Here, and are the usual spherical coordinates, and the are associated Legendre polynomials. We can obtain from by the substitution we will use this relationship later to simplify the calculation. Compared to the equations in Zangwill, the overall sign differs here, and we have inserted a factor of to simplify the implementation. The International System of Units (SI) is used in these units has the same dimensions as. Where and are scalar functions related to the transverse electric and transverse magnetic parts of the solution, respectively, is the speed of light, and is a point in space. The electric field and magnetic field are thereby given by We omit an implicit time dependence in the plane wave traveling in the positive direction with the electric field linearly polarized in the direction. Since the problem is so well studied, we go directly to the solution. Our formulation follows the textbook of Zangwill. The problem has been studied extensively by Bohren and Huffman. The solution for the response of a spherical dielectric particle in a vacuum was given more than 100 years ago. The scattering force comes first because it requires an incident plane wave, whereas the gradient force requires an incident standing wave that is a little more difficult to set up. In this article, we develop code to generate the Mie scattering coefficients and the stress tensor formulas, and we combine them to form first the scattering force and then the gradient force. In addition, we present the code we used to derive the force on a spherical particle used in. The results agree with expressions from Harada and Asakura, as we show in the Appendix. The derivation was too detailed for that article, so we are presenting it here. In the course of that project, we derived the scattering and gradient forces rigorously from Maxwell ’s equations. Recently, we made a theoretical study of a system to sort submicrometer dielectric spheres in the interference field of a laser in slowly flowing air. The limits agree with expressions in the literature. In both cases, the Clausius –Mossotti factor arises rigorously from the derivation without any physical argumentation. Starting from an analytic expression for the force on a spherical particle in a vacuum using the Maxwell stress tensor, as well as the Mie solution for the response of dielectric particles to an electromagnetic plane wave, we derive the scattering and gradient forces. In this article, we derive the expressions including the Clausius –Mossotti factor directly from the fundamental equations of classical electromagnetism. The derivation of the scattering force and the gradient force on a spherical particle due to an electromagnetic wave often invokes the Clausius –Mossotti factor, based on an ad hoc physical model.
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