Physics 2D Lecture Slides Jan 15 Vivek Sharma UCSD Physics - - PowerPoint PPT Presentation

Physics 2D Lecture Slides Jan 15

Vivek Sharma UCSD Physics

Relativistic Momentum and Revised Newton’s Laws

Need to generalize the laws of Mechanics & Newton to confirm to Lorentz Transform and the Special theory of relativity: Example : p

mu =

• 1

2 Before v1’=0 v2’ 2 1 After V’ S’ S 1 2 Before v v 2 1 After V=0 P = mv –mv = 0 P = 0

' ' ' 1 2 ' ' 1 2 1 2 2 1 1 2 2 2 2 2 2 '

' ' before after

2 0, , ' 2 1 1 1 1 1 , 2 2 '

p p

after before

mv p mv m v v v v v V v v v V v v v v v V v v c v v c c c c p mV mv − − − − = = = = = − = + = − − − − + = ≠ + = = −

Watching an Inelastic Collision between two putty balls

Definition (without proof) of Relativistic Momentum

2

1 ( / ) mu p mu u c γ = = −

• With the new definition relativistic

momentum is conserved in all frames

• f references : Do the exercise

New Concepts

Rest mass = mass of object measured In a frame of ref. where object is at rest

2

is velocity of the object NOT of a referen 1 1 ( / ) ! ce frame u u c γ = −

Nature of Relativistic Momentum

2

1 ( / ) mu p mu u c γ = = −

• With the new definition of

Relativistic momentum Momentum is conserved in all frames of references

m

u Good old Newton

Relativistic Force & Acceleration Relativistic Force And Acceleration

2

1 ( / ) mu p mu u c γ = = −

• (

) ( ) ( )

3/ 2 2 2 2 2 2 2 3/ 2 2 3/ 2 2 2 2

1 ( / ) : Relativistic For 1 2 ce ( )( ) 1 ( / ) Since A 2 1 ( / ) 1 ( ccel / ) 1 ( e ) a / r d du d use dt dt du m mu u du F c dt u c u c mc mu mu du F du dt dp d mu F dt dt u c m F u c dt c u c =   − −   = + ×   − −      − +   =   −     = =   −       =   −   

• 3/ 2

2

tion a = Note: As / 1, a 0 !!!! Its harder to accelerate when , F a = you get closer to speed of light 1 ( / ) m d c u t u c u d ⇒ →   −  → 

• Reason why you cant

quite get up to the speed

• f light no matter how

hard you try!

A Linear Particle Accelerator

V

+

• F

E

E= V/d F=eE

3/ 2 3/ 2 2 2 2 2

Charged particle q moves in straight line in a uniform electric field E with speed u accelarates under f F=qE a 1 =

• rce

larger 1 the potential difference V a du F u qE u dt m c m c     = = − −        

• cross

plates, larger the force on particle d

q

Under force, work is done

• n the particle, it gains

Kinetic energy New Unit of Energy

1 eV = 1.6x 10-19 Joules

PEP PEP-

• II accelerator schematic and tunnel view

II accelerator schematic and tunnel view

A Linear Particle Accelerator

3/ 2 2

eE a= 1 ( / ) m u c   −  

Magnetic Confinement & Circular Particle Accelerator

V

• 2

2

Classically v F m r v qvB m r = =

B

F

• B
• r

2 2

( ) (Centripetal accelaration) dp d mu du F m quB dt dt dt du u dt r u m quB mu qBr p qB r r γ γ γ γ = = = = = ⇒ = = ⇒ =

Charged Form of Matter & Anti-Matter in a B Field

Circular Particle Accelerator: LEP @ CERN, Geneve

Magnets Keep Circular Orbit of Particles

Inside A Circular Particle Accelerator @ CERN

Accelerating Electrons Thru RF Cavities

Test of Relativistic Momentum In Circular Accelerator

2

1 ( / ) mu p mu u c qBr m mu qB u r γ γ γ = = = = −

Relativistic Work Done & Change in Energy

x1 , u=0 X2 , u=u

2 2 1 1

3 / 2 2 2 2 2 3 / 2 2 2 2 2 3 / 2 1/ 2 2 2 2 2 2 2

substitute i . . , , 1 1 n W (change in var x u 1 1 1 W rk d )

• x

x x x u u

dp W F dx dx dt du m mu dp dt p dt u u c c du m dt W u c mudu m udt mc c W c c c m mc u u γ = = = ∴ =   − −     ∴ =   −     = = − =     − −     →     −

∫ ∫ ∫ ∫

• 2

2 2 2

• ne is change in Kinetic energy K

K =

• r

Total Ener E= gy mc mc K mc mc γ γ − = +

Why Can’s Anything go faster than light ?

( )

2 2 2 2 2 2 1/ 2 1/ 2 2 2 2 2 2 2 2 4 2 2 2 2 2 2

(Parabolic in Vs ) 1 2 Non-relativistic case: K = 1 ( 1) 1 1 2 1 u K K u c mc mc K mc K mc u u c c u m c K mc c mc mc K mu u m

− −

      = − ⇒ + =         − −                 ⇒ − = + ⇒     −  + = =  ⇒

Lets accelerate a particle from rest, particle gains velocity & kinetic energy

Relativistic Kinetic Energy

When Electron Goes Fast it Gets “Fat”

2

E mc γ =

v As 1, c Apparent Mass approaches γ → → ∞ ∞

Relativistic Kinetic Energy & Newtonian Physics

2 1 2 2 2 2 2 2 2 2 2 2 2

Relativistic KE = 1 When , 1- 1 ...smaller terms 2 1 so [1 ] (classical form recovered) 1 2 2 u u u c c c u K mc mc mc c mc mu γ

−

−   << ≅ − +     ≅ − − =

2 2 2

For a particle Total Energy of a Pa at rest, u = 0 Total Energy E= r m ticle c E mc KE mc γ = = + ⇒

Relationship between P and E

2 2 2 4 2 2 2 2 2 2 2 2 2 2 4 2 2 2 2 2 2 2 2 2 2 2 4 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 4

( ) = ( ) ( ) ........important relati (

• n

For ) 1 p u E mc p mu E p c mc E m c p c m u c E p c m c m u c m c u m c m c c u c m c u u c c γ γ γ γ γ γ γ = = ⇒ = ⇒ = ⇒ − = − = − + − − = − = − =

2 2 2 2 4

E E= pc or p = (light has momentum!) c Re articles with zero rest mass like photon (EM lativistic Invariance : waves) : In all Ref Frames Rest M E p c m c − = ass is a "finger print" of the particle

Mass Can “Morph” into Energy & Vice Verca

• Unlike in Newtonian mechanics
• In relativistic physics : Mass and Energy are the same

thing

• New word/concept : Mass-Energy
• It is the mass-energy that is always conserved in every

reaction : Before & After a reaction has happened

• Like squeezing a balloon :

– If you squeeze mass, it becomes (kinetic) energy & vice verca !

• CONVERSION FACTOR = C2

Mass is Energy, Energy is Mass : Mass-Energy Conservation

be 2 f 2

• re

after 2 2 2 2 2 2 2 2 2 2

2 2 1 Kinetic energy has been transformed E E into mass increase 2 2

• 2

1 1 1 mc mc Mc K m u u c c M M m M m m u c c c c u = + = ⇒ − − ∆ = = = = > − −

2 2 2

mc c       −      

Examine Kinetic energy Before and After Inelastic Collision: Conserved? S 1 2 Before v v 2 1 After V=0 K = mu2 K=0 Mass-Energy Conservation: sum of mass-energy of a system of particles before interaction must equal sum of mass-energy after interaction

Kinetic energy is not lost, its transformed into more mass in final state

Conservation of Mass-Energy: Nuclear Fission

2 2 2 2 3 1 2 1 2 3 2 2 2 1 2 3 2 2 2

1 1 1 M c M c M c Mc u u u c c M M c M M = + − > + + − + ⇒ −

M

M1 M2

M3

+ +

Nuclear Fission < 1 < 1 < 1

Loss of mass shows up as kinetic energy of final state particles Disintegration energy per fission Q=(M – (M1+M2+M3))c2 =∆Mc2

14 236 92 1

• 28

3 55 90 92

+ +3 n m=0.177537u=2.947 U Cs R 1 10 165.4 MeV b kg ∆ × = →

What makes it explosive is 1 mole U = 6.023 x 1023 Nuclei !!

Relativistic Kinematics of Subatomic Particles

Reconstructing Decay of a π Meson