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GR0877 Problem 91

PROBLEM STATEMENT: This problem is still being typed.

SOLUTION: (C) When \nabla \times \mathbf{E} = 0, the line integral of \mathbf{E} around any closed loop, according to Stokes’ theorem, is zero. Because of this, we can unambiguously talk about a scalar function \displaystyle \varphi(r) = - \int_{\mathcal{O}}^{\mathbf{r}} \mathbf{E} \cdot d\mathbf{l} such that \mathbf{E} = - \nabla \varphi.

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Categories: 0877 Tags: , , , ,
  1. Brent
    29.02.2012 at 01:53

    We know that the curl of a gradient is always 0. Given E is the gradient of a scalar, the curl of E must be 0!

    • Stevie
      16.10.2013 at 21:30

      Be careful. The curl of E isn’t always zero. Recall your Maxwell Equations where
      $ latex \nabla \times E=-\frac { \partial B }{ \partial t } $
      which of course can be non-zero if there’s a time-varying magnetic field. But I guess it must be to have a unique solution to $ latex E=- \nabla \phi $ .

      • Stevie
        16.10.2013 at 21:31

        damn! my latex didn’t work! admin, can you please fix it? and thank you, generally, for the wonderful site!

  2. Stevie
    16.10.2013 at 21:34

    Hm… Trying again:
    Be careful. The curl of E isn’t always zero. Recall your Maxwell Equations where
    \nabla \times E=-\frac { \partial B }{ \partial t }
    which of course can be non-zero if there’s a time-varying magnetic field. But I guess it must be to have a unique solution to E=- \nabla \phi .

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