Question

The rest energy of an electron is \(0.511 \mathrm{MeV} .\) What momentum (in MeV/c) must an electron have in order that its total energy be 3.00 times its rest energy?

Short answer

Answer: The momentum of the electron is approximately 1.44 MeV/c.

Step-by-step solution

Write down the relativistic energy-momentum relationship

The relativistic energy-momentum relationship is given by: E² = (mc²)² + (pc)² where E is the total energy, m is the mass, c is the speed of light, and p is the momentum.

Express the total energy in terms of rest energy

We are given that the total energy is 3 times the rest energy, so we can write: E = 3mc²

Substitute the total energy expression into the energy-momentum relationship

Now, substitute the total energy expression into the energy-momentum relationship: (3mc²)² = (mc²)² + (pc)²

Solve the equation for momentum (p)

To find the momentum, we can rearrange the equation and solve for p: (3mc²)² - (mc²)² = (pc)² (8m²c⁴) = (pc)² p = \sqrt{8m²c²}

Substitute the values and calculate the momentum

Now we can substitute the given rest energy value and the speed of light to find the momentum in MeV/c: Rest energy (E₀) = 0.511 MeV c = 1 (when working in units of MeV/c) E₀ = mc² m = \frac{E₀}{c²} = \frac{0.511}{(1)²} = 0.511 Now, substitute these values into the momentum equation: p = \sqrt{8(0.511)²} p ≈ 1.44 \, MeV/c The momentum of the electron when its total energy is 3 times its rest energy is approximately 1.44 MeV/c.