Where Did I mistaken?
Potential applied be V
Mass and charge of electron be M_e & e
Initial Velocity of system be u
Velocity Gained by electron be ?v
Initial energy of electron in system be U
Final energy of electron after acceleration be E
Energy gained by electron due to electric field be ?E
U = 1 / 2 M_e ? u^2
?E = e ? V
1 / 2 M_e ? ?v ^ 2 = e ? V ----For an observer in motion with system
E = 1 / 2 M_e ? ( u + ?v )^2
E = 1 / 2 M_e ? (u) ^ 2 + 1 / 2 M_e ? (?v)^2 + M_e ? u ? ?v
E = U + ?E + M_e ? u ? ?v
Hence, the net energy is greater by M_e ? u ? ?v
Now, the question is where had energy came from?