Vector field, vortex free and sources free

  1. fluidistic

    fluidistic 3,292
    Gold Member

    1. The problem statement, all variables and given/known data
    I must determine whether the following vector fields have sources or vortices.
    1)[tex]\vec A = (\vec x )\frac{\vec a \times \vec x}{r^3}[/tex] where [tex]\vec a[/tex] is constant and [tex]r=||\vec x||[/tex].
    2)[tex]\vec B (\vec x )= \frac{\vec a}{r+ \beta}[/tex] where [tex]\vec a[/tex] and [tex]r[/tex] are the same as part 1) and [tex]\beta >0[/tex].

    2. Relevant equations
    Not sure.


    3. The attempt at a solution
    I think I must calculate whether the divergences and curls of the fields are worth 0, in which case they are free of source and free of vortex.
    I've attempted only part 1) yet (I want to try out part 2 alone once I'm started with part 1).
    The modulus of [tex]\vec A (\vec x )[/tex] is worth [tex]\frac{|\vec a | \sin (\theta )}{ |\vec x|^2}[/tex] and I know its direction is orthogonal to both [tex]\vec a[/tex] and [tex]\vec x[/tex].
    I know how to calculate the div and curl of a field when I have an explicit expression of it but this is not the case in the exercise, hence my attempt to modify the given expression.
    I'd like to know how you'd attempt the problem 1). Thank you.
     
    Last edited: Mar 10, 2011
  2. jcsd
  3. vela

    vela 12,501
    Staff Emeritus
    Science Advisor
    Homework Helper

    Write

    [tex]\vec{A}(x) = \vec{a}\times\left(\frac{\vec{x}}{r^3}\right)[/tex]

    and then use the vector identity [itex]\nabla \cdot (\vec{x}\times\vec{y}) = \vec{y}\cdot(\nabla\times\vec{x}) - \vec{x}\cdot(\nabla\times\vec{y})[/itex].

    To evaluate the cross product, you can write out explicitly what the components of [itex]\vec{x}/r^3[/itex] are in terms of x, y, and z and crank out the curl.

    Similarly, you should be able to look up an identity for the curl of a cross product and evaluate it the same way.
     
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