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[itex]\oint_C \vec{E} \cdot \vec{dr} = -\frac{\partial}{\partial t} \int_S \vec{B} \cdot \vec{dA}[/itex]

but we know that [itex]\vec{E}=-\nabla \varphi[/itex]

and so [itex]\oint_C \vec{E} \cdot \vec{dr} =-\oint_C \nabla \times \nabla \varphi \cdot \vec{dA}=0[/itex] by Stokes' Theorem.

therefore, why isn't the RHS of Farady's Law just 0 all the time?