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I was wondering if there was any relationship between Faraday's law and Maxwell's addition to ampere's law as both have to do with some sort of time varying flux that generates some sort of current. A theoretical, rather than mathematically based explanation would be great!

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  • $\begingroup$ Varying flux does not necessarily generate a current. Varying magnetic flux is associated with induced electric field, which in some cases creates electric current. $\endgroup$ Commented Apr 21, 2020 at 22:01

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If you extend Maxwell's equations to include the possibility of magnetic charge $\rho_M$ and magnetic current $\mathbf J_M$, then Faraday's law and Ampere's law look extremely similar:

$$\nabla \times \mathbf E = -\left(\mu_0\mathbf J_M + \frac{\partial \mathbf B}{\partial t}\right)$$ $$\nabla \times \mathbf B = \mu_0\mathbf J_E + \epsilon_0\mu_0\frac{\partial \mathbf E}{\partial t}$$

See e.g. here.

From the latter, we have that the solenoidal parts of magnetic fields are generated by time-varying electric fields and moving electric charges; from the former, we have that the solenoidal parts of electric fields are generated by time-varying magnetic fields and moving magnetic charges.

All of this lovely symmetry is spoiled by the fact that magnetic charge does not appear to exist - at least not at energy scales which we are currently capable of exploring. The existence of magnetic monopoles at high energies is a fairly robust prediction of many theories which seek to extend the standard model, but as yet there is no experimental evidence for this.

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    $\begingroup$ To add to the similarity argument: in Lorentz-Heaviside units ( en.wikipedia.org/wiki/Lorentz%E2%80%93Heaviside_units) these equations become $$\nabla\times\mathbf E=-\frac 1 c\left(\mathbf J_M+\frac{\partial \mathbf B}{\partial t}\right)$$ and $$\nabla\times\mathbf B=\frac 1 c\left(\mathbf J_E+\frac{\partial \mathbf E}{\partial t}\right)$$ $\endgroup$ Commented Apr 22, 2020 at 12:35

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