# Why does a monopole not radiate energy in electodynamics?

Why there is no monopole radiation in Electromagnetic field? I read somewhere that it is impossible because it violates charge conservation. I don't understand how? How charge conservation gets violated here?

In a multipole expansion of the electric potential, outside of some charge charge distribution $\rho(\mathbf r,t)$, the monopole term is simply

$$V_{mp}(\mathbf r) = \frac{Q_{total}}{4\pi \epsilon_0 r}$$

The associated electric field is then

$$\vec E_{mp} = \frac{Q_{total}}{4\pi \epsilon_0 r^2}\hat r$$

For this term to be time varying at some fixed $r$, the total charge must change with time, i.e., charge must be created or destroyed which is inconsistent with the conservation of electric charge.

So, if there were monopole EM radiation, charge would not be conserved and, further, such radiation would be longitudinal.

It seems relevant to mention that a spherically symmetric solution of Maxwell equations (for a system with a spherically symmetric charge and current distributions) is necessarily static in a (not necessarily thin) vacuum shell (i.e. a region with no charges/matter). This is a consequence of the electromagnetic version of Birkhoff's theorem.

From quantum physics we know that Electrons have angular momentum l and that must be conserved (conservation of momentum) by emitted Photons which have a Spin of 1. There are no photons without spin (no spin= monopole) => therefore dipole radiation.

Hypothetical gravitons have a Spin of 2 (2x2 matrix). So gravitational waves are supposed to have a quadrupole moment

• This doesn't answer the question. – user4552 Oct 11 '14 at 14:00
• The edits don't fix the answer. Any charged particle can emit a photon, not just an electron. We do, for example, have charged particles with spin 0, such as alpha particles, which could emit photons. Photons can have orbital angular momentum as well as their spin 1, so it doesn't follow immediately from any such elementary considerations that a photon can't have total angular momentum zero. Also, this is a classical question with a classical answer, so there's no need to bring in quantum mechanics. – user4552 Oct 11 '14 at 18:52
• This question is about multipole EM radiation. There are classical systems and QM systems. For classical the answers on top are perfect. If you consider QM system then you should consider angular momentum l=1 which is conserved by the emitted photon. for electrons almost always l=1 (dipole), for nucleus l=2 (quadrupole). notice, the spin of the electron/nucleus is neglected (analogue to orbital theory) [link] spektrum.de/lexikon/physik/multipolstrahlung/10038 – Randy Welt Oct 11 '14 at 23:57