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Constructing field lines

  1. Jul 5, 2015 #1
    1. The problem statement, all variables and given/known data
    Determine the field lines of the vector field ##\vec F = (x^2+y^2)^{-1} (-y\hat x + x\hat y)##

    2. Relevant equations
    Definition of a field line
    ##\frac{d\vec r (\tau ) }{d\tau } = C \vec F (\vec r (\tau )) ##

    3. The attempt at a solution
    From the above equation and choosing ##C=1## we get
    \begin{cases}\frac{dx}{d\tau } = \frac{-y}{x^2+y^2}\\
    \frac{dy}{d\tau } = \frac{x}{x^2+y^2}.
    \end{cases}
    Separation of variables yields
    \begin{cases}
    \frac{x^3}{3}+y^2x = -y\tau + c_1\\
    \frac{y^3}{3} + x^2y = x\tau + c_2
    \end{cases}
    somehow this is supposed to tell me the field lines are "horizontal circles centered around the z-axis" but I don't see how. If i multiply the first equation by x and the second by y and substract i end up with
    ##\frac{y^4-x^4}{3} = 2xy\tau + c_2y-c_1x##
    But i still don't see why that is a circle.
     
    Last edited: Jul 5, 2015
  2. jcsd
  3. Jul 5, 2015 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    You can calculate dx/dy. That gives easier formulas.
     
  4. Jul 5, 2015 #3
    Thanks got it now!
     
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