Problem — Particle or photon

Problem

In a laboratory a particle has energy $E=10^{12}$ J and momentum $p=3000$ kg·m/s. Calculate (a) the rest mass, (b) the velocity, (c) determine whether it can be a photon, by comparing with the relation $E=pc$.

Solution

(a) Rest mass — from the energy-momentum invariant. We have $E^2 - (pc)^2 = (mc^2)^2$, so $m = \frac{\sqrt{E^2 - (pc)^2}}{c^2}$ Computing $pc = 3000\cdot 3\cdot 10^8 = 9\cdot 10^{11}$ J, so $E^2 - (pc)^2 = (10^{12})^2 - (9\cdot 10^{11})^2 = 1{,}0\cdot 10^{24} - 0{,}81\cdot 10^{24} = 1{,}9\cdot 10^{23}\ \text{J}^2$ $m = \frac{\sqrt{1{,}9\cdot 10^{23}}}{9\cdot 10^{16}} = \frac{4{,}36\cdot 10^{11}}{9\cdot 10^{16}} \approx \ev{m \approx 4{,}8\cdot 10^{-6}\ \text{kg}}$

(b) Velocity. Dividing $p=\gamma mv$ by $E=\gamma mc^2$ gives $p/E = v/c^2$, so $v = \frac{p\,c^2}{E} = \frac{3000\cdot 9\cdot 10^{16}}{10^{12}} = \frac{2{,}7\cdot 10^{20}}{10^{12}} = 2{,}7\cdot 10^8\ \text{m/s}$ $\ev{v = 0{,}9\,c}$

(c) Is it a photon? A photon has zero mass and satisfies $E = pc$. Here instead $E = 10^{12}\ \text{J} \neq pc = 9\cdot 10^{11}\ \text{J}$ The two values differ, and the rest mass found is nonzero ($\approx 4{,}8\cdot 10^{-6}$ kg): so this is not a photon, but a massive particle travelling at $0{,}9\,c$.

Links

Topics: Special relativity Concepts: Mass-energy equivalence · Relativistic momentum · Spacetime invariant Skills: Symbolic setup