An electron has a total energy of 2.8 MeV. What is its momentum?
In c = 1 units:
E^2 - p^2 = m^2 ------>
p = sqrt[E^2 - m^2] =
sqrt[2.8^2 - 0.511^2] MeV =
2.753 MeV
Put c back by dividing by c
p = 2.753 MeV/c = 1.47*10^(-21) kg m/s
To find the momentum of an electron with a given total energy, we can use the equation that relates energy and momentum for a particle moving at approximately the speed of light:
E^2 = (pc)^2 + (mc^2)^2
Where:
E is the total energy of the electron
p is the momentum of the electron
c is the speed of light in a vacuum
m is the rest mass of the electron
In this case, the electron has a total energy of 2.8 MeV. Since the mass of an electron is approximately 9.11 x 10^-31 kg and the speed of light is approximately 3 x 10^8 m/s, we can substitute these values into the equation:
(2.8 MeV)^2 = (p * c)^2 + (9.11 x 10^-31 kg * c^2)^2
Simplifying the equation, we have:
(2.8 x 10^6 eV)^2 = (p * 3 x 10^8 m/s)^2 + (9.11 x 10^-31 kg * (3 x 10^8 m/s)^2)^2
(2.8 x 10^6 eV)^2 - (9.11 x 10^-31 kg * (3 x 10^8 m/s)^2)^2 = (p * 3 x 10^8 m/s)^2
Taking the square root of both sides:
p * 3 x 10^8 m/s = sqrt[(2.8 x 10^6 eV)^2 - (9.11 x 10^-31 kg * (3 x 10^8 m/s)^2)^2]
Finally, solving for p:
p = sqrt[(2.8 x 10^6 eV)^2 - (9.11 x 10^-31 kg * (3 x 10^8 m/s)^2)^2] / (3 x 10^8 m/s)
Calculating the expression gives us the momentum of the electron.
To find the momentum of an electron, we can use the equation:
Momentum (p) = √(2mE),
where m is the mass of the electron and E is its total energy.
The mass of an electron (m) is approximately 9.11 x 10^-31 kilograms.
Given that the total energy (E) is 2.8 MeV, we need to convert it to joules since the SI unit of energy is the joule.
1 MeV is equal to 1.6 x 10^-13 joules.
Converting the total energy to joules, we get:
2.8 MeV x (1.6 x 10^-13 J/1 MeV) = 4.48 x 10^-13 J.
Now, we can substitute the values into the equation:
Momentum (p) = √(2 x (9.11 x 10^-31 kg) x (4.48 x 10^-13 J)),
Momentum (p) = √(8.09 x 10^-43 kg m^2/s^2).
Finally, we can calculate the momentum:
p ≈ 2.84 x 10^-22 kg m/s.
Therefore, the momentum of the electron is approximately 2.84 x 10^-22 kg m/s.