Infinitely extended cylindrical region in free space has volume charge density

  1. 1. The problem statement, all variables and given/known data
    An infinitely extended cylindrical region of radius a>0 situated in free space contains a volume charge density given by:
    [
    ρ(r)= volume charge density
    ρo=constant=initial volume charge density
    radius=a>0

    ρ(r)=ρo(1+αr^2); r<=a
    ]
    with ρ(r)=0 for r>a

    Questions:
    1. utilize gauss law together with the inherent symmetry of the problem to derive the resulting electrostatic field vector E(r) both inside and outside the cylinder

    2. Use both Poisson’s and Laplace’s equations to
    directly determine the electrostatic potential V(r) both inside
    and outside the cylindrical region. From this potential function,
    determine the electrostatic field vector E(r).

    3. Determine the value of the parameter α for which
    the electrostatic field vanishes everywhere in the region outside
    the cylinder (r > a). Plot Er (r ) and V(r ) as a function of r for
    this value of α.




    2. Relevant equations

    eo=epsilon-not

    gauss' law : divergence of E(r) = ρ(r)/εo


    closed∫{E.nda} = 1/εo*∫∫∫{V{ρ(r)d^3r}




    3. The attempt at a solution
     
  2. jcsd
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