Physics 2D Lecture Slides Lecture 7: Jan 14th 2004 Vivek Sharma - - PDF document

Quiz 1 is This Friday Quiz 1 will cover Sections 1.1-1.6 (inclusive) Remaining material will be carried over to Quiz 2 Bring Blue Book, check calculator battery Write all answers in indelible ink else no grade!

Write answers on consecutive pages: Don’t leave blank pages

All nighters don’t work for these quizzes Get plenty of Sleep….come with a fresh mind!

Physics 2D Lecture Slides Lecture 7: Jan 14th 2004

Vivek Sharma UCSD Physics

Relativistic Force & Acceleration Relativistic Force And Acceleration

2

1 ( / ) mu p mu u c γ = = −

• (

)

3/ 2 2 2 3/2 2

1 ( / ) : Relativistic Force 1 ( / ) Since Acceleration [rate of change of velocity] a = , F a = Note: As / 1, a 0 !!!! Its hard 1 ( / ) e m r dp d mu F dt dt u c m F u c u c du dt du dt u c ⎡ ⎤ − ⎣ ⎦ ⎛ ⎞ ⎜ ⎟ = = ⎜ ⎟ − ⎝ ⎠ ⎡ ⎤ ⎢ ⎥ = ⎢ ⎥ − ⎣ ⎦ ⇒ → →

• to accelerate when you get

closer to speed of light

Reason why you cant quite get up to the speed

• f light no matter how

hard you try! Linear Particle Accelerator : Parallel Plates With Potential Difference

V

+

• F=-eE

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

e-

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

Linear Particle Accelerator : 50 GigaVolts Accelating Potential 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

Antimatter form of electron = Positron (e+) Same Mass but opposite Charge Positron curls the other way from electron in a B Field

Accelerating Electrons Thru RF Cavities

A Circular Accelerator : Using B Field to Confine the electron and RF cavity to power it

Circular Particle Accelerator: LEP @ CERN, Geneve

circular track for accelerating electron Geneva Airport Accelerated electron through an effective voltage of 100 Billion Volts ! To be upgraded to 7 trillion Volts by 2007 Swiss Border French Border

In Tunnel 150m underground, 27km ring of Magnets Keep electron in Circular Orbit

Inside A Circular Particle Accelerator Tunnel : Monorail !

Test of Relativistic Momentum In Circular Accelerator

2

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

• Proof That

p ≠ mu

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 in W (chang . . , , 1 1 1 1 e in var x u) W o 1 rk

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 mc W mc u u udt c m c mc c γ = = = ∴ = ⎡ ⎤ − − ⎢ ⎥ ⎣ ⎦ ∴ = ⎡ ⎤ − ⎢ ⎥ ⎣ ⎦ = = − = ⎡ ⎤ ⎡ ⎤ − − ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ − →

∫ ∫ ∫ ∫

• 2

2 2 2

K = done is change in Kinetic energy K

• r

Total En E= ergy 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 2 2 / 2 1/2 2 2 2 2 2 2 2 4 2 2 2

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

− −

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

Relativistic Kinetic Energy Vs Velocity

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 → → ∞ ∞

New Concept Rest Mass = particle mass when its at rest

2 2

E = Total Energy mc K mc γ = +

Relativistic Kinetic Energy & Newtonian Physics

1 2 2 2 2 2 n 2 2 2

nx n(n-1) (1+x) = (1+ x +small Relativistic KE K = 1 When , 1- 1 ...smaller ter Remember Binomial Theorem for x << er terms) 1! 2! 1; 2 ms s mc u u c c mc u c γ

−

− ⎡ ⎤ ∴ < ⎡ ⎤ ⎢ ⎥ ⎢ < ≅ + + ⎢ ⎥ ⎥ ⎣ + ⎣ ⎦ ⎦

2 2 2 2 2

1

• [1

] (classical form reco 1 2 vered) 2 m u K mc c c u m ≅ + − =

2 2 2

For a partic Total Energy of a le at rest, u = 0 Part Total Energy E= icle m c E mc KE mc γ = = + ⇒

E=mc2 : Sunshine Won’t Be Forever

Q: Solar Energy reaches earth at rate of 1.4kW per square meter of surface perpendicular to the direction of the sun. by how much does the mass of sun decrease per second owing to energy loss? The mean radius of the Earth’s orbit is 1.5 x 1011m.

• Surface area of a sphere of radius r is A = 4πr2
• Total Power radiated by Sun = power received by a sphere whose radius is equal to earth’s orbit radius

r

2 3 2 11 2 26 26 9 2 8 2 30 2 6

(1.4 10 / )(4 )(1.5 10 ) 4.0 10 So Sun loses E=4.0 10 J of rest energy per secon 4 E 4.0 10 J mass decreases by m= 4.4 10 !! ( d Its If the Sun's Mass = 2. 3 0 10 So .0 10 ) how P P P A r A A k W m P W g kg c π π = = × = = × × = × × = × × × long with the Sun last ?

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 : MassEnergy , 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 : Squeeze here, it grows

elsewhere

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

• CONVERSION FACTOR = C2
• This exchange rate never changes !