# The following three particles have the same total energy E: a) a photon b) a proton c) an...

## Question:

The following three particles have the same total energy E:

a) a photon

b) a proton

c) an electron.

Rank the magnitudes of the particles' momenta from greatest to smallest.

## Energy:

Moving matter has kinetic energy. However, by virtue of having mass, particles have a "rest" energy. Therefore, total energy of a particle is given by its motion (i.e. related to its momentum) and also the rest energy from its mass.

## Answer and Explanation:

Energy is the sum of the kinetic and rest energy:

{eq}\displaystyle E^2 = p^2 c^2 + m^2c^4 \Rightarrow p = \dfrac{1}{c} \sqrt{E^2 - m^2 c^4 } {/eq}

A photon is a massless particle. Therefore, its total energy is given by

{eq}p = \dfrac{E}{c} {/eq}

For the bosons, we know that the proton is heaver than the electron, since {eq}m_p = 1.67 \times 10^{-11} {/eq} kg and {eq}m_e = 9.11 \times 10^{-31} {/eq} kg. Therefore, the value

{eq}E^2 - m^2 c^4 \Rightarrow E^2 - m_e^2 c^4 > E^2 - m_p^2 c^4 {/eq}

for the same value of {eq}E {/eq}. Therefore, the electron momentum is greater than the proton momentum. Thus, in order from highest to lowest momenta: photon, electron, proton.

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