Electron orbits

Rutherford's atom is a neutral sphere with a very small, massive, positive nucleus at the centre and electrons revolving round it in dynamically stable orbits.

In hydrogen, the electrostatic attraction supplies the centripetal force: mv²/r = (1/4πε₀)e²/r² (Eq. 12.2). So the orbit radius and the speed are linked by r = e²/(4πε₀mv²) (Eq. 12.3).

Kinetic energy K = ½mv² = e²/(8πε₀r); potential energy U = −e²/(4πε₀r). The minus sign in U says the force points towards the nucleus (along −r).

Total energy E = K + U = −e²/(8πε₀r) (Eq. 12.4). So K = −E and U = 2E: the potential energy is twice the total energy, and the kinetic energy is its magnitude.

E is negative, which means the electron is bound to the nucleus. With a positive total energy the electron would not follow a closed orbit.

Example 12.3: 13.6 eV separates a hydrogen atom into a proton and an electron, so E = −13.6 eV = −2.2 × 10⁻¹⁸ J. Then r = −e²/(8πε₀E) = 5.3 × 10⁻¹¹ m, and with m = 9.1 × 10⁻³¹ kg the speed is v = e/√(4πε₀mr) = 2.2 × 10⁶ m/s.