Explore BrainMass
Share

Explore BrainMass

    The Laplace Equation in Cylindrical Coordinates

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    Please see the attached file for a question taken from the "Classical Electrodynamic 3rd by Jackson".

    © BrainMass Inc. brainmass.com October 10, 2019, 7:46 am ad1c9bdddf
    https://brainmass.com/physics/electricity-magnetism/laplace-equation-cylindrical-coordinates-potential-594333

    Attachments

    Solution Preview

    Laplace equation in cylindrical coordinates is:
    (1.1)
    The boundary conditions are:
    (1.2)
    And
    (1.3)
    Implicitly, we require that the solution will be invariant under full rotations:
    (1.4)
    And inside the cylinder we require a finite (physical) solution - no singular points.
    We start by setting the solution as a product of three independent univariate functions
    (1.5)
    Then the partial differentials become full differentials:
    (1.6)
    Dividing both sides by we get:

    (1.7)
    The left hand side of (1.7) depends on while the right hand side depends on z.
    the variables are independent, hence for (1.7) to be true for any both sides must be equal the same constant:
    (1.8)
    We now solve the equation for
    from the boundary conditions we see that
    (1.9)
    Case 1:
    The equation becomes
    (1.10)
    This is a simple second order equation with constant coefficients, so its general solution is:
    (1.11)
    Where (A,B) are some arbitrary constant ...

    Solution Summary

    The 10-pages solution describes in detail how to find the potential inside a cylinder where both caps are grounded and the envelope is held on some potential V(phi,z). It includes full derivations of the differential equations and the derivation of the expansion coefficients.

    $2.19