Question

Calculate the minimum energy released for each of the given particle-antiparticle annihilations: (a) electron and positron (b) muon and anti-muon (c) tau and anti-tau (d) proton and anti-proton

Answer 1

(a) The minimum energy released for  electron and positron = 1.02 MeV

(b) The minimum energy released for muon and anti-muon = 106.1 MeV

(c) The minimum energy released for tau and anti-tau = 1775 MeV

(d) The minimum energy released for proton and anti-proton = 1876 MeV

(a) electron and positron

Rest  mass of electron and positron = 9.1×$10^{-31}$kg.

E = Δ$mc^{2}$

E = 2$m_{0} c^{2}$

  = 2 × 9.1×$10^{-31}$ × (3×$10^{8}$) × (3×$10^{8}$)

1 MeV = 1.602×$10^{-13}$ J

So, E = 2 × 9.1×$10^{-31}$ × (3×$10^{8}$) × (3×$10^{8}$) / 1.602×$10^{-13}$

E = 1.02 MeV

The minimum energy released for  electron and positron = 1.02 MeV

(b) muon and anti-muon

Rest  mass of muon and anti-muon = 1.89×$10^{28}$kg

E = Δ$mc^{2}$

E = 2$m_{0} c^{2}$

  = 2 × 1.89×$10^{-28}$ × (3×$10^{8}$) × (3×$10^{8}$)

1 MeV = 1.602×$10^{-13}$ J

So, E = 2 × 1.89×$10^{-28}$ × (3×$10^{8}$) × (3×$10^{8}$) / 1.602×$10^{-13}$

         = 17.01 × $10^{-12}$  / 1.602×$10^{-13}$  = 10.61 × 10

E = 106.1 MeV

The minimum energy released for muon and anti-muon = 106.1 MeV

(c)tau and anti-tau

Rest  mass of tau and anti-tau = 3.16×$10^{-27}$kg

E = Δ$mc^{2}$

E = 2$m_{0} c^{2}$

  = 2 × 3.16×$10^{-27}$ × (3×$10^{8}$) × (3×$10^{8}$)

1 MeV = 1.602×$10^{-13}$ J

So, E = 2 ×3.16×$10^{-27}$ × (3×$10^{8}$) × (3×$10^{8}$) / 1.602×$10^{-13}$

         = 28.44 × $10^{-11}$  / 1.602×$10^{-13}$  = 17.75 × $10^{2}$

E = 1775 MeV

The minimum energy released for tau and anti-tau = 1775 MeV

(d)  proton and anti-proton

E = Δ$mc^{2}$

E = 2$m_{0} c^{2}$

  = 2 × 1.67×$10^{-27}$ × (3×$10^{8}$) × (3×$10^{8}$)

1 MeV = 1.602×$10^{-13}$ J

So, E = 2 ×1.67×$10^{-27}$ × (3×$10^{8}$) × (3×$10^{8}$) / 1.602×$10^{-13}$

         = 30.06 × $10^{-11}$  / 1.602×$10^{-13}$  = 18.76 × $10^{2}$

E = 1876 MeV

The minimum energy released for proton and anti-proton = 1876 MeV

Learn more about  minimum energy here;

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