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Potential inbetween two coaxial cylinders

  1. Nov 4, 2012 #1
    1. The problem statement, all variables and given/known data

    There are 2 coaxial cylindrical conductors. The inner cylinder has radius a, while the outer cylinder has radius b. There is no charge in the region a < r < b. If the inner cylinder is at potential Vo and the outer cylinder is grounded, we want to find the potential in the region between the cylinders. We assume L >> b > a and neglect end effects.

    a) Write Laplace's equation in cylindrical coordinates.

    b) Assuming V(r) is a function of the axial distance r alone, integrate the differential equation and use the boundary conditions to find V(r) , a ≤ r ≤ b.



    2. Relevant equations

    2V = 0

    3. The attempt at a solution

    Laplace's equation in cylindrical coordinates would be

    (1/r)(d/dr)(r*dV/dr) =0

    Since there is no phi or z dependence. Also I know it's a partial and not d but I can't type the symbol.

    To solve this

    (1/r)(d/dr)(r*dV/dr) =0

    (d/dr)(r*dV/dr) =0

    r*dV/dr = C

    dV/dr = C/r

    dV = C/r dr

    V = C*ln(r) + K , where C and K are constants.

    Using the boundary conditions:

    0 = C*ln(b) + K

    Vo = C*ln(a) + K

    and using that to solve for constants:

    V = Vo/[ln(a/b)] * ln(r/b)

    Is that right?
     
  2. jcsd
  3. Nov 4, 2012 #2

    mfb

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    2016 Award

    Staff: Mentor

    The result is correct and the other parts look fine, too.

    LaTeX code can be used to write partial derivatives: ##\frac{\partial}{\partial r}## --> ##\frac{\partial}{\partial r}##.
     
  4. Nov 6, 2012 #3

    rude man

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    Homework Helper
    Gold Member

    Confirmed using Gauss' theorem.
     
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