How can an electrically neutral planetary core be geodynamo? The standard theory for the earth's magnetic field is that the planet's core is a geodynamo.
I know moving electric charges can induce a magnetic field, but I also know that opposite charges produce opposite fields, and unless I missed something, normal matter has no net charge on the macro level.
In Maxwell's equations, $\mathrm{\textbf{J}}=0$ means $\mathrm{\textbf{B}}=0$, correct?
So if the Earth's core is electrically neutral, how is geodynamism possible?
 A: The focus on “neutral medium” is a red herring. 
A copper wire is neutral. Move it through a magnetic field, and an EMF is induced. Since copper conducts, that in turn drives a current. And, properly arranged (i.e. a “dynamo” configuration), that current can create more magnetic field and make it all stronger. 
The energy is coming from the motion of the material. So long as the material conducts, a current can flow even though it’s neutral overall. 
It doesn’t even matter whether the current is negative electrons going one way or positive ions going the other. 
A: Maxwell's 4th equation in steady state, states
$$\rm  \vec\nabla \times \vec B= \mu_0 \vec J,$$
so, so far you're right, if $\rm J=0$, then $\rm B=0$. However why do you think that $\rm J$ is zero? The current density is defined over the charge momentum densities, and for a partially or fully ionized fluid consisting only of ions and electrons this is
$$\rm \vec J = \sum_{species} \vec J_s = \vec J_{e} + \vec J_{p}  = n_e q_e \vec{v}_e +n_i q_i \vec{v}_i = n_e q_e(\vec{v}_i - \vec{v}_e)$$
where in the last equality one only uses the ion to electron number ratio and the fact that the electron charge is opposite to that of the ions. Thus, even in the ion-rest-frame (equal to the lab frame in good approximation, due to their slower movement/high inertia) there is a net current, when the net charge
$$\rm Q = \sum_{\rm species} Q_s = n_e q_e + n_i q_i = 0$$
is zero. 
So the answer to your question doesn't even have anything to do with the Maxwell's equations.
Zero net charge does NOT imply zero current.
A: When an electrically conductive medium is set into rotation, it will produce a magnetic field even though it is electrically neutral overall. Geodynamism occurs inside the earth because the electrically conductive liquid ore is in convective motion, because of the temperature difference between the center of the earth and the surface. The dynamics of this process as it takes place inside the earth is complex and involves the earth's rotation, the convective motion of liquid iron, and heat transfer between the core and the outer layers of the earth. 
A: The dynamo on my bicycle is also electrically neutral. Magnetism is caused by current - or spin - not by charge. 
A: If we have a neutral medium containing on average an equal amount of positive and negative charges $N>>1$, then the fluctuation of the total charge is $\sqrt{N}e$. Thus, a neutral medium is formed charge. 
The correct description of the formula for the electric current is described 
AtmosphericPrisonEscape . The conductor with a current on average is electrically neutral. But in order to generate an electric current, a seed electric field strength is needed. This seed intensity is formed due to charge fluctuations, which is described in the formulas I have proposed.
I think this explains the inversion of the Earth’s magnetic field. When the fluctuation of the Earth’s charge reaches a large value, sufficient to change the polarity of the magnetic field, the Earth’s magnetic field is inverted. If the Earth is partly a dielectric, and partly a conductor, then persistent charge fluctuations can form in the Earth.
We note the coincidence of the values of average time between the inversions of the magnetic field $T\approx 10^6 year$ and the characteristic time of fluctuations is equal $T=\frac{N_{av}r_g}{c}=2*10^{13}s$, $N_{av}$ the Avogadro number, $r_g$ the gravitational radius. The frequency of fluctuations, equals on the Earth $\frac{c}{r_g}$, on the Sun is orders of magnitude higher due to the high temperature and thermonuclear reactions and the characteristic time of fluctuations is 9-12 years. The frequency of fluctuations of thermonuclear reactions is much higher in the Sun than the frequency of fluctuations in the Earth. The gravitational radius is replaced by the core radius. The exponential multiplier reduces this frequency. As a result, such a characteristic time of inversion at the Sun is formed. Note that the characteristic time of inversion of the Earth’s magnetic field is not constant when the next fluctuation occurs. The process of formation of fluctuations and its development is long and can last from 200 years to several thousand years. In this case, the magnetic field of the Earth will weaken and the Earth will be unprotected. The last time the inversion of the Earth’s magnetic field occurred 780 thousand years ago. But the period of inversion is not constant and the calculated period is approximate. Thus, no periodicity in the change of poles was detected, and this process is considered stochastic. There are time intervals of tens of millions of years when the inversion did not occur.
