. Author manuscript; available in PMC: 2013 Nov 6.

Published in final edited form as: Phys Rev E Stat Nonlin Soft Matter Phys. 2009 May 13;79(5 0 1):051908. doi: 10.1103/PhysRevE.79.051908

TABLE II.

Magnetic fields of electric multipole moments immersed in an infinite medium of conductivity σ

Dipoles

B (a_{10}) = \frac{1}{4 π} \frac{a_{10}}{r^{3}} (x \hat{y} - y \hat{x})

B (a_{11}) = \frac{1}{4 π} \frac{a_{11}}{r^{3}} (y \hat{z} - z \hat{y})

B (b_{11}) = \frac{1}{4 π} \frac{b_{11}}{r^{3}} (z \hat{x} - x \hat{z})

Quadrupoles

B (a_{20}) = \frac{1}{4 π} \frac{3 a_{20} z}{2 r^{5}} (x \hat{y} - y \hat{x})

B (a_{21}^{1}) = \frac{1}{4 π} \frac{a_{21}}{r^{5}} [- 3 x y \hat{x} + (3 x^{2} - r^{2}) \hat{y}]

B (a_{21}^{2}) = \frac{1}{4 π} \frac{a_{21}}{r^{5}} [3 y z \hat{z} - (3 z^{2} - r^{2}) \hat{y}]

B (a_{21}^{3}) = \frac{1}{4 π} \frac{3 a_{21}}{2 r^{5}} [- x y \hat{x} + (x^{2} - z^{2}) \hat{y} + y z \hat{z}]

B (b_{21}^{1}) = \frac{1}{4 π} \frac{b_{21}}{r^{5}} [3 x y \hat{y} - (3 y^{2} - r^{2}) \hat{x}]

B (b_{21}^{2}) = \frac{1}{4 π} \frac{b_{21}}{r^{5}} [- 3 x z \hat{z} + (3 z^{2} - r^{2}) \hat{x}]

B (b_{21}^{3}) = \frac{1}{4 π} \frac{3 b_{21}}{r^{5}} [(z^{2} - y^{2}) \hat{x} + x y \hat{y} - x z \hat{z}]

B (a_{22}^{1}) = \frac{1}{4 π} \frac{a_{22}}{r^{5}} [(3 {(x + y)}^{2} - 2 r^{2}) \hat{z} - 3 z (x + y) (\hat{x} + \hat{y})]

B (a_{22}^{2}) = \frac{1}{4 π} \frac{a_{22}}{r^{5}} [(2 r^{2} - 3 {(x - y)}^{2}) \hat{z} + 3 z (x - y) (\hat{x} - \hat{y})]

B (a_{22}^{3}) = \frac{1}{4 π} \frac{3 a_{22}}{r^{5}} [- y z \hat{x} - x z \hat{y} + 2 x y \hat{z}]

B (b_{22}^{1}) = \frac{1}{4 π} \frac{2 b_{22}}{r^{5}} [3 x z \hat{x} + (r^{2} - 3 x^{2}) \hat{z}]

B (b_{22}^{2}) = \frac{1}{4 π} \frac{2 b_{22}}{r^{5}} [- (r^{2} - 3 y^{2}) \hat{z} - 3 y z \hat{y}]

B (b_{22}^{3}) = \frac{3}{4 π} \frac{b_{22}}{r^{5}} [z x \hat{x} - z x \hat{y} + (y^{2} - x^{2}) \hat{z}]