Solenoidal field. The magnetic field inside a toroidal coil (Equation 7.7.3 7.7.3) ...

Consider a point source of particles (e.g. a positron c

Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.This overlooked field momentum arises from the Coulomb electric field of the electric charge and the solenoidal magnetic field of the Dirac string. This implies that the monopole-charge system must either: (i) carry a ``hidden momentum" in the string, indicating that the string is real, or (ii) that the monopole-charge system violates the ...external solenoidal field. These gradients are about three times larger than those available with the conventional iron/copper quadrupoles now used in the SLC. Superconducting quadrupoles of two lengths have been specified For the SLC triplets. The effective magnetic length of type Q, is 66.498 + 0.305cm and ofBut a solenoidal field, besides having a zero divergence, also has the additional connotation of having non-zero curl (i.e., rotational component). Otherwise, if an incompressible flow also has a curl of zero, so that it is also irrotational, then the flow velocity field is actually Laplacian. Difference from materialhigher field of 20-T. One can inject more parallel beam into the muon linac by producing pions in the higher field and, according to our simulations, the muon yield in the case of 20-T capture will be 4-times higher compared to the 5-T solenoidal capture (muon yield after the acceleration). The 20-T field can be realized by pulsedThe electron lens is based on a 5–10 keV, 1–2 A electron beam, shaped using a 0.7 m long, 0.8 T solenoidal magnetic field. A cryogen-free superconducting solenoid has been designed to provide this solenoidal field, taking into consideration the constraints on space, utilities, and infrastructure in the IOTA experimental hall.A conservative vector field (also called a path-independent vector field) is a vector field $\dlvf$ whose line integral $\dlint$ over any curve $\dlc$ depends only on the endpoints of $\dlc$. The integral is independent of the path that $\dlc$ takes going from its starting point to its ending point. The below applet illustrates the two-dimensional conservative vector …gradient of a scalar and if in addition the vector field is solenoidal, then the scalar potential is the solution of the Laplace equation. 2 2, irrotational flow 0 , incompressible, irrotational flow ϕ ϕ ϕ =−∇ ∇• =Θ=−∇ ∇• = =−∇ v v v Also, if the velocity field is solenoidal then the velocity can be expressed as thePublisher: McGraw-Hill Education. Introductory Mathematics for Engineering Applicat... Advanced Math. ISBN: 9781118141809. Author: Nathan Klingbeil. Publisher: WILEY. SEE MORE TEXTBOOKS. Solution for A vector field which has a vanishing divergence is called as Rotational field Solenoidal field Irrotational field Hemispheroidal field. Feb 24, 2022 · A solenoidal tangent field, mathematically speaking, is one whose divergence vanishes. They are also called incompressible. I understand why they are called incompressible — a fluid flow is called incompressible when a small fluid parcel retains constant density when it moves along along a streak line. This means that its material derivative ... In the mathematics of vector calculus, a solenoidal vector field is also known as a divergence-free vector field, an incompressible vector field, or a transverse vector field. …Prepare for exam with EXPERTs notes - unit 4 line integrals for utkal university odisha, mathematics-bsch-sem-5I understand a solenoidal vector field implies the existence of another vector field, of which it is the curl: [tex]S= abla X A[/tex] because the divergence of the curl of any vector field is zero. But what if the vector field is conservative instead? I guess in this case it is not necessarly implied the existence of a vector potential.Abstract. The role is explained of toroid moments and average 2 n -power radii in the structure of multipole expansions in the framework of Maxwell-Lorentz electrodynamics. A complete scheme of multipole expansions is given in terms of distribution theory. A number of particular applications of toroid moments is considered in condensed-matter ...Cavities within a multi-Tesla solenoidal field. To determine the effect of very large solenoidal magnetic fields on the generation of dark current, x-rays and on the breakdown voltage gradients of vacuum RF cavities, a test facility has been established at Fermilab in Lab G. This facility consists of a 12 MW 805 MHz RF stationfield and circuit equations based on solenoidal coils for wireless power transfer ISSN 1755-4535 Received on 25th June 2019 Revised 27th October 2019 Accepted on 18th November 2019 E-First on 21st January 2020 doi: 10.1049/iet-pel.2019.0764 www.ietdl.org Song Xianjin1, Liu Guoqiang2, Zhang Chao2, Li Yanhong2, Xu Xiaoyu2The solenoidal field is quasi-stationary in the reference frame of the laser pulse and can be used for guiding electron beams. It can also provide synchrotron radiation beam emittance cooling for ...Aug 1, 1997 · @article{osti_304187, title = {Intense nonneutral beam propagation in a periodic solenoidal field using a macroscopic fluid model with zero thermal emittance}, author = {Davidson, R C and Stoltz, P and Chen, C}, abstractNote = {A macroscopic fluid model is developed to describe the nonlinear dynamics and collective processes in an intense high-current beam propagating in the z-direction ... In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) …solenoid: [noun] a coil of wire usually in cylindrical form that when carrying a current acts like a magnet so that a movable core is drawn into the coil when a current flows and that is used especially as a switch or control for a mechanical device (such as a valve).Solenoidal fields, such as the magnetic flux density B→ B →, are for similar reasons sometimes represented in terms of a vector potential A→ A →: B→ = ∇ × A→ (2.15.1) (2.15.1) B → = ∇ × A →. Thus, B→ B → automatically has no divergence.The magnetic field carried by light can unexpectedly steer waves in structured media through optical spin-orbit interaction, according to researchers from France, Switzerland, and China. ... The longitudinal electric component of Belinfante's elusive spin momentum density is determined, a solenoidal field quantity often referred to as "virtual ...If you are in 2D and if you want more physical setups, I suggest you consider potential flows.. There are various ways to construct these potential flows which are always divergence free and which satisfy certain boundary conditions.. In the wikipedia article, you find the power laws, that describes flows around plates, edges, or in corners.. Also, there …Then the irrotational and solenoidal field proposed to satisfy the boundary conditions is the sum of that uniform field and the field of a dipole at the origin, as given by (8.3.14) together with the definition (8.3.19). By design, this field already approaches the uniform field at infinity. To satisfy the condition that n o H = 0 at r = R,A second explanatory theory is discussed in which radiation from the cloud tops of the "intertropical convergence zone" locally reverses the equatorial solenoidal field to produce two new lines of convergence, one on each side of the equator.View full document. Students also studiedApr 1, 2023 · solenoid: [noun] a coil of wire usually in cylindrical form that when carrying a current acts like a magnet so that a movable core is drawn into the coil when a current flows and that is used especially as a switch or control for a mechanical device (such as a valve). Abstract. The role is explained of toroid moments and average 2 n -power radii in the structure of multipole expansions in the framework of Maxwell-Lorentz electrodynamics. A complete scheme of multipole expansions is given in terms of distribution theory. A number of particular applications of toroid moments is considered in condensed-matter ...The solenoidal field focuses and the gradient (≈0.28 T/m along the magnetic axis) directs outgoing muons (and undecayed pions and antiprotons) into the second section, the transport solenoid. Nonetheless, the overall efficiency is quite high: approximately 0.005 muons/proton-on-target reach the aluminum stopping target; the muons have a peak ...The main equations for compressible flow include the above continuity equation and the momentum equation from the Navier-Stokes equation. The main equation of motion is: In this equation, μ and λ are proportionality constants that define the viscosity and the fluid’s stress-strain relationship. The value of λ is generally a function of ...The peak rf field at the cathode is 150 MV/m with extraction 57 {degree} ahead of the rf peak. A solenoidal field near the cathode allows the compensation of the initial emittance growth by the end of the injection linac. Spatial and temporal shaping of the laser pulse striking the cathode will reduce the compensated emittance even further.Section snippets Formulation. Flows of electrically conducting fluids in magnetic fields are described by the non-dimensional momentum equation and Ohm's law, N − 1 ∂ t + v · ∇ v = − ∇ p + Ha − 2 ∇ 2 v + j × B, j = − ∇ ϕ + v × B. The fluid is assumed incompressible, ∇ · v = 0, and charge conservation ∇ · j = 0 is satisfied by a Poisson …4. [15 points]: Consider a vector field which is spherically symmetric and directed away from the origin everywhere, i.e. v=f(r)r^ 4.1. Show that v is irrotational no matter the form of the function f(r). 4.2. Show that v is solenoidal (for r>0 ) only if the function satisfies f(r)=Brα where α is a constant that you must determine. 4.3.In vector calculus a solenoidal vector field (also known as an incompressible vector field, a divergence-free vector field, or a transverse vector field) is a vector field v with divergence zero at all points in the field: ∇ ⋅ v = 0.We thus see that the class of irrotational, solenoidal vector fields conicides, locally at least, with the class of gradients of harmonic functions. Such fields are prevalent in electrostatics, in which the Maxwell equation. ∇ ×E = −∂B ∂t (7) (7) ∇ × E → = − ∂ B → ∂ t. becomes. ∇ ×E = 0 (8) (8) ∇ × E → = 0. in the ... Conservative and Solenoidal fields# In vector calculus, a conservative field is a field that is the gradient of some scalar field. Conservative fields have the property that their line integral over any path depends only on the end-points, and is independent of the path between them. A conservative vector field is also said to be ...The simplest, most obvious, and oldest example of a non-irrotational field (the technical term for a field with no irrotational component is a solenoidal field) is a magnetic field. A magnetic compass finds geomagnetic north because the Earth's magnetic field causes the metal needle to rotate until it is aligned. Share.A conservative vector field (also called a path-independent vector field) is a vector field $\dlvf$ whose line integral $\dlint$ over any curve $\dlc$ depends only on the endpoints of $\dlc$. The integral is independent of the path that $\dlc$ takes going from its starting point to its ending point. The below applet illustrates the two-dimensional conservative vector field $\dlvf(x,y)=(x,y)$.Cancer is a big risk for astronauts in space, but a shield in development may help. Read more about force fields for spacecraft at HowStuffWorks Now. Advertisement Astronauts face myriad dangers in space, and at least one is perfectly famil...V. A. Solonnikov, "On boundary-value problems for the system of Navier-Stokes equations in domains with noncompact boundaries," Usp. Mat. Nauk, 32, No. 5, 219-220 (1977). Google Scholar. V. A. Solonnikov and K. I. Piletskas, "On some spaces of solenoidal vectors and the solvability of a boundary-value problem for the system of Navier ...This is similar to Poisson's equation but it is terms of a vector potential. e.g. magnetic field within a conductor carrying a steady current, Rotational motion of an incompressible fluid, time varying electromagnetic field in charge free and current free region. Neither irrotational nor solenoidal field for this curl RAbstract: This article is a narrative and systematic review on the electromagnetic optimization literature of superconducting solenoidal magnets and coils. Superconducting solenoids are the basis of magnetic resonance imaging machines and superconducting energy storage systems. As the literature has evolved and many optimization techniques have been used, in this article, we review the ...Jun 4, 2003 · Future linear colliders may require a nonzero crossing angle between the two beams at the interaction point (IP). This requirement in turn implies that the beams will pass through the strong interaction region solenoid with an angle, and thus that the component of the solenoidal field perpendicular to the beam trajectory is nonzero. The interaction of the beam and the solenoidal field in the ... The main equations for compressible flow include the above continuity equation and the momentum equation from the Navier-Stokes equation. The main equation of motion is: In this equation, μ and λ are proportionality constants that define the viscosity and the fluid's stress-strain relationship. The value of λ is generally a function of ...Integrability conditions. If F is a conservative vector field (also called irrotational, curl-free, or potential), and its components have continuous partial derivatives, the potential of F with respect to a reference point r 0 is defined in terms of the line integral: = = (()) ′ (),where C is a parametrized path from r 0 to r, (),, =, =.The fact that the line integral depends on the path C ...Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.Industrial solenoid valves are easy to find when you know where you’re looking. Check out this guide to finding the right industrial solenoid valves for your business so you can order your solenoid valves today.Flat-beam transform configuration. A solenoid generates an axial magnetic field at the location of the cathode. Once the beam leaves the solenoidal field, it encounters three skew quadrupole which remove the x − y correlations. This figure is courtesy of Kip Bishofberger. Reuse & PermissionsScalar potential. In mathematical physics, scalar potential, simply stated, describes the situation where the difference in the potential energies of an object in two different positions depends only on the positions, not upon the path taken by the object in traveling from one position to the other. It is a scalar field in three-space: a ... Another more restrictive representation can be derived if one requests the field F to be solenoidal, as the magnetic field is requested to be because of Eq. 2. Indeed, if a solenoidal field is written in the form of Eq. 45, then S must be harmonic, i.e., it must satisfy Laplace's equation (Eq. 11). However, consider the following identity for ...S2E: Solenoidal Focusing The field of an ideal magnetic solenoid is invariant under transverse rotations about it©s axis of symmetry ( ) can be expanded in terms of the on-axis field as as: See Appendix D or Reiser, Theory and Design of Charged Particle Beams , Sec. 3.3.1 solenoid.png Vacuum Maxwell equations: Imply can be expressed inThe ATLAS toroids use a series of eight coils to provide a magnetic field of up to 3.5 Tesla, used to measure the momentum of muons. There are three toroid magnets in ATLAS: two at the ends of the experiment, and one massive toroid surrounding the centre of the experiment.. At 25.3 m in length, the central toroid is the largest toroidal magnet ever constructed.Solenoid valves are used to replace manual valves so they can be controlled remotely valve, according to Crane Engineering. The electronic controls in the solenoid valve are what makes it possible to control it this way.Problem 1: (a)Suppose A is a vector function given by A = A x(x;y;z)^x+ A y(x;y;z)^y+ A z(x;y;z)^z.Evaluate the quantity r(r A) showing all the steps explicitly. (3 marks) (b)If r and r0are the observation and source position vectors, respectively, and if r = jr r0jis the magnitude of the displacement vector.• For a magnetic field:∇·B= 0, that is there are no sources or sinks of magnetic field, a solenoidal filed. • For an electric field:∇·E= ρ/ε, that is there are sources of electric field.. Consider a vector field F that represents a fluid velocity: The divergence of F at a point in a fluid is a measure of the rateIntegrability conditions. If F is a conservative vector field (also called irrotational, curl-free, or potential), and its components have continuous partial derivatives, the potential of F with respect to a reference point r 0 is defined in terms of the line integral: = = (()) ′ (),where C is a parametrized path from r 0 to r, (),, =, =.The fact that the line integral depends on the …If the magnetic field H is sufficiently weak, the Larmor frequency will be small compared to the frequencies of the finite motion of the system of charges. Then we may consider the averages, over times small compared to the period 2π/Ω, of quantities describing the system. ... Show that ∇φ is both solenoidal and irrotational. 3.6.15. Show ...Divergence at (1,1,-0.2) will give zero. As the divergence is zero, field is solenoidal. Alternate/Shortcut: Without calculation, we can easily choose option “0, solenoidal”, as by theory when the divergence is zero, the vector is solenoidal. “0, solenoidal” is the only one which is satisfying this condition.The simplest and most common inductors are solenoidal coils. These inductors are cylindrical coils wound around air-core or ferromagnetic core. These inductors are easiest to construct. A solenoidal or cylindrical coil can be easily designed to vary inductance by incorporating a mechanism to slide in and out ferromagnetic core of the coil.Example 2 The u velocity component of a steady, two-dimensional, incompressible flow field is uax bxy=−2, where a and b are constants. Velocity component v is unknown. Generate an expression for v as a function of x and y.$\begingroup$ "As long as the current is a linear function of time, induced electric field in the region close to the solenoid does not change in time and has zero curl." ." Also, "If the current does not change linearly, acceleration of charges changes in time, and thus induced electric field outside is not constant in time, but changes inHelmholtz's Theorem. Any vector field satisfying. (1) (2) may be written as the sum of an irrotational part and a solenoidal part, (3) where. (4) (5) See also. …We consider the possibility of using Dirac's ideas of the deformation of Poisson brackets in nonholonomic mechanics. As an example, we analyze the composition of external forces that do no work and reaction forces of nonintegrable constraints in the model of a nonholonomic Chaplygin sphere on a plane. We prove that, when a solenoidal field is applied, the general mechanical energy, the ...The muon collider captures pions using solenoidal fields. The pion are converted to muons as they are bunched in an RF phase rotation system. Solenoids are used to focus the muons as their emitance is reduced during cooling. ... The cooling channel for a neutrino factory consists of a series of alternating field solenoidal cells. The first ...Typically any vector field on a simply-connected domain could be decomposed into the sum of an irrotational (curl-free), a solenoidal (divergence-free) and a harmonic (divergence-free and curl-free) field. This technique is known as Hodge-Helmholtz decomposition and is basically achieved by minimizing the energy functionals for the irrotational ...4.6: Gradient, Divergence, Curl, and Laplacian. In this final section we will establish some relationships between the gradient, divergence and curl, and we will also introduce a new quantity called the Laplacian. We will then show how to write these quantities in cylindrical and spherical coordinates.May 18, 2015 · SOLENOIDAL AND IRROTATIONAL FIELDS The with null divergence is called solenoidal, and the field with null-curl is called irrotational field. The divergence of the curl of any vector field A must be zero, i.e. ∇· (∇×A)=0 Which shows that a solenoidal field can be expressed in terms of the curl of another vector field or that a curly field ... We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain in two and three space dimensions. In contrast to previous works, we study a new model recently developed by Abels et al. for fluids with different densities, which leads to a solenoidal velocity field. The model is given by a non-homogeneous Navier ...gradient of a scalar and if in addition the vector field is solenoidal, then the scalar potential is the solution of the Laplace equation. 2 2, irrotational flow 0 , incompressible, irrotational flow ϕ ϕ ϕ =−∇ ∇• =Θ=−∇ ∇• = =−∇ v v v Also, if the velocity field is solenoidal then the velocity can be expressed as theVALUME 51, NLJMESER j. P H YSICAL RE VIE%" LETTERS 4 JUL+ 1983 Cross Sections for Dielectronic Recombination of B2+ and C3+ via 2s 2p Excitation P. F. Dittner, S ...Considering the quasi-isentropic linear régime, in which the physics of the solenoidal modes is modified in a negligible way by dilatational effects, all results obtained in Sect. 4.3 can be applied immediately to the solenoidal field \({{\varvec{u}}}^s\) and related statistical quantities.Feb 28, 2019 · The solenoidal coils are assumed to be perfect conductors and therefore, the electric field is independent of the magnetic field and approaches zero in the coils. Verify Stoke's theorem for the vector field vector F = (2x - y)i - yz 2 j - y 2 zk over the upper half surface of x 2 + y 2 + z 2 = 1 bounded by its projection on the xy-plane.A Beltrami field is an eigenvector of the curl operator. Beltrami fields describe steady flows in fluid dynamics and force free magnetic fields in plasma turbulence. By application of the Lie-Darboux theorem of differential geoemtry, we prove a local representation theorem for Beltrami fields. We find that, locally, a Beltrami field has a standard form amenable to an Arnold-Beltrami-Childress ...Use Ampere's Law to determine the macroscopic magnetic field B(r) GG a perpendicular distance r away from a (infinitely) long, straight filamentary wire carrying steady current, I. We already know that (here) B ϕˆ G & (i.e. solenoidal/phi field). Use the integral form of Ampere's Law, take an "Amperian" loop contour C, enclosing thePhysics and Measurement CHAPTER OUTLINE 1.1 Standards of Length, Mass, and Time 1.2 Matter and Model-Building 1.3 Dimensional Analysis 1.4 Conversion of Units 1.5 Estimates and Order-of- Magnitude Calculations 1.6 Signifi cant FiguresA solenoid ( / ˈsoʊlənɔɪd / [1]) is a type of electromagnet formed by a helical coil of wire whose length is substantially greater than its diameter, [2] which generates a controlled magnetic field. The coil can produce a uniform magnetic field in a volume of space when an electric current is passed through it. Vector Fields. Quiver, compass, feather, and stream plots. Vector fields can model velocity, magnetic force, fluid motion, and gradients. Visualize vector fields in a 2-D or 3-D view using the quiver, quiver3, and streamline functions. You can also display vectors along a horizontal axis or from the origin.Question: Question \#6) If V⋅B=0,B is solenoidal and thus B can be expressed as the curl of another vector field, A like B=∇×A (T). If the scalar electric potential is given by V, derive nonhomogeneous wave equations for vector potential A and scalar potential V. Make sure to include Lorentz condition in your derivation. This question hasn ...field lines of a solenoidal field have no end points; they must therefore consist of closed loops. And conversely, there can be no vortices in an irrotational field. As we have hinted a number of times, Equations (7.2) to (7.5) are not complete; there are other ways in which the fields can be produced. We will treat electromagnetic inductionConductor moving in a magnetic field. The moving magnet and conductor problem is a famous thought experiment, originating in the 19th century, concerning the intersection of classical electromagnetism and special relativity.In it, the current in a conductor moving with constant velocity, v, with respect to a magnet is calculated in the frame of reference of the magnet and in the frame of ...The solenoidality of the velocity field is valid on the theoretical level, for example on the differential form of governing equations. However, the divergence of the velocity field on an arbitrary numerical setup and process is not strictly zero; therefore, the solenoidal field cannot be strictly applied in practice.We say that a pre-poloidal field is poloidal whenever it is solenoidal. The poloidal-field generator is a second-order differential operator on C ∞ (R ˙ N) given by (8) D = σ σ − r ∂ r ′ ∇ σ, which maps every scalar field f to the poloidal field D f ∈ P (R ˙ N). The following fact is fundamental: Proposition 2.2. Let u: R N → ...A solenoidal field near the cathode allows the compensation of the initial emittance growth by the end of the injection linac. Spatial and temporal shaping of the laser pulse striking the cathode will reduce the compensated emittance even further. Also, to minimize the contribution of the thermal emittance fromVerify Stoke's theorem for the vector field vector F = (2x - y)i - yz 2 j - y 2 zk over the upper half surface of x 2 + y 2 + z 2 = 1 bounded by its projection on the xy-plane.A solenoidal tangent field, mathematically speaking, is one whose divergence vanishes. They are also called incompressible. I understand why they are called incompressible — a fluid flow is called incompressible when a small fluid parcel retains constant density when it moves along along a streak line. This means that its material derivative ...The SiD Detector is one of two detector designs for the future International Linear Collider (ILC) that were validated in 2012 SiD features a compact, cost-constrained design for precision Higgs and other measurements, and sensitivity to a wide range of possible new phenomena. A robust silicon vertex and tracking system, combined with a five Tesla central solenoidal field, provides excellent ...Subscribe to his free Masterclasses at Youtube & discussions at Telegram SanfoundryClasses . This set of Vector Calculus Multiple Choice Questions & Answers (MCQs) focuses on “Divergence and Curl of a Vector Field”. 1. What is the divergence of the vector field at the point (1, 2, 3). a) 89 b) 80 c) 124 d) 100 2.In the mathematics of vector calculus, a solenoidal vector field is also known as a divergence-free vector field, an incompressible vector field, or a transverse vector field. It is a type of transverse vector field v with divergence equal to zero at all of the points in the field, that is ∇ · v = 0. It can be said that the field has no ... The field is more concentrated in the center of the loop than outside the loop. By adding more loops to a coil, you create an increasingly stronger magnetic field. This is known as a solenoid, demonstrated here: Instructions. Observe the coil of wire connected to a battery in a simple circuit. Not the coil is embedded in a table covered with ...Due to the well-known bell-like curve shape of the solenoidal field, the integration step has to include the field maximum. Practically, it means that to be consistent with the traditional matrix the integration interval Δ z has to include the whole effective magnetic field length L , beyond which the field is assumed as null, yielding Δ z = L .. mum field about 1.6 T. Thus the wave numberSolenoidal field . D. Irrotational field. Detailed Magnetic induction, also called electromagnetic induction refers to the production of voltage (or EMF) across an electrical conductor placed inside a varying magnetic field. According to Faraday's law, for a closed circuit, the induced electromotive force is equal to the rate of change of the magnetic flux enclosed by the circuit. Posture can affect a lot of things, including our confidence and ho The magnetic field strength, 𝐵 , inside the center of a solenoid is found using the equation 𝐵 = 𝜇 𝑁 𝐼 𝐿 , where 𝐼 is the current of the solenoid, 𝑁 ...A pressure field is a two-component vector force field, which describes in a covariant way the dynamic pressure of individual particles and the pressure emerging in systems with a number of closely interacting particles. The pressure field is a general field component, which is represented in the Lagrangian and Hamiltonian of an arbitrary physical system including the term with the energy of ... For a purely solenoidal field, the optical effects which a...

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