a

....)

Light changes Refraction of Light _{to object size: M =}

*

speed as it

d. Laws of Geometric O ptics

passes through Fundamental Physical Con tanh

materials with i. The _{image di}mirror equ_{stance and object} ation: The focal length, _{distance are}

Mass of

ditTcrent indices _{described by the following relationship:}

Electron

of refraction; _{1 1 1}

this change in Glass

s+"S'

=y

Mass of Proton

I

mr 1.67x10-27 _kg

speed bends the ii. The object and image distances can also be

light ray as it used to detennine the magnification: Avogadro .

r-passes from n I

s

h

~

_

C

o

_

n

__

st

a

_

n

_

t

_ __

'

h

A

I

6.022x I 023 mol-I

"S'=-1l'=M

to 112 _Elementary

o J f \ ­

e 1.602x10-19 _C

i. The angles of the incident and relracted C. A combination of two thin lenses gives a Charge , 0­

rays are governed by Snell's Law: lens with properties of the two lenses ~ l I " \

-Faraday r n o o =

n, sin

8,

=

n, s in

8

,;

nl, n2: indices of i. The focal length is given by the 5 96.4R5 C mol-I

Constant ~~lf)_

retraction of two materials 1 1 1 rn .... - ­

follo\\ill1g equation' -f

=

-

+ -

1

r u r n =

. 0 n·, f, f, ~DiiiiiiiiiiiiiiiO

c. Internal Reflectance: SID "

=

n; ; Light Speed of Light c

~

ms-I

_{r-=tnJ iiiiiiiiiiiiiiiO_ _ _}

~

3. General Guidelines for Ray Tracing , rn

~ru~tt"'I

passing fromlllaterial of higher n to a lower a. Rays that parallel optic axis pass through _{~~_o}

Molar Gas

nmay be trapped in the material if the angle 'T' R R.314 J mol-I K·I . - N

Constant

b. Rays pass through center of the lens n l O _ r t ' I

r-=!,...:!=N

of incidence is too large unchanged I I

===:4'

4. Polarized Light: The E tield of th.: c. Image: Formed by convergence of ray Boltzmann 2 2

===.­

CD CD = c o

k

1

1.3RX

10-23 JK -I

electromagnetic wave is not spherically tracings Constant

~~-r---Ray Tracing I

d. Illustration of ray tracing for a

symmetric (EX: plane (linear) polarized light, 0­

Gravitation

circularly polarized light) _{G 16.67XIO-llm3 kg·IS·}1

C on verging Constant

a. One way to generate a polarized wave is by

Lens

retlecting a beam on a surface at a precise

Permeability of

angle, called

B

,

1

4

;<

x 10-7 _N A-l

Space

b. The angle depends on the relative indices of

l. Goal: Examine

refraction and is defined by Brewster's

constructive and Permittivity of

B n·,

I

R.R5 x 10.12 F 111. 1