Chapter2_all - PHY3063 R. D. Field Relativistic Energy and...

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PHY3063 R. D. Field Department of Physics Chapter2_1.doc University of Florida Relativistic Energy and Momentum ( Summary ) Relativistic Energy: The total relativistic energy is the sum of the kinetic energy ( energy of motion ) plus the rest mass energy ( RME = m 0 c 2 ). 2 0 c m KE RME KE E + = + = Also, the relativistic energy is equal to the relativistic mass, m, times c squared. 2 0 2 c m mc E γ = = with 0 m m = where 2 1 / 1 / β = = c v . Relativistic Kinetic Energy: 2 0 1 2 0 2 0 2 0 2 2 1 ) 1 ( ) 1 ( v m c m KE c m c m mc RME E KE → = = = = << Relativistic Momentum: The relativistic momentum p is the relativistic mass, m, time the velocity. v m v m p r r r 0 = = 0 m m = Energy Momentum Connection: 2 2 0 2 2 ) ( ) ( c m cp E + = with 2 2 2 z y x p p p p + + = Speed β of a particle: The speed of an object with rest mass m 0 is given by 2 2 0 2 ) ( ) ( c m cp cp E cp c v + = = = . Relativistic Mass Mass of the object at rest Classical KE Relativistic energy and momentum are conserved!
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PHY3063 R. D. Field Department of Physics Chapter2_2.doc University of Florida Relativistic Kinetic Energy ( derivation ) Relativistic Force: The force is equal to the rate of chance of the (relativistic) momentum as follows: v dt dm dt v d m dt v m d dt p d F r r r r r + = = = ) ( where 0 m m γ = is the relativistic mass. Relativistic Kinetic Energy: The kinetic energy of a particle is (as classical) the total work done in moving particle from rest to the speed v as follows: RME E c m mc dm c dm v dm v c dm v mvdv mv vd dx dt mv d Fdx KE m m m m = = = + = + = = = = 2 0 2 2 2 2 2 2 0 0 ) ) (( ) ( ) ( ) ( where I used ) / 1 ( 2 2 0 2 2 m m c v = and dm v c mvdv ) ( 2 2 = . Energy Momentum Connection: ) / 1 /( 2 2 2 0 2 c v m m = and 2 0 2 2 2 ) / 1 ( m c v m = which implies that 2 0 2 2 2 2 / m c v m m = and 4 2 0 2 2 2 4 2 c m c v m c m = thus 2 2 0 2 2 ) ( ) ( c m cp E + = . Speed β of a particle: Since mv p = and 2 / c E m = we get 2 / c Ev p = and thus 2 2 0 2 ) ( ) ( c m cp cp E cp c v + = = = β . Classically this term is zero and F = ma Time t = 0 F Particle at rest: v = 0, m = m 0 Later time t v Particle moving at speed v, m = γ m 0
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PHY3063 R. D. Field Department of Physics Chapter2_3.doc University of Florida Particles with Zero Rest Mass (Photons) Case 1 (m 0 > 0): In this case 2 2 0 2 2 ) ( ) ( c m cp E + = with 2 2 2 z y x p p p p + + = and 1 ) ( ) ( 2 2 0 2 < + = = = c m cp cp E cp c v β . Case 2 (m 0 = 0): In this case cp E = with 2 2 2 z y x p p p p + + = and 1 = = = E cp c v . Also, c p c E m = = 2 . Photons: Photons have zero rest mass and ( as we will see ) there energy is related to the frequency, f, of their oscillations as follows: ω h = = hf E , where h is Planck’s constant s eV s J h × = × = 15 34 10 136 . 4 10 626 . 6 with ) 2 /( π h = h and where ω = 2 π f is the “angular” frequency. The momentum of a photon is given by k h p h = = λ , where λ is the wavelength of the photon and k is the wave number (k=2 π / λ ). We see that 2 2 c hf c E m = = and c p E h E p h f = = = . Relativistic Mass m 0 > 0 implies β < 1!
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Chapter2_all - PHY3063 R. D. Field Relativistic Energy and...

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