Why does a monopole not radiate energy in electodynamics?
Solution 1
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.
Solution 2
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.
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user135580
Updated on January 13, 2021Comments
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user135580 almost 3 years
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?
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Admin about 9 yearsThis doesn't answer the question.
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Admin about 9 yearsThe 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.
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Randy Welt about 9 yearsThis 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