Faraday's law -- How is the RHS required for all surfaces?

[millo]

Homework Statement

My question is how do we know that faraday's law right side is required for all surfaces.

Relevant Equations

∫_c Edl =-d/dt ∫_s Bda

∫_c Edl =-d/dt∫_sBda

 

[Delta2]

do you mean why the right side is independent of the surface s as long as we keep the curve c , the boundary of the surface the same? This is a consequence of Stoke's Theorem (or curl theorem).

Since $\nabla\cdot \vec{B}=0$ (Gauss's law for magnetism) we can set $\vec{B}=\nabla\times\vec{A}$. Stokes theorem tell us that $$\iint_S \vec{B}\cdot d\vec{S}=\iint _S(\nabla\times \vec{A})\cdot d\vec{S}=\oint_C\vec{A}\cdot d\vec{l}$$ so the surface integral of B over S will equal the line integral of A over the curve C, thus it remains constant for all surfaces S with the same boundary C.

 

[millo]

Yes this is what I meat, thanks for answer.