Motion of Charged Particles in Electric and Magnetic Fields

• Miklos Szilagyi
Chapter
Part of the Microdevices book series (MDPF)

Abstract

In order to be able to use the Lagrangian equations (1-33) and (1-35) we have to define the Lagrangian L. We know that the force acting on our charged particle is defined by Eq. (1-14) and we also know that the force in general is defined by the Newtonian law of motion
$$\frac{{d{\text{p}}}}{{dt}} = \frac{d}{{dt}}(m{\text{v) = F}}$$
(2-1)
where p = mv is the particle’s momentum and
$$m = {m_0}/{\left( {1 - {v^2}/{c^2}} \right)^{1/2}}$$
(2.2)
is the relativistic mass of the particle (m0 is its rest mass). Substituting the Lorentz force (1-14) and Eqs. (1-11) and (1-12) into Eq. (2-1) we obtain
$${{d(m{\rm{v)}}} \over {dt}} = Q({\bf{E}} + {\bf{v}} \times {\bf{B}}) = Q\left( { - {\rm{grad}}\;{u}{{\partial \bf{A}} \over {\partial t}} + \rm{v} \times {\rm{curl}}\;{\bf{A}}} \right)$$
(2.3)
We know from vector analysis that for any vectors A and v
$$\rm{grad}(\bf{A} \cdot \bf{v})=(\bf{A} \cdot \nabla) \bf{A}+\bf{A} \times \rm{curl}\; \rm{v} + \rm{v} \times \rm{curl}\;\bf{A}$$
(2.4)
where
$$\nabla = \sum\limits_{i = 1}^3 {{{{{\bf{e}}_i}} \over {{h_t}}}{\partial \over {\partial {q_i}}}}$$
(2.5)
is the symbolic nabla vector operator with which Eqs. (1-13), (1-5), and (1-6) become
$$\rm{grad}\;u = \nabla u$$
(2.6)
$$\rm{div}\bf{V} = \nabla \bf{V}$$
(2.7)
and
$$\rm{curl} \bf{V} = \nabla \times \bf{V}$$
(2.8)
respectively.

Keywords

Magnetic Field Charged Particle Lorentz Force Electrostatic Field Relativistic Potential
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Plenum Press, New York 1988

Authors and Affiliations

• Miklos Szilagyi
• 1
1. 1.University of ArizonaTucsonUSA

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