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Polarized half-cylinder

  1. Feb 14, 2015 #1

    ShayanJ

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    Gold Member

    Consider a cylinder(height ##l##, radii a) cut in half by a plane parallel to its axis(z-axis). Now imagine it has a polarization density ## \vec P=P_0 \hat z## and so its two semi-circular surfaces have bound charge densities ## \sigma_u=-\sigma_d=P_0 ##. I want to calculate the electric potential of these charge densities at an arbitrary point of space.
    The charge elements is clearly (using cylindrical coordinates) ## dq=\sigma \rho d\varphi d\rho ##(with ## 0\leq \rho \leq a ## and ##0\leq \varphi \leq \pi ##).
    Now I should find out the distance between an arbitrary point on the semi-circular surface of the cylinder and an arbitrary point in space(##(R,\phi,Z)##) which can be written as ##\sqrt{R^2+Z^2+\rho^2+(\frac l 2)^2-2\sqrt{(R^2+Z^2)(\rho^2+(\frac l 2)^2)} \cos\gamma} \ \ ## where ##\gamma## is the angle between the two position vectors. We can write(using spherical coordinates) ## \cos\gamma=\cos \theta \cos\vartheta +\sin\theta \sin\vartheta \cos(\varphi-\phi) ##. So I can put this into the distance formula and write the integral I should calculate. But the problem is, now I have an integral w.r.t. cylindrical coordinates that contains some spherical coordinates. So I should either transform the integral to spherical coordinates or write the spherical coordinates in terms of cylindrical coordinates. I figured that the second option turns the integral into an intractable mess so I want to pursue the first option(Then I can use Legendre polynomials and spherical harmonics to do the integral). I calculated the Jacobian of the transformation from cylindrical to spherical coordinates(its ## \frac 1 r ## where ##r=\sqrt{\rho^2+(\frac l 2)^2}## is the radial component of the spherical coordinates of the point on the cylinder ). But I don't know how should I do the transformation. I'm confused here. Can anyone help?
    Thanks
     
    Last edited: Feb 14, 2015
  2. jcsd
  3. Feb 14, 2015 #2
    If I wanted to know potentials of such charge distributions I would use software package like Ansoft Maxwell.
     
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