An electron and a positron are moving toward each other with equal speeds of 3 x 106 m/s. The two particles annihilate each other and produce two photons of equal energy. (a) Do you need to use relativity for this problem? Support your answer numerically, and comment intelligently. (b) What were the deBroglie wavelengths of the electron and positron? (c) Find the energy of each photon. (d) Find the momentum of each photon. (e) Find the wavelength of each photon.
a )
yes need to have relativity
given
speed is v = 3 X 106 m/s
v = 0.01 c ( since c = 3 X 108 m/s )
b )
deBroglie wavelength of electron is ele = h / m v
= 6.67 X 10-34 / 9.1 X 10-31 X 3 X 106
= 2.44 X 10-10 m
positron is also have same mass
deBroglie wavelength of electron is positron = h / m v
= 6.67 X 10-34 / 9.1 X 10-31 X 3 X 106
= 2.44 X 10-10 m
c )
KE = mc2 ( 1 /(1-v2/c2)1/2 - 1 )
= mc2 ( 1 /(1-0.012c2/c2)1/2 - 1 )
= mc2 ( 1 /(1-0.012c2/c2)1/2 - 1 )
KE = 5 X 10-5 X m c2
= 5 X 10-5 X 9.1 X 10-31 X 9 X 1016
KE = 4.095 X 10-18 J
E = moc2 + KE
= 9.1 X 10-31 X 3 X 108 + 4.095 X 10-18
= 4.095 X 10-18 J
d )
momentum
P = m v
= 9.1 X 10-31 X 3 X 106
= 2.73 X 10-24 kg.m/s
both are same for electron and positron .
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