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Do the Maxwell equations definitively rule out the existence of magnetic monopoles?

Your thinking is the wrong way around. The direction is not: Maxwell's laws say that there are no monopoles, so there are none. The correct direction is: There are no monopoles, so Maxwell's laws say ...
• 2,703

Why can't we directly relate the magnetic field $\bf B$ to the electric field $\bf{E}$ using the equation $∇×\bf{B}= μ_0σ \bf{E}$?

In the scenario where the electric field is static ($\frac{\partial \vec{E}}{\partial t}=0$) and only present inside a conductor where Ohm's law $\vec{J}=\sigma \vec{E}$ holds, then yes you could ...
• 52.3k
Accepted

Ampere-Maxwell Law for Resistor in AC circuit

The "ideal resistor" is just what it says: an idealization. Typically we can get adequately accurate solutions to circuits by assuming that resistors are ideal and have no capacitance, but ...
• 28.8k

Does the “wave” solution contain every solution to the Maxwell’s equations?

The wave solution, as is usually meant, is only for the case of zero charge density and zero current density. As such, it can never cover the case of non-zero densities in either charge or current or ...
1 vote

Does the “wave” solution contain every solution to the Maxwell’s equations?

All free space solutions to Maxwell's equations are indeed also solutions to the electromagnetic wave equation. This is because the wave equation is deduced directly from Maxwell's equations. An ...
• 2,387
1 vote

Ampere-Maxwell Law for Resistor in AC circuit

I think this is merely a reshuffling of contributions that is not going to be a big deal. Before we get there, let's review Ohm's law, which is typically remembered as $$\tag1V=IR$$ If a wire or ...

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