Electromagnetic Drag on a Magnetic Dipole Interacting With a Moving Electrically Conducting Sphere

Abstract

In this paper, we report an analytical study of the forces and torques acting upon a magnetic dipole interacting with a moving electrically conducting sphere. The work is motivated by the question whether Lorentz force velocimetry [Thess , Phys. Rev. Lett., vol. 96, 2006, 164501]-a noncontact flow measurement technique for liquid metals and electrolytes-can be applied to granular materials as well. We derive explicit expressions for all forces and torques for the case of low magnetic Reynolds number and small particle size. After a discussion of symmetry and reciprocity relations among the forces and torques, we apply the general theory to the particular cases of a translating (nonrotating) and rotating (nontranslating) sphere. The analysis for the purely translating sphere leads to the conclusion that the force is proportional to <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> / <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">8</sup> where <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</i> is the radius of the sphere and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</i> is its minimum distance to the magnetic dipole. This result indicates that Lorentz force velocimetry can indeed be applied to granular metallic materials. The analysis for the purely rotating sphere leads to the result that the torque is proportional to <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> / <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">h</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">6</sup> . This result can be applied to derive a rigorous solution for a rotary Lorentz force flowmeter interacting with a rotating sphere. This solution implies that, contrary to intuitive expectation, a frictionless rotary Lorentz force flowmeter rotates with only 4/5 of the angular velocity of the sphere with which it interacts rather than undergoing synchronous rotation.