How much work must be done to increase the speed of an electron from 0.200c to 0.210c? In (MeV)
The energy is given by E = gamma m c^2. Take the difference of this for
v= 0.210c and 0.200 c
CountIblis is correct.
gamma is 1/sqrt[1 - (v/c)^2]
gamma changes from 1.0206 to 1.0461
Multiply the difference by the rest mass energy.
is rest mass energy
9.11e-31 * (3e8)^2 ??
and do I need to convert from J to ev to KeV?
Thanks for the help
Yes, that is the rest mass energy in Joules.
Since they do ask for the answer in MeV, use the conversion factor
1 MeV = 1.602*10^-12 J
To calculate the amount of work required to increase the speed of an electron from 0.200c to 0.210c in units of MeV (Mega-electron Volts), we need to use the relativistic kinetic energy equation.
The formula to calculate kinetic energy (KE) of a relativistic particle is given by:
KE = (γ - 1) * m * c^2
where γ is the Lorentz factor, m is the mass of the particle, and c is the speed of light.
To calculate γ, we need to use the following equation:
γ = 1 / sqrt(1 - (v/c)^2)
where v is the velocity of the electron.
First, let's calculate γ for an electron with a velocity of 0.200c:
γ1 = 1 / sqrt(1 - (0.200c/c)^2)
= 1 / sqrt(1 - 0.04)
= 1 / sqrt(0.96)
≈ 1 / 0.9798
≈ 1.0202
Next, let's calculate γ for an electron with a velocity of 0.210c:
γ2 = 1 / sqrt(1 - (0.210c/c)^2)
= 1 / sqrt(1 - 0.0441)
= 1 / sqrt(0.9559)
≈ 1 / 0.9776
≈ 1.0228
Now that we have the values of γ1 and γ2, we can calculate the work done to increase the speed of the electron. The change in kinetic energy is given by:
ΔKE = (γ2 - γ1) * m * c^2
Since the mass of an electron is m = 9.10938356 × 10^-31 kg, and c = 3 × 10^8 m/s, we can substitute the values into the equation:
ΔKE = (1.0228 - 1.0202) * (9.10938356 × 10^-31 kg) * (3 × 10^8 m/s)^2
ΔKE ≈ (0.0026) * (9.10938356 × 10^-31 kg) * (9 × 10^16 m^2/s^2)
ΔKE ≈ 2.203 × 10^-12 Joules
To convert Joules to MeV (Mega-electron Volts), we need to use the conversion factor:
1 Joule ≈ 6.242 × 10^18 MeV
Therefore, the amount of work done to increase the speed of the electron would be approximately:
ΔKE ≈ 2.203 × 10^-12 Joules * 6.242 × 10^18 MeV/Joule
ΔKE ≈ 1.375 MeV
So, the amount of work required to increase the speed of the electron from 0.200c to 0.210c is approximately 1.375 MeV.