Question

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.

Answer 1

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 = m_oc² + 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 .