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Faraday's law says that a variable magnetic flux produces an induced emf. A varying flux can be due to a varying magnetic field or a varying surface. The Faraday-Maxwell equation refers to a curl electric field produced by a variable magnetic field. Does this equation exclude the emf induced by a constant magnetic field and a variable surface? That is, is this equation less general than Faraday's law?

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In a given reference system, the Maxwell-Faraday differential equation $$ \nabla \times {\bf E} = -\frac{\partial {\bf B}}{\partial t} $$ only implies the integral relation $$ \int_{\partial \Sigma}{\bf E}\cdot{\mathrm d}{\bf l}=- \int_{\Sigma}\frac{\partial}{\partial t}{\bf B}\cdot{\mathrm d}{\bf S},\tag{1} $$ where $\Sigma$ is a surface and $\partial \Sigma$ its boundary (a closed curve in the space).

However, it is possible to rewrite this formula in terms of variations of flux due to a time variation of $\Sigma$, by noticing that the variation of the flux of ${\bf B}$ at constant ${\bf B}$ can be written as a line integral over $\partial \Sigma$: $$ \int_{\partial \Sigma}{\bf v}\times {\bf B}\cdot{\mathrm d}{\bf l}=- \frac{{\mathrm d}}{{\mathrm d} t}\int_{\Sigma(t)}{\bf B}\cdot{\mathrm d}{\bf S}.\tag{2} $$ The velocity ${\bf v}$ is the local value of the velocity of the line element ${\mathrm d}{\bf l}$. By combining the two formulas $(1)$ and $(2)$, we get the formula $$ \int_{\partial \Sigma}\left({\bf E}+{\bf v}\times {\bf B})\right)\cdot{\mathrm d}{\bf l}=- \frac{{\mathrm d}}{{\mathrm d} t}\int_{\Sigma(t)}{\bf B}(t)\cdot{\mathrm d}{\bf S},\tag{3} $$ where the left hand side is the emf due to the non-conservative electric field induced by the time variation of ${\bf B}$ and a term that could be re-interpretated as the circuitation of the Lorentz force per unit charge over $\partial \Sigma$.

Therefore, the original integral form of the Faraday-Lenz formula can be derived from the Faraday-Maxwell partial derivative equation and it should not be considered a more general equation.

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  • $\begingroup$ When you "rewrite" the (1) equation to obtain (2), you go from a fem produced by electric force (E.dl) to a fem produced by magnetic force (vxB).dl. Isn't it something more than rewriting an equation? $\endgroup$
    – Darkmatter
    May 24, 2021 at 14:48
  • $\begingroup$ @Txema Well, I choose the term rewriting because it is equivalent to add the same quantity, although written differently, on both sides of the equation. On the other hand, you are right. There is something more gained in the process: as you observe, the formula stresses the connection between emf due to non-conservative electric field and the one due to the movement of charges in the magnetic field. $\endgroup$ May 24, 2021 at 19:27
  • $\begingroup$ Ok, understood after thinking a little more. Thank you. $\endgroup$
    – Darkmatter
    May 26, 2021 at 7:12

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