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

  • Thread starter millo
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  • #1
millo
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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
 

Answers and Replies

  • #2
Delta2
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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.
 
  • #3
millo
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Yes this is what I meat, thanks for answer.
 
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