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Quadrupole
A quadrupole or quadrapole is one of a sequence of configurations of things like electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure reflecting various orders of complexity. Mathematical definition The quadrupole moment tensor ''Q'' is a rank-two tensor—3×3 matrix. There are several definitions, but it is normally stated in the traceless form (i.e. Q_ + Q_ + Q_ = 0). The quadrupole moment tensor has thus nine components, but because of transposition symmetry and zero-trace property, in this form only five of these are independent. For a discrete system of \ell point charges or masses in the case of a gravitational quadrupole, each with charge q_\ell, or mass m_\ell, and position \vec_\ell = \left(r_, r_, r_\right) relative to the coordinate system origin, the components of the ''Q'' matrix are defined by: : Q_ = \sum_\ell q_\ell\left(3r_ r_ - \left\, \vec_\ell \rig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadrupole Magnet
Quadrupole magnets, abbreviated as Q-magnets, consist of groups of four magnets laid out so that in the planar multipole expansion of the field, the dipole terms cancel and where the lowest significant terms in the field equations are quadrupole. Quadrupole magnets are useful as they create a magnetic field whose magnitude grows rapidly with the radial distance from its longitudinal axis. This is used in particle beam focusing. The simplest magnetic quadrupole is two identical bar magnets parallel to each other such that the north pole of one is next to the south of the other and vice versa. Such a configuration will have no dipole moment, and its field will decrease at large distances faster than that of a dipole. A stronger version with very little external field involves using a ''k''=3 Halbach cylinder. In some designs of quadrupoles using electromagnets, there are four steel pole tips: two opposing magnetic north poles and two opposing magnetic south poles. The steel is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadrupole
A quadrupole or quadrapole is one of a sequence of configurations of things like electric charge or current, or gravitational mass that can exist in ideal form, but it is usually just part of a multipole expansion of a more complex structure reflecting various orders of complexity. Mathematical definition The quadrupole moment tensor ''Q'' is a rank-two tensor—3×3 matrix. There are several definitions, but it is normally stated in the traceless form (i.e. Q_ + Q_ + Q_ = 0). The quadrupole moment tensor has thus nine components, but because of transposition symmetry and zero-trace property, in this form only five of these are independent. For a discrete system of \ell point charges or masses in the case of a gravitational quadrupole, each with charge q_\ell, or mass m_\ell, and position \vec_\ell = \left(r_, r_, r_\right) relative to the coordinate system origin, the components of the ''Q'' matrix are defined by: : Q_ = \sum_\ell q_\ell\left(3r_ r_ - \left\, \vec_\ell \rig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Focusing
In accelerator physics strong focusing or alternating-gradient focusing is the principle that, using sets of multiple electromagnets, it is possible to make a particle beam simultaneously converge in both directions perpendicular to the direction of travel. By contrast, weak focusing is the principle that nearby circles, described by charged particles moving in a uniform magnetic field, only intersect once per revolution. Earnshaw's theorem shows that simultaneous focusing in two directions transverse to the beam axis at once by a single magnet is impossible - a magnet which focuses in one direction will defocus in the perpendicular direction. However, iron "poles" of a cyclotron or two or more spaced quadrupole magnets (arranged in quadrature) can alternately focus horizontally and vertically, and the net overall effect of a combination of these can be adjusted to focus the beam in both directions. Strong focusing was first conceived by Nicholas Christofilos in 1949 but not pub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multipole Expansion
A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similarly to Taylor series, multipole expansions are useful because oftentimes only the first few terms are needed to provide a good approximation of the original function. The function being expanded may be real- or complex-valued and is defined either on \R^3, or less often on \R^n for some other Multipole expansions are used frequently in the study of electromagnetic and gravitational fields, where the fields at distant points are given in terms of sources in a small region. The multipole expansion with angles is often combined with an expansion in radius. Such a combination gives an expansion describing a function throughout three-dimensional space. The multipole expansion is expressed as a sum of terms with progressively finer angular ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electromagnet
An electromagnet is a type of magnet in which the magnetic field is produced by an electric current. Electromagnets usually consist of wire wound into a coil. A current through the wire creates a magnetic field which is concentrated in the hole in the center of the coil. The magnetic field disappears when the current is turned off. The wire turns are often wound around a magnetic core made from a ferromagnetic or ferrimagnetic material such as iron; the magnetic core concentrates the magnetic flux and makes a more powerful magnet. The main advantage of an electromagnet over a permanent magnet is that the magnetic field can be quickly changed by controlling the amount of electric current in the winding. However, unlike a permanent magnet that needs no power, an electromagnet requires a continuous supply of current to maintain the magnetic field. Electromagnets are widely used as components of other electrical devices, such as motors, generators, electromechanica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octupole
A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles used in the spherical coordinate system (the polar and azimuthal angles) for three-dimensional Euclidean space, \R^3. Similarly to Taylor series, multipole expansions are useful because oftentimes only the first few terms are needed to provide a good approximation of the original function. The function being expanded may be real- or complex-valued and is defined either on \R^3, or less often on \R^n for some other Multipole expansions are used frequently in the study of electromagnetic and gravitational fields, where the fields at distant points are given in terms of sources in a small region. The multipole expansion with angles is often combined with an expansion in radius. Such a combination gives an expansion describing a function throughout three-dimensional space. The multipole expansion is expressed as a sum of terms with progressively finer angular feature ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electric Dipole Moment
The electric dipole moment is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI unit for electric dipole moment is the coulomb- meter (C⋅m). The debye (D) is another unit of measurement used in atomic physics and chemistry. Theoretically, an electric dipole is defined by the first-order term of the multipole expansion; it consists of two equal and opposite charges that are infinitesimally close together, although real dipoles have separated charge.Many theorists predict elementary particles can have very tiny electric dipole moments, possibly without separated charge. Such large dipoles make no difference to everyday physics, and have not yet been observed. (See electron electric dipole moment). However, when making measurements at a distance much larger than the charge separation, the dipole gives a good approximation of the actual electric field. The dipole is represented ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-port Network
A two-port network (a kind of four-terminal network or quadripole) is an electrical network ( circuit) or device with two ''pairs'' of terminals to connect to external circuits. Two terminals constitute a port if the currents applied to them satisfy the essential requirement known as the port condition: the electric current entering one terminal must equal the current emerging from the other terminal on the same port.Gray, §3.2, p. 172Jaeger, §10.5 §13.5 §13.8 The ports constitute interfaces where the network connects to other networks, the points where signals are applied or outputs are taken. In a two-port network, often port 1 is considered the input port and port 2 is considered the output port. It's used in mathematical circuit analysis. Application The two-port network model is used in mathematical circuit analysis techniques to isolate portions of larger circuits. A two-port network is regarded as a " black box" with its properties specified by a matrix of n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its '' ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperplanes are the 1-dimensional lines. This notion can be used in any general space in which the concept of the dimension of a subspace is defined. In different settings, hyperplanes may have different properties. For instance, a hyperplane of an -dimensional affine space is a flat subset with dimension and it separates the space into two half spaces. While a hyperplane of an -dimensional projective space does not have this property. The difference in dimension between a subspace and its ambient space is known as the codimension of with respect to . Therefore, a necessary and sufficient condition for to be a hyperplane in is for to have codimension one in . Technical description In geometry, a hyperplane of an ''n''-dime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electric Permittivity
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter ''ε'' (epsilon), is a measure of the electric polarizability of a dielectric. A material with high permittivity polarizes more in response to an applied electric field than a material with low permittivity, thereby storing more energy in the material. In electrostatics, the permittivity plays an important role in determining the capacitance of a capacitor. In the simplest case, the electric displacement field D resulting from an applied electric field E is :\mathbf = \varepsilon \mathbf. More generally, the permittivity is a thermodynamic function of state. It can depend on the frequency, magnitude, and direction of the applied field. The SI unit for permittivity is farad per meter (F/m). The permittivity is often represented by the relative permittivity ''ε''r which is the ratio of the absolute permittivity ''ε'' and the vacuum permittivity ''ε''0 :\kapp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle Accelerator
A particle accelerator is a machine that uses electromagnetic fields to propel charged particles to very high speeds and energies, and to contain them in well-defined beams. Large accelerators are used for fundamental research in particle physics. The largest accelerator currently active is the Large Hadron Collider (LHC) near Geneva, Switzerland, operated by the CERN. It is a collider accelerator, which can accelerate two beams of protons to an energy of 6.5 TeV and cause them to collide head-on, creating center-of-mass energies of 13 TeV. Other powerful accelerators are, RHIC at Brookhaven National Laboratory in New York and, formerly, the Tevatron at Fermilab, Batavia, Illinois. Accelerators are also used as synchrotron light sources for the study of condensed matter physics. Smaller particle accelerators are used in a wide variety of applications, including particle therapy for oncological purposes, radioisotope production for medical diagnostics, ion imp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Charged Particle Beam
A charged particle beam is a spatially localized group of electrically charged particles that have approximately the same position, kinetic energy (resulting in the same velocity), and direction. The kinetic energies of the particles are much larger than the energies of particles at ambient temperature. The high energy and directionality of charged particle beams make them useful for many applications in particle physics (see Particle beam#Applications and Electron-beam technology). Such beams can be split into two main classes: # ''unbunched beams'' (''coasting beams'' or ''DC beams''), which have no longitudinal substructure in the direction of beam motion. # ''bunched beams'', in which the particles are distributed into pulses (bunches) of particles. Bunched beams are most common in modern facilities, since the most modern particle accelerators require bunched beams for acceleration. Assuming a normal distribution of particle positions and impulses, a charged particle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
