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How to prove the radius of curvature at any point on a line?

  1. Aug 8, 2009 #1
    In a magnetic field, field lines are curves to which the magnetic induction B is everywhere tangetial. By evaluating dB/ds where s is the distance measured along a field line, prove that the radius of curvature at any point on a line is given by

    density symbol-->p= B^3 / [ B x( B * del) B]

    where do i start with this?? I have absolutely no idea. please help with start with a direction to go.
     
    Last edited: Aug 8, 2009
  2. jcsd
  3. Aug 8, 2009 #2

    gabbagabbahey

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    Hi catheee, welcome to PF!:smile:

    This forum supports [itex]\LaTeX[/itex], which allows you to write equations in a clear manner....

    [tex]\rho=\frac{B^3}{\textbf{B}\times(\textbf{B}\cdot\mathbf{\nabla})\textbf{B}}[/tex]

    ^^^Is this what you mean?

    A good place to start might be with the (mathematical) definition of radius of curvature for a general curve....what equation(s) do you have for that?
     
  4. Aug 8, 2009 #3
    yes! thats the equation I have, but its the only equation I was given and I was told to solve it, I don't see anything about the radius of curvature equation in the textbook I have. So, what you're saying is that I should get the radius of curvation equation and then use it to solve the problem?

    btw, thanks for answering my question!
     
  5. Aug 8, 2009 #4

    gabbagabbahey

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    Really?!:bugeye:

    [itex]\textbf{B}\times(\textbf{B}\cdot\mathbf{\nabla})\textbf{B}[/itex] is a vector whereas [itex]\rho[/itex] and [itex]B^3[/itex] are both scalars...how exactly does one divide a scalar by a vector to produce another scalar?!


    Out of curiosity, what textbook is this problem from? It might be easier to see what they expect you to do if I scan through the text quickly.

    Of course! How on earth would you find what the radius of curvature for a certain curve is without knowing the definition of 'radius of curvature'? Surely, basic vector calculus is a prerequisite for studying whatever course this text is for?
     
  6. Aug 8, 2009 #5
    im using introduction to electrodynamics by griffiths 3rd edition.
    oh and sorry about double posting i didnt know that that was against the rules.
    :)
    im looking up the def of radius of curvature but it doesn't look like any of the equations would apply to this problem. Ughh
     
  7. Aug 13, 2009 #6

    gabbagabbahey

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    I am familiar with that text, but it won't be of much help for this problem.


    Okay, if I gave you the equation of some parameterized curve [itex]\textbf{r}(u)=x(u)\hat{x}+y(u)\hat{y}+z(u)\hat{z}[/itex], could you calculate the radius of curvature? Could you calculate the (unnormalized) tangent vector?

    If so, then let [itex]\textbf{r}(u)[/itex] describe one of your field lines....what does the fact that [itex]\textbf{B}\left(\textbf{r}(u)\right)[/itex] is tangent to [itex]\textbf{r}(u)[/itex] tell you?
     
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