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Physics notes for HKAL students. Final review on major topics. Out-Of-Syllabus stuff is indicated and sifted.
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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 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
Lsw 2011/04/16 rev Contact: [email protected]
2 - AL PHY
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
Lsw 2011/04/16 rev Contact: [email protected]
3 - AL PHY
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
Lsw 2011/04/16 rev Contact: [email protected]
4 - AL PHY
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
Lsw 2011/04/16 rev Contact: [email protected]
5 - AL PHY
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.
Lsw 2011/04/16 rev Contact: [email protected]
6 - AL PHY
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
Lsw 2011/04/16 rev Contact: [email protected]
7 - AL PHY
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
Lsw 2011/04/16 rev Contact: [email protected]
8 - AL PHY
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
Lsw 2011/04/16 rev Contact: [email protected]
9 - AL PHY
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 .
Lsw 2011/04/16 rev Contact: [email protected]
10 - AL PHY
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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11 - AL PHY
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
Lsw 2011/04/16 rev Contact: [email protected]
12 - AL PHY
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
Lsw 2011/04/16 rev Contact: [email protected]
13 - AL PHY
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
Lsw 2011/04/16 rev Contact: [email protected]
14 - AL PHY
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
Lsw 2011/04/16 rev Contact: [email protected]
15 - AL PHY
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
Lsw 2011/04/16 rev Contact: [email protected]
16 - AL PHY
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
Lsw 2011/04/16 rev Contact: [email protected]
17 - AL PHY
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.
Lsw 2011/04/16 rev Contact: [email protected]
18 - AL PHY
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
Lsw 2011/04/16 rev Contact: [email protected]
19 - AL PHY
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
Lsw 2011/04/16 rev Contact: [email protected]
20 - AL PHY
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
Lsw 2011/04/16 rev Contact: [email protected]
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APPENDIX
Appendix 1: Voltage for common objects
By HyperPhysics