Question

electron and positron (an electron's antiparticle, having the same mass but opposite charge) form a bound system orbiting a common center. The particles are a distance b from each other and move with the same speed. What is the orbital period of the particles?

Answer 1

Since this is an orbiting system. it means that Kepler's third law will be valid. Kepler's third law states that $T^{^{2}}=\frac{4\pi ^{^{2}}r^{3}}{GM}$

We need to convert this formula for electrical orbits. Since, from our basic knowledge of gravity we know that $\frac{GM}{r^{2}}$ is acceleration. In electrostatics acceleration is defined by - $\frac{e^{2}}{4\pi \epsilon _{o}r^{2}m}$ , therefore using this information we know that $GM = \frac{e^{2}}{4\pi \epsilon _{o}m}$ .

Putting this value in the formula we have for Kepler's third law -

$T^{^{2}}=\frac{4\pi ^{^{2}}r^{3}}{\frac{e^{2}}{4\pi \epsilon _{o}m}} = \frac{16\pi^{3}r^{3}\epsilon _{o}m}{e^{2}}$

where r is the separation between the particles, which in this case is b and m is the reduced mass which is given as $\frac{m_{e}m_{p}}{m_{e}+m}_{p}$ but since the mass are equal, reduced mass will be equal to $m_{e}/2$ .

Hence the orbital period of time for the electron and positron system is given as - $T = \sqrt{\frac{16\pi^{3}b^{3}\epsilon _{o}m_{e}}{2e^{2}}}$