Physics 2D Lecture Slides Oct 8 Vivek Sharma UCSD Physics - - PDF document

Physics 2D Lecture Slides Oct 8

Vivek Sharma UCSD Physics

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

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 = 1.6x10-19 Joules 1 MeV = 1.6x10-13 Joules 1 GeV = 1.6x10-10 Joules

Parallel Plates

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

Accelerating Electrons Thru RF Cavities

Circular Particle Accelerator: LEP @ CERN, Geneve

Magnets Keep Circular Orbit of Particles

Inside A Circular Particle Accelerator @ CERN

Test of Relativistic Momentum In Circular Accelerator

2

1 ( / ) mu qB mu p mu qB u r mu c r p 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 ) Work 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

K =

• ne is change in

E= Kinetic energy K

• r

Total Energy mc mc K mc mc γ γ − = +

But Professor… 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

− −

⎛ ⎞ ⎜ ⎟ ⎜ ⎟ = − ⇒ + = ⎜ ⎟ ⎡ ⎤ ⎡ ⎤ ⎜ ⎟ − − ⎢ ⎥ ⎢ ⎥ ⎜ ⎟ ⎣ ⎦ ⎣ ⎦ ⎝ ⎠ ⎡ ⎤ ⎡ ⎤ ⇒ − = + ⇒ ⎢ ⎥ ⎣ ⎦ − ⎦ + = = ⎣ ⇒

Relativistic Energy

A Digression: How to Handle Large/Small Numbers

• Example: consider very energetic particle with very large Energy E
• Lets Say γ = 3x1011, Now calculate u from
• Try this on your el-cheapo calculator, you will get u/c =1, u=c due

to limited precision.

• In fact u ≅c but not exactly!, try to get this analytically

2 2 2 2

1 E mc K K mc mc mc γ + = = = +

1/ 2 2

1 1 u c γ ⎡ ⎤ = − ⎢ ⎥ ⎣ ⎦

2 24 2

1 1 (1 )(1 ) 1 u Since = 1, 1 2 c 1 2 1 1 1 5 10 , 2 0.999 999 999 999 999 999 999 995c !! Such particles are routinely produced in violent cosmic collisions u c u γ β β β β β γ β β β γ

−

= = − + − ≅ + = ≈ − ⇒ − = = × = ⇒ =

In Quizzes, you are Expected to perform Such simple approximations

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

1 ( ) = ( ) ( ) ........important relation F (

• r

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

2 2 2 2 4

E E= pc or p = (light has momentu particles with zero rest mass like pho m!) c Relativistic Invariance ton (EM waves) : In all Ref Frames Rest : E p c m c − = Mass 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 , just like spacetime
• 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
• This exchange rate never changes !

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