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Conservitive electric fields

  1. Nov 18, 2008 #1
    I'm being taught electromagnetism at university, but there's one definition that has been left slightly ambiguous for an electric field to be conservative i've been taught that
    1) [tex]\nabla[/tex]xE = 0
    But ive also been taught that
    2) [tex]\oint[/tex] E.dl = 0

    But im not sure if 1) & 2) have to be true for it to be conservitive.
    OR that if 1) is true then 2) is true (and visa versa) ie, 1) [tex]\Leftrightarrow[/tex] 2)

    So do i have to test for both to check the field is conservitive, or just one and say it implies the other.
  2. jcsd
  3. Nov 18, 2008 #2


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    2. is a definition, which is equivalent to saying a vector field E is conservative if is can be written as the gradient of some scalar field.
    1. can be shown to be equivalent to 2. by writing [itex]\mathbf{E}=\nabla\phi[/itex] and taking the curl.

    So, in short, no you don't need to test both. Most of the time it's a lot easier to calculate 1. to check whether the vector field is conservative.
  4. Nov 18, 2008 #3
    Thanks dude :)
  5. Nov 18, 2008 #4


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    Gold Member

    In addition to what cristo said, knowing that (2) is true for a conservative field can make later calculations easier, but as cristo said, (1) is usually easier to calculate. So if you're asked to prove whether a vector field is conservative it is usually best to use (1), but it is also useful to know (2) as well.
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