AL Phy Notes

39
Lsw 2011/04/16 rev Contact: [email protected] 1 - AL PHY Notes for Physics AL I) Dynamics 1. Extrapolation to determine absolute Zero By Pressure Law/Charles’s Law, PV T . Plot P or V against o C and extrapolate it Pressure Law, PT : Keep Volume constant by a fixed mass flask Boyle’s Law, 1 P V : Keep temperature constant by slow motion Charles’ Law, V T : Keep pressure constant by mercury piston Note that to reduce volumetric error, use as short tube as possible Linear temperature scale: linear interpolation Corrected Law: 2 2 ( )( ) ; : : : ln : ( ) By system per molecule T a P V nb nRT V a attractive Intermolecular forces V nb total volume of molecules Helmholtz free energy A U TS kT Q dA dU SdT TdS First Law dU U W TdS pdV dA pdV SdT A NkT P v V N 2 2 ' ' Na b V 2. Internal energy = PE+KE KE per molecule= 3 3 2 2 A R T kT N . Rotational, translational, vibrational PE negligible in ideal gas (no bonding, Van der Waals force, no attraction) 3. Ideal gas: High temp, Low pressure Low pressure=> low density=> volume of container dominates High temp=> Kinetic energy dominates 4. Kinetic theory assumptions Elastic collision, negligible size, no interaction between molecules, short duration, random 5. 2 1 3 PV NMc Average change in momentum= mu t (Considering x-axis for one molecule - intermolecular force negligible) ; Newton’s 2 nd law yields: Force on molecule by wall = 2 mu l N3 rd law yields: Force on wall = Force on molecules. Large population gives x y z u u u

description

Physics notes for HKAL students. Final review on major topics. Out-Of-Syllabus stuff is indicated and sifted.

Transcript of AL Phy Notes

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Notes for Physics AL

I) Dynamics

1. Extrapolation to determine absolute Zero

By Pressure Law/Charles’s Law, PV T . Plot P or V against oC and extrapolate it

Pressure Law, P T : Keep Volume constant by a fixed mass flask

Boyle’s Law, 1

PV

: Keep temperature constant by slow motion

Charles’ Law,V T : Keep pressure constant by mercury piston

Note that to reduce volumetric error, use as short tube as possible

Linear temperature scale: linear interpolation

Corrected Law:

2

2

( )( ) ;

:

:

: ln

:

( )

By system

per molecule

T

aP V nb nRT

V

aattractive Intermolecular forces

V

nb total volume of molecules

Helmholtz free energy A U TS kT Q

dA dU SdT TdS

First Law dU U W TdS pdV

dA pdV SdT

A NkTP

v V N

2

2

'

'

N a

b V

2. Internal energy = PE+KE

KE per molecule=3 3

2 2A

RT kT

N . Rotational, translational, vibrational

PE negligible in ideal gas (no bonding, Van der Waals force, no attraction)

3. Ideal gas: High temp, Low pressure

Low pressure=> low density=> volume of container dominates

High temp=> Kinetic energy dominates

4. Kinetic theory assumptions

Elastic collision, negligible size, no interaction between molecules, short duration, random

5. 21

3PV NMc

Average change in momentum=mu

t

(Considering x-axis for one molecule - intermolecular

force negligible) ; Newton’s 2nd law yields: Force on molecule by wall = 2mu

l

N’ 3rd law yields: Force on wall = Force on molecules. Large population gives x y zu u u

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6. Pressure definition: Force exerted by molecules rebounding from surface and calculated in

average rate of change of momentum of molecules per unit area.

Saturated vapour pressure: pressure exerted by vapour in equilibrium with liquid

7. First Law of Thermodynamics ΔU=Q+W

As conservation of energy

Internal energy U =3

2RT neglecting rotational, vibrational, etc

Heat is non-mechanical exchange of energy between system that are of different temp.

Q = ml or mcΔT (Steam hurts more as they carry latern heat)

Work done (transfer of energy in and out) by gas= d ( )P V Fds

Isothermal: 2

1

/ ln

V

V

nR nRnR T is constant WD dV V

TV T

Adiabatic (no heat exchange, rapid motion) .P

V

C

CU W PV const

,V PC C being molar heat capacity at constant volume/pressure

Isobaric (Pressure constant) : WD P V

8. Second law of Thermodynamics

No perfect heat engines for entirely doing work

ln , , " ",A

Q RS k W for reversible process W as ways k

T N

1

1

W N q

W q

Entropy always increases. See that transferring quanta has greater effect (entropy change)

on the object with less energy

9. Conduction rate

( )hot coldA T TQ

t d

Ud

as conductance ~

ACapcitance

d

1 2 3

1

1 1 1....

all layersU

U U U

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10. Lattice model (Left)

F=dU

dr

,

2

.,

between atomsslope of f r graph k Max fE UTS

r r r

For certain temperature (>oK) , they vibrate in between A and B with mean r1. For higher

temperatures, KE and thus total energy increases and they vibrate shifting to larger mean

separation.

Asymmetry with displacement larger on extension side.

Lattice modelling: Short range repulsive and large range attractive force ; Restore upon

extension /compression, No net force at eq.

Note that molecules in systems above 0K vibrate about their mean position.

11. Stress-Strain model (Right)

A: proportional limit

B: elastic limit

C: UTS (ultimate tensile stress)

D: breaking stress

AD: Elastic deformation; >B: Plastic deformation

Find Y by two parallel steel wire connected as to eliminate effects of change in support and

temperature.

~YA A

k Cl d

;

Force extension StressStress Strain E

Area natural length Strain

12. Adhesive force and cohesive force

Adhesive: attraction between unlike bodies(eg, electrostatic) => tend to spread

Cohesive: Intermolecular force inside the liquid

13. Viscosity

Tangential force between layers of liquid of relative motion: u

F Ay

, tends to reduce P

Zero velocity on sides, maximum at centre

Viscosity falls as temperature increases (liquid)

14. Steady flow

Liquid elements at points follow the same path and velocity and do not vary with time

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15. Bernoulli’s principle

Work done on fluid = work gain by fluid (no energy lost)

Incompressible yields 1 1 i iAv Av

1 1 1 2 2 2

2 2

2 1

2

( )

( )2

1( ),

2

Work done P Av P A v t

Vv VvEnergy gained Vg h no E lost

Cancel V Contiunity P gh v Const

2

2

max

:

2

2

( ), 2

. 0

2( )

A o

B B o

C o C

C o C

B

atm B

Siphon P P gd

vP gh P

vP P gh

Since P P opened end v gh

Max velocity when P

P ghv

16. Electric vehicles

Environmental friendly: no CO2 emerged, O2 intake

Efficiency: Do not consume energy when idling

Source: Electricity could be generated by multi-sources

Short range, heating effect

All energy finally degrades into “internal energy”, which is low-graded

And hardly turns into other forms

17. Gas discharge tube

Mean free path comparable to tube length and avalanche effect

18. Hydroelectric power

mgh VPower gh

t t

19. Stefan’s law

4 4 2: 4Total energy

Emissive power T spherical L T RA unit time

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20. Black body radiation

A black body abosrbs all wavelengths and thus emit all wavelengths (continous spectrum)

21. Surface tension

Molecules in the surface area escape and are more spaced. Molecular concentration

decreases, while the attractive force between molecules produce the surface layer under

tension.

WD to separate further the molecules against molecular attraction force and increases PE of

the liquid.

Capillary effect: fluid flows against gravity in narrow tubes that intermolecular attractive and

surface tension acts on it.

2 cosh

gr

1 1

( )

:

4,

: :

, 0 .

x y

x y

PR R

for spheres R R

for real bubble two sides PR

ALT from free energy dF PdV dA

PdV dA dF at eq

22. Weaks

Deep water: , . ,gk def of k by wave theory

Shallow water: ck gh k ; Apex angle = arcsin( )c

for v c onlyv

23. Upthrust/ Buoyancy

Pressure difference = fluid displacedgV

Archimedes’ principle: Any floating object displaces its own weight of fluid.

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II) Theoretical Mechanics

24. CG expt

A rigid body consists of infinite point masses and all of them are subjected to G-Field forces.

Resultant of these forces forms single weight of a rigid body and acts through the position

Centre of Gravity. (CG equals CM if the gravitational field intensity equals, at all points)

<- Join the intersection of two drawing.

25. Centre of Mass

int

2

int int2, ( ) ; 0

( :)

,

ext

i

i ext

all i all i all i

ext

i i

all i

i i i

all i

i i

all i

Every particles are acted by F F

d rThen m F F F

dt

Set F MR

m rrdm

RM M

CG M g OG m g r

MR a m r a

For the sector with uniform density

2

, , 0

2 1cos

3 2

,

( , , )

by symmetry y

r r dx dmx

M M

For non uniform density

x y z x dVx

M

By Newton’s second law, The CM accelerates at F/M no matter where does the force act at.

An body’s linear motion could be treated as if it is concentrated at CM

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26. Tension

i

l xT m g

l

T Tlv

M

27. Momentum conservation

Elastic situations that A hits stationary B

ma=mb: va=0, vb=ua. Largest KE of B

ma>>mb: va≈ua , vb≈2ua. Largest speed of B

ma<<mb: va≈ua, vb≈0. Largest momentum of B

1 1 2 2 1 1 2 2

1 2 2

2 2 2 2

1 2 1 2

1 2

, , /

0

0

Recall that m v m v m u m u u v are initial final velocities

if elastic and m m and u

v v u u

v v

Ie, collision of equal masses with initial stationary objects gives perpendicular final velocities.

2 10

T

Impulse Fdt mv mv By Newton’s 3rd (reaction law), the force acting on each object during impact equals:

1 2 2 2 2 1 1 2 1 10 0

,T T

F dt F dt m v m v m v m v T is short enough

Non-elastic collision: Total Kinetic energy is not conserved giving some energy in other form

Momentum is conserved, in all systems.

Newton’s coefficient law: 1 2 1 2( ) , 1v v e u u e when elastic

Say initially the upper sphere is static,

: cos ( cos )

. : cos cos

: sin sin

Coeff law w v e u

Cons of momentum u v w

no change in momentum at u w

28. Energy derivation

2

2

Fds mgdx mgh

dv mvFds m ds mvdv

dt

2

2

kxFds kxdx

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29. Equilibrium and pseudo force

Equilibrium if and only if Net force=Net moment=0

A pseudo force of magnitude ma and reversed direction is ‘acted on CM (dynamic eq.)

In the figure:

;

;

f ma

R mg

ma L R s mg x

Equilibrium breaks at A first (slipping) if

A B

A B

F F

R R

30. Centripetal acceleration = 2v

r

22

0 0lim limt t

v v for small

v v vv r

t t r

The acceleration is given and perpendicular to vA and towards the center 2

1 2

2

2

2 2

2

2 1

cos ( ) ( )

sin ( )

( )

( )

dT mg ma a is radius

dt

dmg ma

dt

d xmg T m

dt

dT T I

dt

2

2

: 2 ( )

1( )

0

Tangential acceleration r r Polar coordinates

dr

dt r

r conserved if Tangential force I conserved if no net torque

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31. Application of Circular motion

In vertical closed tubes:

2 2

min

1 1(1 cos ), ,

2 2

, 4

mu mv mgr is angle with vertical u as velocity at lowest pt

put u gr

When u is at minimum, the particle just reaches the top. For 4u gr it completes the circle

In strings:

2

min

2

1

, ,

5

, cos ,

2 ,

2 5 , 0, cos (

at the toppest point for lowest velocity centripetal force is provided entirely by mg

mvmg

r

u gr

mvand for all T mg

r

for u gr it oscillates about the bottom

ufor gr u gr it follows free projectile at T

2 2

)3

gr

gr

Further, it reaches 2 2 2 2 4 2

3 2

2 2 5 4(1 )

3 3 18

u gr u gr g r u grur

gr g r

above the ground

32. Centrifuge

Consider the part of liquid between A and B, Pb>Pa as to provide centripetal force inwards

required.

For that part of liquid, force due to pressure differences exactly equals cf. (centripetal force)

needed.

If the part is replaced by smaller density (i.e. mass), force is too large to move inwards

(towards centre of rotation) much effective than leaving suspension* as2r g

Practical uses:

Milk cream separation, solid from suspension, laundry driers spin to remove water

(The drum of a drier has many holes in it which reaction from the circumference provides

adequate cf. acting inwards. However, no such reaction in the holes and water rippled out)

*The denser portion will sink to the bottom due to the pressure difference (Weight>Upthrust)

which is given by: PA gV . Now the force by centrifuge is 2PA Vr .

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33. Oscillators and energies

Pressure restoring force:

2

22

mx gA x

VPeriod

gA

Upthrust:

2 2

watermx Agx

V hPeriod

Ag g

h is height of the column

Assume Repulsive force only:2 2

2 2

2 22 3 2 3

3

3

2

4 ( ) 4 ( )

( 2 2 )4

2

( )

o o

o o o o

o

o

o

Ze Zemx

r x r x

Ze Zer xr r xr x

r

m rPeriod

Ze

provided that x r

34. Resistance related SHM

Period is larger when compared to non-dump

Amplitude is exponentially decreasing

35. SHM kinematics

SHM: acceleration acts in proportional to x but in oppose direction with it, i.e.

x x (See app5.1)

2

2 22 2

2 2 2

2 2 2 2

cos( )

sin( )

cos( )

1 ( )

( . )

1 1( ) ( )

2 2

( )

x A t

x A t

x A t

x xv A x ellipse

A A

x x st line

KE mv m A x quadratic

We call x k x const as general form of SHM

SHM could be visualized as projection of a circular motion’s Vertical AB (or horizontal )

component:2

cos

cos

x r

a r

a x

SHM energy of a vertical spring: Taking GPE into account and total energy is given by (Taking eq. pt as GPE=0) :

2 2 2 2 21 1 1 1 1( )

2 2 2 2 2mv k x e mgx mv kx ke

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36. Forced oscillations

2 1

2 22 2 2 22

1 22 2 2 2

( ) sin , ,

sin( tan ( ))1 1

{ cos sin }4 4 ( )

kt

total

dxsetting f t a pt resistive force mk as natural freq

dt

kpa pt

px e A k t A k t

p k p

Transient solution: dependent on natural frequencies, the 2

kt

e

part

Steady solution: dependent on driving force, the part depends on “p” on right

Maximum amplitude as

2 21

2p k

(For sake of convenience, we say “resonance” when p=ω)

At resonance, F leads displacement by 1

2 2lim tan ( )

2p

kp

p

Power input=output, maximum power transferred.

Amplitude remains finite when there’s resistive force

Analogy to LCR circuit, max. power at resonance: 2 2

2 2

.1

( )

sourceV Ravg Power

RR L

C

37. Rolling

Pure rolling, no friction/ slipping:

v r Spin faster and slows linearly: v r

Spin slower and accelerates: v r 2

2

, , :

dFr I

dt

F mx

F mg as slipping

As it is rolling deformation occured reaction points to oppose motion rolling friction

An rigid body’s general motion is comprised of motion ofCM and its own rotation about CM.

2 21 1

2 2Energy mv I

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38. Vector force treatment (see app 4.2)

1 2 1 2, , , , , ,

.

: ,

,

.

n n

i

i i

i i

Forces f f f acting on r r r is equivalent to

R f acts through the Line Of Action

Line Of Action Let r be a point on LOA then

r R r f

Notice that r on LOA r R constant

Couple is indp of the Point of

Moment taken

0to any point on LOAQ R M

*For equilibrium, add a force –R acting at Q and a moment M That equals couple by R and –R

39. Polar coordinates

2 2

2 2

2 2

2 2

cos sin ; sin cos

;

( ) (2 )

[( ) (2 ) ] (

r

r

r

r

r

r

r

e i j e i j

dedee e

dt dt

Displacement r e

dr dVelocity e r e

dt dt

d r d dr d dAcceleration r e r e

dt dt dtdt dt

dmv d r d dr d dRecall F m r e r e co

dt dt dt dtdt dt

2

2 2 2

2 2 2

)

. /

1 1( ( ) )

2 2

1; ( ) ( )

2

b b

a a

nst mass

dAngular momentum conserved if no ext torque force in e mr

dt

KE mv m r r

Area r d Arc length rd dr

40. Relative motion

Relative velocity: : AB A Bvel of A relative to B V V V

Effect of common acceleration (eg, gravity) could be cancelled

in dealing with their relative motion

Apparent velocity: apparent wind my motionV v v

Closest approach to the object when the relative velocity

does not lie on the path is : d sinθ

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41. General projectile treatment:

2

22

2

, :

cos

sin2

:

tan sec2

x y indpendence

x u t

gty u t

General form

gxy x

u

2 2

2

4 2 2 2

2 2

sin

2

2 sin

sin 2

4 4( 2 )

0

, 0, ,

( , , )

uH

g

uT

g

uR

g

u g x u gy

g x

for reaching a point

See that the collection of points which give under fixed u will be the envelop of safety

The points above no matter what be cannot be striked

y

2

2

2

2 2 2

2

2

2 2

, ' ( , )

' ,

, ( )

:

1cos sin

2

1sin cos

2

g ux

u g

For wall problems put wall s top point coordinate a b

when there s no real root of u then the particle cannot pass it

Also u g b a b for any

Inclined motion

x u t g t

y u t g t

time of fl

2

2

2 sin

cos

sin(2 ) sin{ }

cos

90max

2

o

uight between one collision

g

uRange

g

when

Showing +ve x side only, envelope of safety models the enclosed region with fixed initial velocities

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42. Power Fv

dWP

dt

d Fds

dt

dsF Fv

dt

43. Conservative force:

Path independent: Energy used= final initialU U

0Closed

F ds

0F

dUF

dr

44. Gravity

Classical Newtonian G-Force=2

gGMm

r

Total energy in an circular orbit= 21

2 2

GMmmv

r

Weak equivalence principle: Gravitational field strength does not depend rest mass

G-Force point to the instant position, but not the “retarded” position

45. 2

r

ext

GMm GMmU F dr dr

r r

22

2 2 2; og RGM Rg

R r r

46. Reduced mass algorithm

1 1 2 2

1 2

1 2

1 2 1 2

1 2

2 21 21 2

1 2

,

1( )

1 1

1 1( )

2 2

mutual

centre of mass relative

CM

F m a m a

Let F a where a is relative acceleration a a

m mF FF

m m m m

m m

m mKE m m v v

m m

v should not change unless net external force

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47. Moment of Inertia 2

2

/ / :

: ( ) ,

:

:

def

x G

z x y

I mr

axis thm

Any rigid body I m GX I G is centre of gravity

axis thm

Lamina I I I

22 2 2 2

2

2 2 22

: ( ) ,4

2

4 5

about G

a

a

rFor sphere I m x x r a

m r x

a x MaI x dm

48. Rotation

Every particle in a rigid body experiences:

2

,

2

,

,

2 2

2

( ) ;

( )1

( )

C i i i

iT i i

i

T i i

all i

i to Axis

i Axis

all i

dF m r

dt

d rF m could be comprised of Internal forces

r dt

F r Total external torque about Axis

d r dm I

dt dt

: (1) and (2) have different total external torque. (2)

and (3) have different distribution of masses and

thus different Moment of inertia. 2

2sin

2

dMg I as a SHM

dt

IT

Mg

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III) Wave Theory

49. cos( )y A t kx . Propagating to right: t

Phase velocity=k

Group velocity:gv

k

(envelope velocity, different from Phase velocity when speeds of

different frequencies are not equal) =v for particles

Speed of sound:C

c

(Elasticity/Inertial properties)

50. Superposition

Resultant displacement=sum of corresponding displacements

51. Polarization

Selection of direction of disturbance from two or more choices

Only transverse waves could be polarized

Brewster’s angle = 2

1

arctan(1 2) arctan( )n

nn

Sunglasses aligned to block the s-polarized glaze reflected ray

Scattering: re-radiation in all directions of EM Waves=> reduction in initial axis

Save bandwidth by polarization in antenna

LCD: Twist angle to control amount of light

Wire grid: E field induces motion of e- or reflected

Phase Shift of reflecting ray

(air to glass, i tn n ):

Phase Shift of reflecting ray

(glass to air, i tn n ):

Transmittance

2

2

cos[ ]

cos

t t t

i i

n ET

n E

; Reflectance R=

2( )r

i

E

E. R+T=1 (consv. of energy)

/ /0 0

' ( ) :

cos cos

cos cos

i t t i

r i

i t t i

Fresnel equation s solutions same

n nE E

n n

//0 //0

2 cos

cos cos

i it i

i t t i

nE E

n n

0 0

cos cos

cos cos

i i t tr i

i i t t

n nE E

n n

0 0

2 cos

cos cos

i it i

i i t t

nE E

n n

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52. Reflection and transmission

Followed by Fresnel equations, at normal incidence:

0

0

:

2:

i t

i t

i

i t

n nReflected A A

n n

nTransmitted A A

n n

53. Classical Huygen’s principle

Every point on AB is regarded as source of secondary wavelet. Common

tangent CD of the spherical wavelets of radius ct. Constructed wavefront

CD is parallel to AB.

54. Wavefront diagram, Snell’s Law

Different in speed, AB not parallel to A’B’

55. Simple refraction

Refractive index increases with frequencies. (group velocities varies)

Due to differences in speed: EM wave photons are being forced into

“modes/phonon” with lattice.

If the photon matches the phonon mode, the photon is absorbed=> heat

If not, re-emitted with delay and slowed phase velocity of the propagation. Freq. stays const.

Phonon: Quantum of vibrational energy taking levels 1 2 3

, , ,2 2 2

h h h

56. Double silt expt

Requirements: Monochromic light, narrow vertical slits, close together, parallel to source,

coherent sources, D>>d, strong enough source

Coherent time: Constant phase relationship within

emitted photons

Compose of diffraction and interference pattern.

More slits give sharper effects.

Core: In phase arrival gives maximum.

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57. Diffraction of narrow gaps or aperture

Effect due to superposition that on unrestricted part of a wavefront that have been

obstructed by an obstacle or aperture.

Sounds of lower frequencies could be diffraction more effectively by Large speaker cones

58. Intensity, dB

2

10

,

( )4

( ) 10log ( )o threshold

PI consequence of energy conservation

r

IIntensity level dB

I

59. Sound wave as Pressure wave

Reflection from air to solid: Compression-> Compression (High P->High P)

Kundt tubes: Reflect and superimpose and standing waves.

Powders swirls away from antinodes and heaps are formed at nodes.

Count the powder’s number and thus separation=2

and

use v f . Use dry tube and thin layers.

60. Standing wave

Superposition of two trains of waves in opposite direction with near A and f.

-Nodes points are always destructive.

-Energy is confined in st. wave

-In phase for all particles in adjacent nodes, but are of different amplitudes

Acoustic devices emerge sound waves with Superposition of its natural frequencies.

Always form st. wave when there’s one side of reflection only

String and open tube: 0 0 0 0,2 ,3 ,4 ,.....f f f f

Closed tube: 0 0 0 0,3 ,5 ,7 ,....f f f f

f0 is fundamental note; Harmonics are multiple of f0 and overtones are the resonant

frequencies. Below: Drum mode 0 ,2

L1+c=λ/4 ; l2+c=3λ/4 first position that a loud sound is heard: fundamental freq. l2-l1=λ/2

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61. Lens Law with Real is Positive:

1 1 1

u v f

Convex Lens Concave Lens Convex mirror Concave mirror

f + - - +

v Real: >f, inverted Virtual: <f, upright

Virtual for all, upright

Virtual for all, upright

Real: >f, inverted Virtual: <f, upright

62. Diffraction grating

Hold a diffraction grating against one end. View

through the grating’s vertical filament of the ray-

box lamp placed about 1m from meter rules.

Move the pencil until it is in line with the middle

of red color in first order spectrum.

Measure x and thus sinand . Apply

sin ( )d n constructive ,n=1 and

d=diffraction slit separation.

Make sure the filament of lamp is vertical.

Error: uncertainty of determining location of max. red line.

Diffraction grating outweighs prism as they offer boarder spectrum and sharper images.

63. Interference theory

Cancellation/Reinforcement due to the phase

difference.

Slinky springs to demonstrate interference:

Observable interference pattern for Light:

-phase difference constant (pd <coherent length)

-use a single source (as light is emitted in quanta

of energy and two sources could not offer a stable

phase difference)

-pd comparable to wavelength

Observable interference pattern for Sound:

-phase difference constant

-could be dual-source

*Pd: path difference

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64. Interference applications

->Oil film: As viewing angle varies,

constructive interference’s satisfied

for different colors=> Shows different

Color as head turns or strips of color when viewing from top.

Not valid for thick film: pd >coherent length/

multitude of colors CI at the same time.

Air wedge:

Newton’s ring (Concentric circles):

Interference occurred from reflected lower surface and upper ray. Separation decreases as

gradient of thickness of air film increases; Central black as phase shift between media

Newton’s ring is best observed as normal incident of light since pd is smallest and intensities

between the two reflected ray are most comparable

Air wedge:

At normal incidence, optical pd= 2t n . n is refractive index for material between

Reflected rays (red and blue) superimpose together

Due to phase shift:

: 2 0.5 ,1.5 ,2.5 ,...

: 2 ,2 ,3 ,...

CI nt

DI nt

Fringe separation s : tan2 2 tan

sns n

At oblique incidence,

Pd= 2 cos( )nt r

65. Beats

Interference between two sounds of slightly different frequencies, perceived as periodic

variation of volume whose rate is the difference of frequencies: 1 2beatf f f

Used in tuning folks, speed detectors

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66. Medium dependent Doppler’s Effect

s

o s

c vApparent wavelength

f

c cf f

c v

Re

, o s

lative velocity c v

c vby v f f f

c

Only the red component arises Doppler’s effect as it is approaching.

Radar send microwave of frequency fo to a travelling car and reflected

waves (f1) are superimpose with fo to form beats. 1 0f f f .

Beat

frequency depends on car speed.

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IV) Classical Electrodynamics

67. Electrostatics (See app 3.5)

E-field as EF

q.

0V reference: infinite far. Potential: potential difference from inf.

:

( )

r

ext

work done in taking Q for x

F x Q V

dVE conservative field

dx

U F dr

,2 2

,

( ): ; :

, : ;4 2

S

sphere outside cyclinder

inside parallel plate

Q VMaxwell eqt E Guass Law E dA

Q rBy symmetry E E

r R

QE

A

2

1 1

4 2 4 2

i ji i

all pairs i ji ijNot repeating

QQQ qPotential energy of system V dv

r r

-When any charged particle is accelerated, it emits EM wave

-Earnshaw’s theorem: A charge acted on by electrostatic forces cannot rest in stable

equilibrium in an electric field

68. Potential for common objects: See app 1

69. Electric shield (Faraday’s cage)

Conducting metal’s inside E-field =0

Charges in the conducting material will

redistribute themselves as to cancel fields’

effect in cage interior.

Coating WIFI-antenna blocks its signal

70. Flame Probe

Earth the electroscope. Flame probe as to investigate

potential, whose calibrated electroscope is of the same

potential and rise of golden leaf indicates potential.

Conducting sphere is charged to 1kV by EHT and gains

positive charges. The flame ionize surrendering area as to

provide ions for neutralizing the needle (charges on the

probe). Potential is unaltered. Verify 1

Vr

and its

equiopotential(r constant).

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71. Electrical breakdown

Electrostatic discharge: Spark. Ionized conductive channel in air above limits of voltage

72. Capacitor (Condenser)

2

,2

: ; 0

;

:

(1 )

~ 5

:

1,

2

2

in between outside

emf

t

RCo

o

QFor ONE plane E

A

QParallel planes E E

A

Qd AV C

A d

qdc connections V ir

C

i I e

t RC as fully charged up

Parallel LC circult

Q diir L f

C dt LC

CVEnergy QdV

73. Applications of capacitor

-Smoothing circuit

-Blocking dc. (AC coupling)

-Integrator in analogs (integrating voltage:0

1 1t

c inV V dt forRC RC

)

-Storing energy (Snubber condenser)

-Flash units

Reed switch experiment: during half cycle the capacitor is charged and discharge through

protective resistor. Light beam galvanometer measures average current=Qf

Use low enough freq. and high enough resistor but further reduce it will not increase current

reading

Constant rate charge up: Q=It

Use voltmeter as shown: (high impedance)

74. Energy transformation in electric circuits

E: Force/columb acting on free electrons

Emf: Energy imparted by source/columb

Pd: Energy converted to other forms/columb

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75. Electric signals in circuits, drift velocity

I nAveI

JA

l

RA

v~ 4 510 ~ 10 m/s;

n as charge density;

J as current density

signal speed: c

Drift velocity: electrons are accelerated in E-field by experience collisions and thus have net

small displacement only.

22 1, , .

; . , 7.00

F

electron d

F F

F F

E eE d ne dv v

m m v mv

v is Fermi velocity E is material dp property eV for Cu

76. Joule Heating

Lattice ions gain vibrational energy as they are collided with moving and accelerating

electrons. Internal energy rises as temperature rises.

77. Kichhoff’s laws:

0k

at Junction

I

0k

Loop

V

21

1 2 2 1

1

2

3 3 2 1

1

5 0

0

i dtdiL i R i R e t

dt C

i dti R i R

C

78. Superposition of currents

79. Temperature dependence of resistivity

Metal: temp increases with resistances

Semiconductor: resistance decreases as temp increases. (Commonly)

80. Energy conversion

1kWh=3600000J;

1eV= 191.6 10 J

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81. Magnetic flux links

Magnetic field lines are always closed : 0 ; 0S

B dA B

Magnetic field is NOT a force field.

But WD by magnetic field on charges in always 0

Maxwell correction: changing E-field generates magnetic fieldE

B Jt

Which gives EM wave equations:

22

2

22

2

( ) 0

( ) 0

1

Et

Bt

and phase velocity c

82. General magnetic field laws - Biot Savart law:2

ˆ

4

I dl rB

r

2

2

2

, ,

sin

4

4

3 ,

sinsin ( )

4

m m

i

m i

which can be shown equivalent to F BIL

Consider an object m and current I r apart

m l IF Hm F

r

mH

r

By rd Law F F

mIlF HIl BIl qvB right angles

r

' : enclosed

C

Ampere s Law B dl I

Helmholtz coil: B= 3/24( )5

nI

R

Single wire loop: B=2

2 2 3/22( )

IR

R x

Solenoid: B=2

defNi N N AL

l I l

Straight infinite wire: B=2

I

r

Earth field: Magnetic South = Geographical North

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76. Ammeter structures with eddy currents

Couple by current coil, turns and stops by restoring

hair torque. Current being proportional to deflection.

Radial field for linear scale. NABI k

Eddy current to provide critical damping (eddy

current always oppose motion)

77. CRO and Root mean square(see Appendix 2.1,4.2)

2( )a period

rms

I t dtI

T

78. Mutual force between current carriers

F= 1 2

2

I ILength

r

, BI Hl

,

N VL

diI

dt

79. 7 11 2 10A Nm

Mutual force of current in vaccum

80. Magnetic force

Hold equilibrium by riders and adjust current by rheostat. Add magnet/ Increase current.

Restore equilibrium by adding riders

Shield it, align East-West (magnet), avoid overheating, stray magnetic fields

NI mmf

1 2

24Mutual between magnets

m mF

r ; B H

Ferromagnet: Remains permanent magnets

Paramagnet: Occurred only at external magnetic field

81. Hall’s effect and Hall probe

Force will act at right angles to charges carries when a current-carrying conductor is placed

perpendicularly to uniform B-field. Concentration across one end will be higher and an

Electric field is set up in between. h

BIV

nqt

Hall probe is thin slice of semiconductor with low charge density. Steady current, B-field

perpendicular, accumulation of charges and deflection ceases whenever balance. CRO used.

Search coils for ACs.

Hysteresis effect

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82. Dc motor, generators fine structure

dc motor/gen; ac motor/gen

Back emf induced against increase in current, Start

rotation at sinBANI t resistive torque

Magnetic flux linkage changes over time and back emf

to oppose the rotation. VI=useful power by motor.

sin

cos

sin

BANI t

BA t

dNback emf BAN t

dt

83. Simple transformers

Induced emf: Vrms=4.44 N f Φ

Magnetizing current magI : current used to keep B/H flux in core

Core loss: Work-done in core as resistors/Hysteresis loss

Flux leakage: Self-inducing effects, air linked

Coil loss: Resistance in coils

Real Transformer equivalent:

:

: ,

: ,

C

P S

P S

Core loss R

Flux leakage X X

Coil resistance R R

As useful flux in core is kept constant (little variation about 2%)

MMF conserved and primary current rises with secondary current

Back emf varies about 0.05% to 1% only, but could contribute a fluctuation of current

Laminated cores to prohibit formation of eddy current, toroidal core to reduce reluctance/air

gaps (Not easy to distort at ends) , coolants:

Neglecting Xs, Rs, Xp, Rp

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84. Induction theory (Oppose changes)

Faraday’s law: Induced emf generated is proportional to rate of change of magnetic flux

linkage (dN

dt

)

Lenz’s law: Induced current flows as to oppose motion of change causing it (negative)

Lorentz force: ( )

sin

F q E v B

F BI L BI for magnitude

which could be shown equivalent to Faraday’s law

85. Induction applications

Changing ac with magnetic flux, induces emf and float a conductor when gravity = magnetic

force.

Force on coil to force vibrating the paper cone. Set up sound waves in surrounding air

Sound wave: compressed surrounding air collides with other molecules and passes its

momentum. When diaphragm is pulled back, extra spaces in air molecules formed. Expansion

is created and molecules fill in. Repeated process of pressure wave is formed, which is the

combination of rarefaction and compression

Electromagnet provides current through sea water. Sea water experience backward force

and 3rd law yields forward force on boat.

2

: .

;

( )

1

2 2

. , .

Cyclotron accelerate electrons through a gapmv

force qvB qvBr

mtime t to complete semi circular orbit indp v

qBqB

Alternating voltaget m

mvr as v increases r increased

qB

Mass spectrometer: Different ions give different r: '

'

mEr

qBB

86. Inductor

When dc is closed, current tried increase and being opposed by Lenz’s law and a back emf is

developed. Current could only increase steady from zero to E/R (exponentially). As current

breaks, the drop in current is large and rate of change of flux is large. Large induced emf of

magnitude ~100V which causes sparking due to electrostatic discharge (occurred at 4kv/cm).

2

1:

1

:

1

2

L

i

i

dIBack emf L

dt

X L

Parallel

L

Series L

Energy stored LI

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87. Eddy current

Current induced in conductor as to induce magnetic fields that oppose the one created it

-Induction heating

-Break

-levitation

Minimized using laminations: charges accumulates on the boundaries, producing E-field to

oppose further accumulation

88. Ac phasor

2 2

2 2

:

arctan( )

( )

:

1 1

arctan( )1

1 1 1( ) ( )

L c

total L C

L c

total

L C

series

X X

R

V X X R

parallel

X X

R

VR X X

89. AC resonance

: L Cseries X X

Largest current flows if XL=XC

Parallel: LC in parallel, exchange between

electric and magnetic energy

damped oscillations (used in radios)

Emfs of various frequencies are induced

from the aerial, current flows in aerial. By

mutual induction, current of same

frequencies will be induced in LCR circuit.

By adjusting C, resonant frequency of LCR is changed. Large pd of that

frequency will develop across C.

(Analogy to forced oscillation: If the source is

superimposed with multitude of freq. ,

the one closed to natural frequency dominates)

Practical I-t curve:

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90. Ripple discharge

Storing capacitor acts to release charges and energy when input voltage decrease. Time

constant is large compared and the voltage cannot follow and a more or less dc is developed.

91. Electron deflects in E-field and B-field

Thermionic emission: heating a metal to supply enough energy for electrons escaping

attractive force of metallic bond. ,E as work function

2

2

: ; :

, :

e

e B

e

B E

m vEeE field a B field F evB

m r

Eif e is subjected to E B field for no deflection F F v

B

eE lD L

m v

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V) Modern Physics

92. Rutherford scattering

The atom is highly positive with charge Ze and electrons surround

Spaces around the atom. Direct closest approach: 2

02

2

14

2initial

Zer

m v

The imaginary cone up-slide model:4

ZQq GMmmgh

r r

Alpha particles lost their KE as being repelled by E-force, acquiring initial KE when returned

93. Line spectrum

Absorption/Sun spectrum: Cooler gas around the sun, lines with exact wavelengths in

emission spectrum. The atoms absorbs light they can emit, re-radiate photon in all directions,

and reduce in original direction. Called Fraunhofer lines

All lines are distinct and compare lines with those of hydrogen, helium in laboratories

Emission spectrum: luminous gas at low pressure. Electrons are excited from low to high level

and drop from higher level.

Compared: Continuous spectrum by hot solids, high pressure tube, filament lamp

Atoms are closed and interact with each other forming all , f As known as “atomic scattering” or “atomic resonance”

94. Photoelectric

stoppingE hf V e

One to one: immediate emission

E=hf: no dependence on intensity .With E<hf no absorption

Work function: minimum energy supplied to enable an electron escape from surface

Max KE depends on hf and work function solely. No faster electron for more intense wave

Sound track: varying sound tracks varies light intensity and thus photoelectric current

95. Thermionic emission and cathode ray tube

Thermionic emission: External work against the metal’s attractive force and free electrons by

exceeding the work function. Accelerating electrons by voltage: 21

2eV mv

Cathode ray

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96. X-ray Theory: ionizing, high penetrating power

Bombardment of energetic electrons. Gradual energy lost by collisions. EM wave emitted by

classical theory.

Max. energy of X ray= All energy lost in one atom : c

h eV

Line characteristic: High energy e knocking out inner shell e.

Vacancies are then being filled by outer electrons: c

E h

97. Bohr’s Model:

Angular momentum:

2

nhmvr

2

2

2 2

2 2 2

2 2 2 2 2

2 2 2

4

2 2 2 2

2

' :4

4 4

1,

2 4 8

1 1, 13.6

8

e

nhv

mr

mv ZQqBohr s assumption

r r

m n h ZQq

r m r r

n h ZQq Z Q q mr energy

mZQq r n h

e mFor Hydrogen E eV

h n n

98. Frank-Hertz expt

Discrete E-Level:

V<1: electron has not enough KE to overcome retarding

voltage

V<5: more and more electron gains enough energy to reach

anode

V=5: some electrons gain enough energy (>EP)and undergo

inelastic collision with Hg. Remaining energy could not

overcome retarding

V>5: remaining energy>retarding

Energy level assumption: Electrons with energy greater than the gap 1 2E E E could

be absorbed by probability. Only Photons with discrete energy hf E could be absorbed.

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99. Electron diffraction

Electrons are accelerated by:

2

2

2

2

1

2 21

2

sin

sin tan , min .

2 22

2

o o

o

o

mveV

eVv

m

h h

eV m eV m eV

m c

relate a m for dark fringes

yy is spread of first

D

D hDBright region width y

a a m eV

Can be used as electron microscope->

100. Energy equivalent in Special relativity

2 2 2( ) ( )oE m c pc

Rest mass: Closed and relatively rest system’s Newtonian mass

Relativistic mass: Total energy/c2

Relativistic momentum is conserved in 4-dimensional fields

2

2

21

o

E pc

c v

m cE

v

c

Photon: Put mo=0, E=pc=hf

2

21

om vp

v

c

101. Wave-Particle duality

:h

de Brogile wavelengthp

All matter exhibits wave-properties

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102. Statistical and numerical treatment of radioactive spontaneous decay

Cannot tell which and when the nuclide decays.

Statistical reference: large N (see app 2.1)

1/2

1/3

( . )

ln 2

ln3

kt

o

A A

define probability of decay in unit time k

aka fraction of total number which decay in unit time

dNkN

dt

N N e

tk

tk

MassActivity kN knN k N

molarmass

103. Radioactive hazard:

Direct consumption into/exposure to human bodies, cancers

Destroy body cells, Mutations

104. Unit Sv, dose equivalent: a function of energy

, ,T R R T RH w D

105. Solar fusion:

4 2 2 2 4p e He 106. Mass defects and binding energy curve

2E mc

Binding energy is energy required to split the nucleus completely.

Fe-56 carries maximum BE/nucleon and is most stable.

Eg, 235 144 89 3 177U n Ba Kr n MeV

When nucleus of large mass is split into two daughter nuclei, energy is released

107. Nuclear Plants

Fission is often accompanied by Chain reaction.

Slow neutrons are favorable for reactions. U-235 captures slow neutrons

Fuel rod: enriched U-235

Moderator: water/graphite as to slow down neutron

Control rod: boron-coated steel as to absorb neutrons and control fission rate

Coolant: Pressurized water under critical temperature, boil water in secondary circuit

108. Solar cell as photoelectric cell

Array of solar cells converts EM-wave to currents

Photovoltaic effect: Generated electrons are transferred in material and set up a voltage

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109. Brief Radioactivity

C-14 dating: 14 14N n C p

By cosmic ray imparted (neutrons)

C-14 remains throughout constant (Carbon intake=output)

Check activity -> time after death, assuming living samples’ activities equals the testing

sample’s initial activity

Ionization power:

Radioactive particles attract nearby particles and ionize them by giving up energy.

Cloud chamber reveals their path

Ionization chamber: ion pairs produced are accelerated towards anode and cathode.

GM tube: Electrons emitted and accelerated and undergo avalanche

Neutron ionization: though neutron are neutral, But they can:

-Be absorbed and emit gamma or electron

-Recoil proton

Common reactions: 2 2: 238 234

: 137 137

: 11 11

: 26 26

e

e

e

U Th

Cs Ba

C B

e capture Al e Mg

2

:

[ ] : ( 2 )

[ ] : ' (1 cos )

[ ] :

e

e

stopping potential

Annihilation e e

High freq photon strike Pair production e e hf m c

hMid freq photon strike Compton scattering

m c

cLow freq photon strike Photoelectric h eV

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VI) Modern technologies

110. Astronomy: Kepler’s 3 laws

110.1 Sun as in focus

110.2 Equal areas swept(conservation of AM)

110.3 3rd Law: 2

2

3 2

2

2 32

( )2 1

4

mv GMm

r r

GMm T GMr

m

aT

GM

111. Telescopes

Objectives are used to collect large

amount of light and form a

intermediate image at its focus.

Eyepiece acts as to magnify image

and produce a virtual image.

Eyes ring: position to collect most

light

112. Light Microscopes

11 2

1

', ( 1)( 1)

o e

hh h v DM m m

D h h f f

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113. Apparent weightlessness

Normal reaction=o

When all of the weight is used for centripetal acceleration, reaction ceases

114. Relativistic Doppler’s Effect and applications

Light requires no medium to travel in and we are considering their relative speeds only.

2

2

2

2

2

2

2 2

2 2

2

2

1

11

1

1

(1 )

1

o

o s

Time measured in source frame that two wavefronts reach observerc v

Time measured in observerTime measured in source

v

c

v

c c v f

c c v c vf f

c c vv

c

c v c v c v c v c vf f f f

c v c v c v

v

c

2

2

1 1 1

11

v v

vc cf f fcv

c

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115. Red shift

Red shift is the feature that distant stars/galaxies emitting light which is similar to that of

stars in our galaxy but with wavelength of spectrum shifted to the Red end (higher

wavelength). By relativistic Doppler’s effect, they are moving away from us and it possibly

suggests that universe is expanding.

116. Microscope resolving power

sin 1.220D

Images are up to : Aberrations (refraction); Diffractions

Airy disk: as first minimum, diffraction pattern

117. Lasers and fluorescent

Laser:

Stimulated emission from population inversion. A meta-stable

state is required. When a photon incident with correct energy

gap, inducing electrons of higher energy fall and form

coherent (same frequency) , intense waves(constructive

interference, same direction).

Mirrors could be employed for further interference.

Fluorescent:

Fluorescent material absorbs energy and decay in steps. The

excited molecule firstly give up energy by collision with other

molecules. When it is returned to ground states, a photon of

lower energy and frequency is emitted.

Radiation energy<absorbed energy

Mercury lamps: mercury gives UV and being absorbed by

fluorescent materials on the coating, visible light turns out.

118. X-ray: Application

X-ray intensity drops as they interact with matter. Ax

oI I e A = absorption coefficient, =mass absorption coefficient

Attenuation length/ mean free path: depth into a material that intensity decreased to 37%

1 1x

A

119. CT image

As a map of attenuation coefficients of body parts. Beer’s Law:

exp{ }o i i

parts i

I I A x

Grey level: attenuation

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APPENDIX

Appendix 1: Voltage for common objects

By HyperPhysics