The potential for a homogeneous cylinder in a cylindrical coordinate system

Abstract

Some special cases of the potential for a homogeneous cylinder in a cylindrical coordinate system may be treated by virtue of simple integrals, for example, the potential for a straight rod or wire segment and that for a homogeneous cylinder at the point on its axis. However, because of the involved mathematical operations, the analytical formula of the potential for a homogeneous cylinder at an arbitrary point has not been seen from others. In order to solve the problem, the author has taken the following steps: (1) expanding Green’s function e i k | r ′ − r | / | r ′ − r | {e^{ik\left | {r’ - r} \right |}}/\left | {r’ - r} \right | in the cylindrical coordinate system; (2) transforming Green’s function e i k | r ′ − r | / | r ′ − r | {e^{ik\left | {r’ - r} \right |}}/\left | {r’ - r} \right | into Green’s function 1 / | r ′ − r | 1/\left | {r’ - r} \right | by setting the wave number k k to be zero and integrating the separated azimuthal function cos n ( ϕ ′ − ϕ ) {\cos ^n}\left ( {\phi ’ - \phi } \right ) ; (3) using the integral recursion relation for the function r ′ 2 m + 1 / [ ( z ′ − z ) 2 + r ′ 2 + r 2 ] ( 4 m + 1 ) / 2 {r’^{2m + 1}}/{\left [ {{{\left ( {z’ - z} \right )}^2} + {{r’}^2} + {r^2}} \right ]^{\left ( {4m + 1} \right )/2}} with respect to r ′ r’ and those for the functions 1 / [ ( z ′ − z ) 2 + r 2 ] ( 2 m − 1 ) / 2 1/{\left [ {{{\left ( {z’ - z} \right )}^2} + {r^2}} \right ]^{\left ( {2m - 1} \right )/2}} and 1 / [ ( z ′ − z ) 2 + r 2 + a 2 ] ( 4 m − 2 l − 1 ) / 2 1/{\left [ {{{\left ( {z’ - z} \right )}^2} + {r^2} + {a^2}} \right ]^{\left ( {4m - 2l - 1} \right )/2}} with respect to z ′ z’ , then we can complete the integrals for the function 1 / | r ′ − r | 1/\left | {r’ - r} \right | and obtain the analytical expression of the potential for the cylinder in the cylindrical coordinate system. For numerical comparison, we have calculated the potentials for the cylinder and the prolate or oblate spheroid with equivalent volume and same high aspect ratio at far field point. The results are satisfactory.