All Problems Electromagnetic Waves. Radiation
Problem 4.211
Demonstrate that a closed system of charged non-relativistic particles with identical specific charges emits no dipole radiation.
4.211. The electric dipole moment of the system is \(\mathbf{p}=\sum e \mathbf{r}_{i}=\) \(=(e / m) M \mathbf{r}_{C},\) where \(M\) is the mass of the system, \(\mathbf{r}_{C}\) is the radius vector of its centre of inertia. Since the radiation power \(P \propto \ddot{p^{2}} \alpha\) \(\propto \dot{\mathbf{r}}_{C}^{2},\) and in our case \(\ddot{\mathbf{r}}_{C}=0, P=0\) too.