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 � � ⎛ ⎞ � � dp d mu = = ⎜ ⎟ F � � ⎜ ⎟ mu − dt dt = = γ 2 ⎝ ⎠ 1 ( / ) u c p mu − 2 1 ( / ) u c ⎡ ⎤ m du ⎢ ⎥ = ⎢ F : Relativistic Force ( ) ⎥ Relativistic 3/2 − dt 2 1 ( / ) u c ⎣ ⎦ � Force � du Since Acceleration a = , [rate of change of velocity] And dt Acceleration � � F 3/ 2 ⎡ ⎤ ⇒ − 2 a = 1 ( / ) u c ⎣ ⎦ m � Reason why you cant → → Note: As / u c 1, a 0 !!!! quite get up to the speed Its hard e r to accelerate when you get of light no matter how closer to speed of light hard you try! Linear Particle Accelerator : Parallel Plates With Potential Difference F=-eE - Parallel Plates + e - E= V/d E F= -eE d V Charged particle q moves in straight line � � Under force, work is done in a uniform electric field E with speed u on the particle, it gains � � Kinetic energy accelarates under f orce F=qE � � � New Unit of Energy 3/2 3/2 ⎛ ⎞ ⎛ ⎞ � 2 2 du F u qE u = = − − ⎜ ⎟ ⎜ ⎟ a 1 = 1 1 eV = 1.6x10 -19 Joules 2 2 ⎝ ⎠ ⎝ ⎠ dt m c m c 1 MeV = 1.6x10 -13 Joules larger the potential difference V a cross 1 GeV = 1.6x10-10 Joules plates, larger the force on particle

Linear Particle Accelerator : 50 GigaVolts Accelating Potential � � eE 3/ 2 ⎡ ⎤ − 2 a = 1 ( / ) u c ⎣ ⎦ m PEP PEP- -II accelerator schematic and tunnel view II accelerator schematic and tunnel view Magnetic Confinement & Circular Particle Accelerator � � Classically V B 2 v = F m r 2 v � = qvB m r F B r γ dp d ( mu ) du = = = γ = F m quB dt dt dt 2 du u = (Centripetal accelaration) dt r 2 u γ = ⇒ γ = ⇒ = m quB mu qBr p qB r r

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

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 � � � mu = = γ p mu − 2 1 ( u / ) c γ = = Proof That mu qB r p p ≠ mu qB r γ = mu Relativistic Work Done & Change in Energy � � x x � � 2 2 dp ∫ ∫ = = W F dx . . dx dt x x 1 1 du � m mu dp dt = ∴ = X 2 , u=u p , substitute in W , 3 / 2 ⎡ ⎤ dt 2 2 u u − − 1 ⎢ 1 ⎥ 2 2 ⎣ ⎦ c c du m dt udt u ∫ ∴ = → (chang e in var x u) W 3 / 2 ⎡ ⎤ x 1 , u=0 2 u 0 − ⎢ 1 ⎥ 2 ⎣ ⎦ c u 2 mudu mc ∫ = = − = γ − 2 2 2 W mc mc m c 3 / 2 1 / 2 ⎡ ⎤ ⎡ ⎤ 2 2 u u 0 − − ⎢ 1 ⎥ ⎢ 1 ⎥ 2 2 ⎣ ⎦ ⎣ ⎦ c c W o rk done is change in Kinetic energy K γ − 2 2 K = mc mc or γ = + 2 2 Total En ergy E= mc K mc

But Professor… Why Can’s ANYTHING go faster than light ? 2 ⎛ ⎞ ⎜ ⎟ 2 ( ) ⎜ 2 ⎟ mc mc 2 = − ⇒ + = ⎜ 2 2 K mc K mc ⎟ 1 / 2 1/2 ⎡ ⎤ ⎡ ⎤ 2 2 u u ⎜ ⎟ − − ⎢ 1 ⎥ ⎢ 1 ⎥ ⎜ ⎟ 2 2 ⎣ ⎦ ⎣ ⎦ ⎝ ⎠ c c ⎡ ⎤ 2 u − 2 ⎡ ⎤ ⇒ − = + 2 4 2 ⎢ 1 ⎥ m c K m c ⎣ ⎦ 2 ⎣ ⎦ c K K ⇒ = − + − 2 u c 1 ( 1) (Parabolic in Vs u ) 2 2 m c mc → ∞ → As K , u c 1 2 K ⇒ = 2 Non-relativistic case: K = 2 m u u m Relativistic Kinetic Energy Vs Velocity

A Digression: How to Handle Large/Small Numbers • Example: consider very energetic particle with very large Energy E + 2 E mc K K γ = = = + 1 2 2 2 mc mc mc 1/ 2 ⎡ ⎤ 1 u = − • Lets Say γ = 3x10 11 , Now calculate u from � ⎢ 1 ⎥ γ ⎣ 2 ⎦ c • 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 1 1 γ = = In Quizzes, you are − β + β − β 2 (1 )(1 ) 1 Expected to perform u β ≅ + β = Such simple Since = 1, 1 2 c approximations 1 γ ≈ − β 2 1 1 − ⇒ − β = = × = β 24 1 5 10 , u c γ 2 2 ⇒ = u 0.999 999 999 999 999 999 999 995c !! Such particles are routinely produced in violent cosmic collisions When Electron Goes Fast it Gets “Fat” γ = + 2 2 Total Energy E = mc K mc = γ 2 E mc v → γ → ∞ As 1, c ∞ Apparent Mass approaches New Concept Rest Mass = particle mass when its at rest

Relativistic Kinetic Energy & Newtonian Physics γ − 2 2 Relativistic KE K = mc mc ⎡ ⎤ Remember Binomial Theorem ⎢ ⎥ nx n(n-1) ⎢ ⎥ + n 2 for x << 1; (1+x) = (1+ x +small er terms) ⎣ ⎦ 1! 2! 1 − ⎡ ⎤ 2 2 u 2 1 u ∴ < < ≅ + + ⎢ ⎥ When u c , 1- 1 ...smaller ter ms 2 2 ⎣ ⎦ c 2 c 2 1 u 1 ≅ + − = 2 2 2 s o K mc [1 ] m c m u (classical form reco vered) 2 2 c 2 = γ = + 2 2 Total Energy of a Part icle E mc KE mc ⇒ 2 For a partic le at rest, u = 0 Total Energy E= c m E=mc 2 : 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 10 11 m. r Surface area of a sphere of radius r is A = 4 π r 2 • • Total Power radiated by Sun = power received by a sphere whose radius is equal to earth’s orbit radius P P = = π = × π × 2 3 2 11 2 P A 4 r (1.4 10 W m / )(4 )(1.5 10 ) A A = × 26 P 4.0 10 W × 2 6 So Sun loses E=4.0 10 J of rest energy per secon d × 26 E 4.0 10 J = = × 9 Its mass decreases by m= 4.4 10 k g !! × 2 8 2 c ( 3 .0 10 ) × 30 If the Sun's Mass = 2. 0 10 kg So how long with the Sun last ?

= γ ⇒ = γ 2 2 2 2 4 E mc E m c Relationship between P and E = γ ⇒ = γ 2 2 2 2 2 2 p mu p c m u c ⇒ − = γ − γ = γ − 2 2 2 2 2 4 2 2 2 2 2 2 2 2 2 E p c m c m u c m c ( c u ) 2 2 2 4 m c m c − = − = 2 2 2 2 2 4 = ( c u ) ( c u ) m c − 2 2 2 u c u − 1 2 c = + 2 2 2 2 2 E p c ( m c ) ........important relation F or particles with zero rest mass like pho ton (EM waves) E E= pc or p = (light has momentu m!) c − = 2 2 2 2 4 Relativistic Invariance : E p c m c : In all Ref Frames Rest 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 = C 2 • This exchange rate never changes !