In a particle accelerator such as the one at CERN a proton i

In a particle accelerator such as the one at CERN, a proton is moving in a circular orbit of radius r=10 km at a speed close to the speed of light. Neglecting relativistic effects, compute the amount of radiated energy per second. Compare that to the mass of the proton: E_p = m_pc² = 930 MeV

Solution

Total power radiated P = (2/3)Kq² a ² / c ³

Where K = 8.99 x10 ⁹ Nm ²/ C ²

            q = Charge of proton = 1.6 x10 ^-19 C

             a = accleration of the proton = v ² / r

              v = Speed ~ Speed of light = c

              r = radius = 10 km = 10000 m

Substitute values you get a = c ² / 10000

                                          = 10 ^-4 c ²

Substitute values in power equation you get ,

P = (2/3) Kq ² a ² / c ³

    = (2/3)(8.99x10 ⁹)(1.6x10 ^-19)(10^-4 c ²) ² / c ³

    = 9.589 x10 ^-18 c ⁴ / c ³

    = 9.589 x10 ^-18 x c

   = 9.589 x10 ^-18 x 3 x10 ⁸

   = 2.8768 x10 ^-9 J/s