Coulomb force on a line charge

  1. 1. The problem statement, all variables and given/known data

    We have a source charge that is a uniform sphere with a radius a (centered at origin) and uniform charge density, [tex]\rho[/tex]. There is a line charge with a length L that begins at Z0 and ends at Z0 + L (lies on the Z axis). This line charge has a uniform charge density of [tex]\lambda[/tex].

    2. Relevant equations

    Ill combine this with my attempt.

    3. The attempt at a solution

    First, I am resolving the source charge as a point. The sphere is of a uniform charge density and centered on the origin. So the 'q' for this source charge is [tex]\frac{4}{3}\pi[/tex]*a2*[tex]\rho[/tex].

    Second, I am calling the line charge L*[tex]\lambda[/tex].

    My solution so far : Fq'onq(sphere/point on line charge)= [tex]\frac{\lambda*\rho*a^{3}*L}{3*\epsilon}*\frac{\vec{R}}{R^{3}}[/tex]
    My problem (assuming the above is correct) is that I am uncertain how to express the vector, R, from the source charge to the line charge.

    In general, I am unsure of how to express a vector from a point to a continuous distribution of charges, or even from a continuous distribution to another.
     
  2. jcsd
  3. Redbelly98

    Redbelly98 12,017
    Staff Emeritus
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    I don't think you can treat the line charge as a point charge, as you can do with spheres.
     
  4. Ok I've got it.My line charge was indeed wrong.

    Basically, I chose my vector to be simple. The position with respect to the source was Z*Z(hat) and I ended up integrating from Z0 to Z0+L with 1/Z2 as the integrand ( lambda is constant, was pulled out.)

    Some quick simplification results in the answer in the back of the book.

    I was thinking too hard about the vector I suppose.

    Thank you!
     
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