Question

An electron has a total energy of 6.5 MeV. What is its momentum (in MeV/c)?

Short answer

Answer: The momentum of the electron is approximately 6.44 MeV/c.

Step-by-step solution

Write down given values and relativistic energy-momentum equation

Write the relativistic energy-momentum equation, the mass of an electron, and the given total energy: \(E^2 = (mc^2)^2 + (pc)^2\) \(E = 6.5 \,\text{MeV}\) \(m = 0.511 \,\text{MeV}/c^2\)

Rewrite equation in terms of momentum (p)

Solve the relativistic energy-momentum equation for momentum \(p\), by isolating it on one side of the equation: \((pc)^2 = E^2 - (mc^2)^2\)

Input the given values and calculate the momentum

Substitute the given total energy and electron mass into the equation: \((p \cdot c)^2 = (6.5 \,\text{MeV})^2 - (0.511 \,\text{MeV}/c^2 \cdot c^2)^2\) Now, calculate the value of the right side of the equation: \((p \cdot c)^2 = (6.5 \,\text{MeV})^2 - (0.511 \,\text{MeV})^2\)

Determine the momentum p

To find the momentum \(p\), take the square root of both sides of the equation: \(p \cdot c = \sqrt{(6.5 \,\text{MeV})^2 - (0.511 \,\text{MeV})^2}\) Calculate the value of the right side: \(p \cdot c ≈ 6.44 \,\text{MeV}\) So, the momentum of the electron is approximately 6.44 MeV/c.