Force Lorentz

As the most important example of a Minkowski force in special relativity we will now explicitly consider the Lorentz force on a charged particle, which has already been encountered in section 2.4. In IS, where the particle is at rest, it experiences a force 

Now the electric field in IS may be expressed by the components of the electric and magnetic fields in IS, 

Note that we have artificially extracted the electric field in the last equality of Eq. (3.134), which represents an exact relation. Inserting this expression for the electric field into the relation for the Minkowski force as given by Eq. (3.131) directly yields the four-dimensional Lorentz force / in IS, 

Insertion of the force / into the equation of motion (3.124) yields the relativistic equations of motion for a charged particle subject to an external electromagnetic field. For = 0 it reads 

We close this section by mentioning that the familiar Lorentz force as given by Eq. (3.137) is the correct nonrelativistic, i.e., low-velocity approximation to the Newtonian force F since. 

In the 19 century, H. A. Lorentz found experimentally that a charge with q moving with a velocity of v in a magnetic field (B) felt the following force (F) (1) The magnitude of F was found to be proportional to vflsin 0. (2) F was found to be directed at right angles to the plane of spanned by v and B as shown in Fig. 1-2. Thus, F can be represented by the vector product of v and B. 

If a is assumed to be a dimensionless constant having a=l, B can be defined as follow  

More generally, the force induced on a charge in the presence of both the electric (E) and magnetic fields is given by 

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The source is brought to a. positive poteptial (I/) of several kilovolts and the ions are extracted by a plate at ground potential. They acquire kinetic energy and thus velocity according to their mass and charge. They enter a magnetic field whose direction is perpendicular to their trajectory. Under the effect of the field, Bg, the trajectory is curved by Lorentz forces that produce a centripetal acceleration perpendicular to both the field and the velocity. 

Magnetic Sector Field. In a magnetic field B an ion with the velocity v and the charge q experiences a centripetal force, the Lorentz force P ... 

It gets worse with magnetic properties, and the Lorentz force... 

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 functions A are vector potentials of the field, and the

scalar potentials from which the field can be derived through (5) and (9). An infinite number of potentials leading to the same field can be constructed from (6) and (10). Using (5) and (9) the Lorentz force defined in terms of potentials... 

See also Legislation Regulations Layered HTS, complex Lorentz force in,... 

Magnetic sector B Deflection of a continuous ion beam separation by momentum in magnetic field due to Lorentz force... 

Ion cyclotron resonance ICR Trapped ions separation by cyclotron frequency (Lorentz force) in magnetic field... 

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]... 

Fig. 4.17. The Right Hand Rule (I thumb, B index finger, Fl middle finger) to determine the direction of the Lorentz Force (a) the current corresponds to the direction where positive charges move, i.e., the figure directly applies for positive ions, (b) A real magnet yoke without coils and flight tube. With kind permission of Thermo Electron (Bremen) GmbH, (left) and Waters Corporation, MS Technologies, Manchester, UK (right).

As we know from the discussion of magnetic sectors, an ion of velocity v entering a uniform magnetic field B perpendicular to its direction will move on a circular path by action of the Lorentz force (Chap. 4.3.2), the radius of which is determined by Eq. 4.13 ... 

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

The combination of the transport current and magnetic field surroimd-ing superconductors creates stresses on the assembly through the generation of Lorentz forces. Calculation of coil stresses for final coil... 

The so-called Aharonov-Bohm effect is observed with another experimental setup. A solenoid is placed immediately after the plate, between the slits, and its axis is parallel to the slits, and therefore normal to the beam trajectory beam. If the solenoid is long enough, the magnetic field remains confined in it as a consequence, the magnetic field is shown to have a null value in the region crossed by electrons beamed on either sides of the solenoid. The Lorentz force exerted on the electrons is expected to be null in the absence of any external electrical field. 

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