Analytical formula of induced electric fields in a spherical conductor by an ELF dipole magnetic field source

An analytical formula of induced electric field E in a spherical conductor by an ELF dipole magnetic field source ${\it M} = {\it M}_{\|} + {\it M}_{\perp}$ is mathematically derived in vector form based on the equivalent mutlipole moment method with reexpansion technique (RE‐EMMM), where M ‖ and ${\it M}_{\perp}$ are parallel and perpendicular components of M , respectively. The validity of the formula is confirmed in the following three ways: (i) the derivation of the formula from the Sarvas equation with the reciprocity theorem derived by Eaton; (ii) the convergence of the formula to that of homogeneous magnetic field when M ‖ is located at the infinite distance; (iii) comparison of the analytical solutions with numerical solutions by RE‐EMMM. Furthermore, a formula for the trajectory, which satisfies E = 0 , is derived for the field by M ‖ . © 2008 Wiley Periodicals, Inc. Electr Eng Jpn, 166(3): 8– 17, 2009; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/eej.20739