MagneticForcesOnMovingCharges 2006-04-27 12:43:53 2006-04-27 12:52:56 Topic % this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Charged particles are trapped in the Earth's magnetosphere, supposedly by the "mirror effect". The general motion of charged particles in magnetic fields is said to be covered by Lorentz's Force. I believe there is a simpler reason for the "trapped" particles than the mirror effect. The particles are trapped by the scalar parallel force of the magnetic field. This parallel field seems to have been overlooked, theoretically and experimentally. This article addresses the parallel force and its impact on moving charges. Lorentz's Force is given as: \begin{equation} F = e \mathbf{v} \times \mathbf{B} \end{equation} where "e" is the \htmladdnormallink{Electric Charge}{http://planetphysics.org/encyclopedia/Charge.html} traveling with \htmladdnormallink{velocity}{http://planetphysics.org/encyclopedia/Velocity.html} "v" in the "B-field". The general Electromagentic force should include a scalar and a vector force: \begin{equation} F= e \mathbf{vB} = -e \mathbf{v \cdot B} + e\mathbf{v \times B} \end{equation} The scalar force is parallel to the magnetic field and the magnetic field acts like a hamonic oscillator. This action, in effect traps the charge parallel to the field. Example for positive charge e=1, v= I + J and B= J F = (I + J)J = -1 + K F = (-1 + K) J = -J - I = -(I + J) F = -(I + J) J = 1 - K F = (1 - K)J = I + J repeating, etc, etc. The vector force rotates AROUND the B field (I, K, -I, -K) CCW around B. The scalar force oscillates ALONG the B field (J, -1, -J, 1). In general whenever the velocity is oblique to the B field , there are two forces, a scalar force parallel to the field and a vector force perpendicular to the field. Conservation of energy requires these two velocities to exist when oblique and therefore, two forces should exist. I have not seen any experiments confirming this scalar force. It could be that no one is looking given the influence of the Lorentz Force "Law". Obviously, negative charges rotate in the opposite direction from positive charges. In general, I believe fields in physics are quaternion, after Hamilton, and thus the more complex product of veleocity and filed takes place: \begin{equation} \mathbf{v_qB_q = (v_sB_s - v_v \cdot B_v) + (v_sB_v + v_vB_s + v_v \times B_v)} \end{equation} I welcome discussion on this topic and any experimental confirmation or refutation of the paralllel magnetic force.