# Does Faraday-Maxwell equation exclusively refer to a variable magnetic field instead to a variable magnetic fux?

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?

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 $$\partial \Sigma$$.