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I'm confused about two conflicting ideas: the first is that monopoles cannot emit EM radiation (see here), and the second is that an accelerating point charge does produce EM radiation (see here). I was under the impression that point charges could not produce EM waves? What is the significance of dipoles in EM radiation if point charges accelerating also produce radiation?

Edit: For more context, this has all really been motivated by some vague statements I've heard about how to have gravitational radiation, you need a quadrupole moment, as opposed to in EM where you just need a dipole moment.

Qmechanic
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Chris
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4 Answers4

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Please read the exact wording in the cited source carefully. The forbidden thing is the monopole radiation, not radiation from a monopole.

"Monopole radiation" is the radiation due to time dependent electric field, which in turn is produced by the monopole term in the multipole expansion of the potential. This is forbidden, as it would require time dependent charge.

On the other hand, one can make a monopole radiate by simply moving it. Think of a moving (positive) charge as a static charge plus a time dependent dipole (the negative charge of the dipole is not moving and is sitting on top of the static positive charge). Then the radiation is coming from the time dependent electric field coming from the dipole term in the multipole expansion.

2012rcampion
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John
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I'm confused about two conflicting ideas: the first is that monopoles cannot emit EM radiation ([see here][1]), and the second is that an accelerating point charge does produce EM radiation ([see here][2]).

I was under the impression that point charges could not produce EM waves?

A stationary point charge certainly won't.

What is the significance of dipoles in EM radiation if point charges accelerating also produce radiation?

The potential due to a moving point charge (for simplicity in the non-relativistic limit, and in some convenient unspecified units) looks like: $$ \phi(\vec r) = \frac{q}{|\vec r - \vec s(t)|}\;, $$ where $\vec s(t)$ is the position of the charge as a function of time.

The multipole expansion of this potential is: $$ \phi(\vec r) = \sum_{\ell=0}^\infty \frac{q r_\lt^\ell}{r_\gt^{(\ell + 1)}}P_\ell(\cos(\theta))\;, $$ where $r_\lt$ is the lesser of $|\vec r|$ and $|\vec s|$, $r_\gt$ is the greater, and $\theta$ is the angle between $\vec r$ and $\vec s$.

Suppose, as an example, that the magnitude of the observation point $|\vec r|$ is always larger than $|\vec s|$. And suppose that $\vec r$ is in the $\hat z$ direction. In this case: $$ \phi = \frac{q}{r} + \frac{qs(t)}{r^2}\cos(\theta_q(t)) + \ldots $$

The first term is a monopole term, which is, as expected, time independent. The next term is a dipole term, and there are higher order terms indicated by the "$\ldots$".

hft
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I have an intuitive answer. Accelerating a charge induces a changing magnetic field in response which results in radiation. (There is no way the magnetic field is like a monopole since $\nabla\cdot B = 0$. )

Isaiah Curtis
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It depends on what you mean by a "monopole". If by a monopole you mean a fluctuating amount of charge at a fixed position then that kind of a radiator cannot exist for it violates the most basic conservation law. If by a "monopole" you mean charge(s) moving and accelerating/decelerating, that charge will radiate. If by "monopole" you mean an antenna like this monopole, then that is among the most common antennas. There the charges run around back and forth and radiate because they accelerate.

hyportnex
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