Lorentz force equation

Table G Definitions of the Electric Field E, the (Di)electric Polarization P, the Electric Displacement D, the Magnetic Field H, the Magnetization M, the Magnetic induction or flux density B, statement of the Maxwell equations, and of the Lorentz Force Equation in Various Systems of Units rat. = rationalized (no 477-), unrat. = the explicit factor 477- is used in the definition of dielectric polarization and magnetization c = speed of light) (using SI values for e, me, h, c) [J.D. Jackson, Classical Electrodynamics, 3rd edition, Wiley, New York, 1999.]. For Hartree atomic u nits of mag netism, two conventions exist (1) the "Gauss" or wave convention, which requires that E and H have the same magnitude for electromagnetic waves in vacuo (2) the Lorentz convention, which derives the magnetic field from the Lorentz force equation the ratio between these two sets of units is the Sommerfeld fine-structure constant a = 1/137.0359895...

We have performed numerical experiments using a three dimensional relativistic kinetic electromagnetic particle-in-cell code The code works from first principles by solving the Lorentz force equation for the particles and the Maxwell s equations for the electromagnetic fields. 

ICR-MS [17] is based on the phenomenon of cyclotron motion of the ions in the analyzer cell. This motion is caused by a combination of magnetic (B-) and electric (E-) fields (trapping plates). The ions are forced to circulate as a result of the Lorentz force (equation 1 m, v, q are mass, velocity and charge of the ions). 

The force that is exerted on a charged particle carrying a charge q, possessing an instantaneous velocity v and submitted to the action of a magnetic induction 6, is given by the Lorentz force equation ... 

It is possible to derive the potential corresponding to F directly from (3.41) (see Goldstein 1950). Here we take a somewhat simpler approach and start with a suggested form of the potential and show that the application of (3.43) to this satisfies the Lorentz force equation. We choose the potential... 

The Ether is not useful to teach MT. The EM field is most effectively viewed as an irreducible entity completely defined by Maxwell s equations. (If one wants to make the interaction with point charges in N.M or QM explicit, one can add the Lorentz force or the minimal coupling.) All physical properties of th EM field and its interaction with matter follow from Maxwell s equations and the matter equations. 

If there is also an electric field present, the electric force qE must be taken into account as well. The complete force equation for a charged mass point, also known as the Lorentz force, is... 

The Lorentz Force Law can be used to describe the effects exerted onto a charged particle entering a constant magnetic field. The Lorentz Force Fl depends on the velocity v, the magnetic field B, and the charge of an ion. In the simplest form the force is given by the scalar equation [3,4,70,71]... 

Maxwell s equations, as well as the Lorentz force, can be derived from the Lagrangian density... 

This equation can be deduced from the general Laplace formula that expresses the force exerted on a conductor of length dl, through which a current / passes, in a magnetic field of intensity B. The orientation of the Lorentz force (F = I dl AB) can be found by different approaches such as the right-handed three-finger rule or using the orientations of a direct trihedron. 

In equations 5-8, the variables and symbols are defined as follows p0 is reference mass density, v is dimensional velocity field vector, p is dimensional pressure field vector, x is Newtonian viscosity of the melt, g is acceleration due to gravity, T is dimensional temperature, tT is the reference temperature, c is dimensional concentration, c0 is far-field level of concentration, e, is a unit vector in the direction of the z axis, Fb is a dimensional applied body force field, V is the gradient operator, v(x, t) is the velocity vector field, p(x, t) is the pressure field, jl is the fluid viscosity, am is the thermal diffiisivity of the melt, and D is the solute diffiisivity in the melt. The vector Fb is a body force imposed on the melt in addition to gravity. The body force caused by an imposed magnetic field B(x, t) is the Lorentz force, Fb = ac(v X v X B). The effect of this field on convection and segregation is discussed in a later section. 

The behavior of such a resonant discharge can be described by solving the continuum momentum equations for electron velocity, assuming a constant frequency ve. The force on the electrons is both due to the electric field and the Lorentz force caused by the magnetic field. The average power input per unit volume to the plasma is found to be10... 

Table 2.7 Maxwell s and Constitutive Equations, and Lorentz Force in Various Unit Systems3 ...

Unit System Maxwell Equations Constitutive Equations Lorentz Force... 

This is incompatible with Maxwell s equations, as shown below by using Gauss s law, Eq. (2.7.16), and the Lorentz force, Eq. (2.7.24). Assume that the two systems S and S move at velocities v and v and relative velocity V= v — v. If we use the Galileian transformation and assume that the charge q and the electric displacement D is the same in the two systems ... 

One of the most widely used steric parameters is molar refraction (MR), which has been aptly described as a "chameleon" parameter by Tute (160). Although it is generally considered to be a crude measure of overall bulk, it does incorporate a polarizability component that may describe cohesion and is related to London dispersion forces as follows MR = 4TrNa/By where N is Avogadro s number and a is the polarizability of the molecule. It contains no information on shape. MR is also defined by the Lorentz-Lorenz equation ... 

The first achievement of the study of Holas and March [99] is to establish the differential form of the above virial theorem [their Eq. (2.15)]. Again, as in the zero field case treated above, this differential virial theorem is interpreted as a force-balance equation. The well-known Lorentz force of electromagnetism then appears quite naturally in this equation. 

The Lorentz force, qvB (q, charge v, velocity), can be equated to the centripetal force... 

The theoretical content of classical electrodynamics can be summarized by the Lorentz force law and Maxwell s equations. The Lorentz force law describes the force on a charge q moving with velocity v in the presence of an electric field E and a magnetic field B ... 

The remaining parts of the lagrangian L describe the free field and fi-ee particles, and are not needed here. The requirements of V are that under lagrangian variation of the fields (for fixed partiele variables) it should contribute the appropriate terms to the Maxwell equations, and that variation of the particle variables (for a fixed electromagnetic field) gives the Lorentz force on the particles. 

The working equation for ICR can be quickly derived by equating the centripetal force and the Lorentz force experienced by an ion in a magnetic field ... 

An ion of mass m and charge q travelling with a velocity v in an electric field with strength E and a magnetic field with flux density B experiences a force F defined by the Lorentz force law. This equation can be combined with Newton s laws to yield the acceleration a as... 

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