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Two problems involving vector calculus-

  1. Jul 1, 2011 #1
    1. The problem statement, all variables and given/known data
    (Sorry for the confusing font- I tried to figure out using the latex reference to indicate a vector, but couldn't do it- please view the "superscripted" variable as a vector. Thanks)

    A. The electrostatic potential at P arising form a dipole of unit strength at O, oriented in the direction of the unit vector n, is given by V= n[itex]\stackrel{OP\rightarrow}{}[/itex]/r^3, where r=OP and n coincides with the direction of the positive z axis. Find the value of

    del^2V

    B. Let F be the gravitational field of a mass oriented at the origin O and is expressed by,

    F=c[itex]\stackrel{OA\rightarrow}{}[/itex]/r^3
    where c is a constant and r is the distance OA.
    Find the value of
    Curl F
    2. Relevant equations
    del^2V=d^2v/dx^2+d^2v/dy^2+d^2v/dz^2

    the curl of a function involves the determinant of the matrix [i j k; deriv_x deriv_y deriv_z; f_x f_y f_z] (wrote in matlab form)


    3. The attempt at a solution
    Sorry about putting two problems together- I figured since it is of similar origin, and the problem itself put it as if it is two parts to a problem. If it must be separated, that can be done.

    A. I have managed to get to the point where I can get the expression of V= <0,0,1>[itex]\cdot[/itex]<x,y,z> /Magnitude of OP ^3.
    Now do I just have to get the 2nd partial derivative of this function based on the three coordinate systems x y and z? I did it but looked pretty ugly (couldn't simplify), and wanted to make sure.

    B. For this part, it looked like the function was:
    F=c*<x,y,z>/(magnitude of OA)^3
    I went through and calculated the curl of this function based on the formula above, but got 0. is this correct?
     
  2. jcsd
  3. Jul 1, 2011 #2

    hunt_mat

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    Homework Helper

    A) Potentials are not vector fields but scalar fields, are you saying that [itex]v=r^{-3}[/itex]? you know that [itex]r=\sqrt{x^{2}+y^{2}+z^{2}}[/itex], so now you have a function of x,y and z.

    b)not done this calculation but curl is an indication of vorticity which is now much the field rotates (sort of...), if everything is coming from the centre...
     
  4. Jul 4, 2011 #3
    hunt mat- thanks for your input!
    a) I think my attempt at latex input failed miserably..
    the function of v is
    v= (n_unitvector*OP_vector)/r^3, and yes, r=(√x2+y2+z2)^3, so the final expression is
    z/((√x2+y2+z2)^3), where x2=x^2- getting the del^2 of this function is what I need to do right?
     
  5. Jul 5, 2011 #4

    hunt_mat

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    Homework Helper

    Basically you're on the right tranck.
     
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