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    Laplace equation in cylindrical coordinates

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    In this problem, you will find the electrostatic potential inside an infinitely long, grounded, metal cylinder of unit radius whose axis coincides with the z-axis (See figure below). In cylindrical coordinates, the potential, V(r, theta, z), satisfies Laplace's equation... <i>Please see attached</i>... Let us assume that the potential is known at z = 0, so that
    V(r, theta, z)= f(r, theta)
    is a given function.
    Because of the symmetry in the problem, V(r, theta, -z) = V(r, theta, z), so we only have to consider the semi-infinite cylinder in the domain z>0. Since the cylinder is grounded, the potential there must be zero, meaning that V(1, theta, z) = 0...

    Please see attached for full question.

    © BrainMass Inc. brainmass.com June 3, 2020, 5:29 pm ad1c9bdddf
    https://brainmass.com/math/calculus-and-analysis/laplace-equation-cylindrical-coordinates-28506

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    Hi there!

    The solution is attached both in Word format and in pdf format (the last one for the benefit of the solution library).

    Check out the following site:
    http://mathworld.wolfram.com/HelmholtzDifferentialEquationCircularCylindricalCoordinates.html

    For a nice solution of the Helmholtz equation in cylindrical coordinates.

    The Laplace equation:

    Can be solved using separation of variables:

    Substituting it back into the equation one obtains:

    Multiplying by the equation becomes separated:

    Since we require periodic solution for the polar function Θ we can write:

    Which is the harmonic equation:

    With the ...

    Solution Summary

    The 6 pages solution shows how to solve Lapalace equation in cylindrical coordinates and how to apply the boundary conditions.

    $2.19

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