Interaction Force Between an Axial Magnetic Dipole and a Magnetic Sphere

Abstract

In this article, we address the fundamental problem of a magnetic sphere of radius a experiencing the magnetic field produced by an axial magnetic dipole located at (0, 0, c), c > a. We provide analytic solutions for the scalar magnetic potentials and exact expressions for the interaction force in terms of the non-dimensional permeability k = (μ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sup> /(μ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">e</sup> + μ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sup> )), 0 ≤ k ≤ 1 and the dipole-sphere separation c/a. An approximate expression for the force is proposed which exactly captures the actual force characteristics qualitatively. By modeling the tip of a magnetic force microscope as a dipole and the sphere as a magnetic material, the numerical values of the force are calculated, using the theoretical results presented herein, for samarium cobalt (SmCo) and Alnico 5 (Alcomax) magnetic materials. Our estimates predict forces as high as 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-13</sup> N for SmCo magnets and 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-12</sup> N for Alcomax types of magnetic materials. Larger force estimates are attainable by altering the tip-to-sphere separation and other parameters as well.