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53-508: The A8X has three cores clocked at 1.5 GHz, a more powerful GPU compared to the A8 and it contains 3 billion transistors. With an extra 100 MHz and an additional core, the A8X performs around 13% better on single threaded and 55% better on multithreaded operations than the A8 inside the iPhone 6 and iPhone 6 Plus. Further comparison to the A8 shows that the A8X uses a metal heat spreader , which

106-474: A r g m a x n ^ n ^ p d q d t ( A , p , n ^ ) . {\displaystyle \mathbf {I} (A,\mathbf {p} )={\underset {\mathbf {\hat {n}} }{\operatorname {arg\,max} }}\mathbf {\hat {n}} _{\mathbf {p} }{\frac {\mathrm {d} q}{\mathrm {d} t}}(A,\mathbf {p} ,\mathbf {\hat {n}} ).} In this case, there

159-408: A surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics . For transport phenomena , flux is a vector quantity, describing the magnitude and direction of the flow of a substance or property. In vector calculus flux is a scalar quantity, defined as the surface integral of the perpendicular component of a vector field over

212-423: A (single) scalar: j = I A , {\displaystyle j={\frac {I}{A}},} where I = lim Δ t → 0 Δ q Δ t = d q d t . {\displaystyle I=\lim _{\Delta t\to 0}{\frac {\Delta q}{\Delta t}}={\frac {\mathrm {d} q}{\mathrm {d} t}}.} In this case

265-692: A 1998 patent. On October 14, 2015, a district judge found Apple guilty of infringing U.S. patent US 5781752   , "Table based data speculation circuit for parallel processing computer", on the Apple A7 and A8 processors. The patent is owned by Wisconsin Alumni Research Foundation (WARF), a firm affiliated with the University of Wisconsin . On July 24, 2017, Apple was ordered to pay WARF $ 506 million for patent infringement. Apple filed an appellate brief on October 26, 2017, with

318-440: A central quantity and proceeds to derive the now well-known expressions of flux in terms of temperature differences across a slab, and then more generally in terms of temperature gradients or differentials of temperature, across other geometries. One could argue, based on the work of James Clerk Maxwell , that the transport definition precedes the definition of flux used in electromagnetism . The specific quote from Maxwell is: In

371-402: A closed surface, in other words the net outflux from a 3D region, is found by adding the local net outflow from each point in the region (which is expressed by the divergence ). If the surface is not closed, it has an oriented curve as boundary. Stokes' theorem states that the flux of the curl of a vector field is the line integral of the vector field over this boundary. This path integral

424-467: A flux according to the electromagnetism definition, the corresponding flux density , if that term is used, refers to its derivative along the surface that was integrated. By the Fundamental theorem of calculus , the corresponding flux density is a flux according to the transport definition. Given a current such as electric current—charge per time, current density would also be a flux according to

477-420: A hotter source to a colder heat sink or heat exchanger . There are two thermodynamic types, passive and active. The most common sort of passive heat spreader is a plate or block of material having high thermal conductivity , such as copper , aluminum, or diamond. An active heat spreader speeds up heat transfer with expenditure of energy as work supplied by an external source. A heat pipe uses fluids inside

530-494: A magnetic field opposite to the change. This is the basis for inductors and many electric generators . Using this definition, the flux of the Poynting vector S over a specified surface is the rate at which electromagnetic energy flows through that surface, defined like before: The flux of the Poynting vector through a surface is the electromagnetic power , or energy per unit time , passing through that surface. This

583-532: A positive point charge can be visualized as a dot radiating electric field lines (sometimes also called "lines of force"). Conceptually, electric flux can be thought of as "the number of field lines" passing through a given area. Mathematically, electric flux is the integral of the normal component of the electric field over a given area. Hence, units of electric flux are, in the MKS system , newtons per coulomb times meters squared, or N m /C. (Electric flux density

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636-430: A sealed case. The fluids circulate either passively, by spontaneous convection, triggered when a threshold temperature difference occurs; or actively, because of an impeller driven by an external source of work. Without sealed circulation, energy can be carried by transfer of fluid matter, for example externally supplied colder air, driven by an external source of work, from a hotter body to another external body, though this

689-420: A surface. The word flux comes from Latin : fluxus means "flow", and fluere is "to flow". As fluxion , this term was introduced into differential calculus by Isaac Newton . The concept of heat flux was a key contribution of Joseph Fourier , in the analysis of heat transfer phenomena. His seminal treatise Théorie analytique de la chaleur ( The Analytical Theory of Heat ), defines fluxion as

742-410: A surface. By contrast, according to the electromagnetism definition, flux is the integral over a surface; it makes no sense to integrate a second-definition flux for one would be integrating over a surface twice. Thus, Maxwell's quote only makes sense if "flux" is being used according to the transport definition (and furthermore is a vector field rather than single vector). This is ironic because Maxwell

795-559: A very high thermal conductivity. Synthetic diamond is used as submounts for high-power integrated circuits and laser diodes. Composite materials can be used, such as the metal matrix composites (MMCs) copper–tungsten , AlSiC ( silicon carbide in aluminium matrix), Dymalloy (diamond in copper-silver alloy matrix), and E-Material ( beryllium oxide in beryllium matrix). Such materials are often used as substrates for chips, as their thermal expansion coefficient can be matched to ceramics and semiconductors. In May 2022, researchers at

848-404: Is j  cos  θ , while the component of flux passing tangential to the area is j  sin  θ , but there is no flux actually passing through the area in the tangential direction. The only component of flux passing normal to the area is the cosine component. For vector flux, the surface integral of j over a surface S , gives the proper flowing per unit of time through

901-417: Is also called circulation , especially in fluid dynamics. Thus the curl is the circulation density. We can apply the flux and these theorems to many disciplines in which we see currents, forces, etc., applied through areas. An electric "charge," such as a single proton in space, has a magnitude defined in coulombs. Such a charge has an electric field surrounding it. In pictorial form, the electric field from

954-408: Is drawn by curves (field lines) following the "flow"; the magnitude of the vector field is then the line density, and the flux through a surface is the number of lines. Lines originate from areas of positive divergence (sources) and end at areas of negative divergence (sinks). See also the image at right: the number of red arrows passing through a unit area is the flux density, the curve encircling

1007-408: Is maximized across the disk that is perpendicular to it. The unit vector thus uniquely maximizes the function when it points in the "true direction" of the flow. (Strictly speaking, this is an abuse of notation because the "arg max" cannot directly compare vectors; we take the vector with the biggest norm instead.) These direct definitions, especially the last, are rather unwieldy. For example,

1060-399: Is no fixed surface we are measuring over. q is a function of a point, an area, and a direction (given by a unit vector n ^ {\displaystyle \mathbf {\hat {n}} } ), and measures the flow through the disk of area A perpendicular to that unit vector. I is defined picking the unit vector that maximizes the flow around the point, because the true flow

1113-430: Is not exactly heat transfer as defined in physics. Exemplifying increase of entropy according to the second law of thermodynamics, a passive heat spreader disperses or "spreads out" heat, so that the heat exchanger(s) may be more fully utilized. This has the potential to increase the heat capacity of the total assembly, but the additional thermal junctions limit total thermal capacity. The high conduction properties of

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1166-469: Is sometimes referred to as the probability current or current density, or probability flux density. As a mathematical concept, flux is represented by the surface integral of a vector field , where F is a vector field , and d A is the vector area of the surface A , directed as the surface normal . For the second, n is the outward pointed unit normal vector to the surface. The surface has to be orientable , i.e. two sides can be distinguished:

1219-604: Is the concentration ( mol /m ) of component A. This flux has units of mol·m ·s , and fits Maxwell's original definition of flux. For dilute gases, kinetic molecular theory relates the diffusion coefficient D to the particle density n = N / V , the molecular mass m , the collision cross section σ {\displaystyle \sigma } , and the absolute temperature T by D = 2 3 n σ k T π m {\displaystyle D={\frac {2}{3n\sigma }}{\sqrt {\frac {kT}{\pi m}}}} where

1272-407: Is the vector area  – combination A = A n ^ {\displaystyle \mathbf {A} =A\mathbf {\hat {n}} } of the magnitude of the area A through which the property passes and a unit vector n ^ {\displaystyle \mathbf {\hat {n}} } normal to the area. Unlike in the second set of equations,

1325-509: Is the electric flux per unit area, and is a measure of strength of the normal component of the electric field averaged over the area of integration. Its units are N/C, the same as the electric field in MKS units.) Two forms of electric flux are used, one for the E -field: and one for the D -field (called the electric displacement ): This quantity arises in Gauss's law – which states that

1378-530: Is zero. As mentioned above, chemical molar flux of a component A in an isothermal , isobaric system is defined in Fick's law of diffusion as: J A = − D A B ∇ c A {\displaystyle \mathbf {J} _{A}=-D_{AB}\nabla c_{A}} where the nabla symbol ∇ denotes the gradient operator, D AB is the diffusion coefficient (m ·s ) of component A diffusing through component B, c A

1431-529: The D -field flux equals the charge Q A within it. Here the expression "flux of" indicates a mathematical operation and, as can be seen, the result is not necessarily a "flow", since nothing actually flows along electric field lines. The magnetic flux density ( magnetic field ) having the unit Wb/m ( Tesla ) is denoted by B , and magnetic flux is defined analogously: with the same notation above. The quantity arises in Faraday's law of induction , where

1484-615: The U.S. Court of Appeals for the Federal Circuit , that argued that Apple did not infringe on the patent owned by the Wisconsin Alumni Research Foundation. On September 28, 2018, the ruling was overturned on appeal and the award thrown out by the U.S. Federal Circuit Court of Appeals. The patent expired in December 2016. Heat spreader A heat spreader transfers energy as heat from

1537-556: The University of Illinois at Urbana-Champaign and University of California, Berkeley devised a new solution that could cool modern electronics more efficiently than other existing strategies. Their proposed method is based on the use of heat spreaders consisting of an electrical insulating layer of poly (2-chloro-p-xylylene) ( Parylene C) and a coating of copper. This solution would also require less expensive materials. Flux density Flux describes any effect that appears to pass or travel (whether it actually moves or not) through

1590-498: The A8 does not, and it doesn't use the package on package configuration with included RAM which the A8 does. This is similar to how the older "X" variants, the A5X and A6X , were designed. Instead the A8X in the iPad Air 2 uses an external 2 GB RAM module. In a first for Apple, the A8X is reported to have a semi-custom GPU. The A8X uses an 8-cluster GPU based on Imagination Technologies PowerVR Series 6XT architecture. Officially,

1643-424: The area at an angle θ to the area normal n ^ {\displaystyle \mathbf {\hat {n}} } , then the dot product j ⋅ n ^ = j cos ⁡ θ . {\displaystyle \mathbf {j} \cdot \mathbf {\hat {n}} =j\cos \theta .} That is, the component of flux passing through the surface (i.e. normal to it)

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1696-499: The area of that cross section, or the amount of sunlight energy that lands on a patch of ground each second divided by the area of the patch, are kinds of flux. Here are 3 definitions in increasing order of complexity. Each is a special case of the following. In all cases the frequent symbol j , (or J ) is used for flux, q for the physical quantity that flows, t for time, and A for area. These identifiers will be written in bold when and only when they are vectors. First, flux as

1749-406: The arg max construction is artificial from the perspective of empirical measurements, when with a weathervane or similar one can easily deduce the direction of flux at a point. Rather than defining the vector flux directly, it is often more intuitive to state some properties about it. Furthermore, from these properties the flux can uniquely be determined anyway. If the flux j passes through

1802-428: The case of fluxes, we have to take the integral, over a surface, of the flux through every element of the surface. The result of this operation is called the surface integral of the flux. It represents the quantity which passes through the surface. According to the transport definition, flux may be a single vector, or it may be a vector field / function of position. In the latter case flux can readily be integrated over

1855-445: The curve ∂ A {\displaystyle \partial A} , with the sign determined by the integration direction. The time-rate of change of the magnetic flux through a loop of wire is minus the electromotive force created in that wire. The direction is such that if current is allowed to pass through the wire, the electromotive force will cause a current which "opposes" the change in magnetic field by itself producing

1908-457: The field, the flux for the sides is zero and there is an equal and opposite flux at both ends of the tube. This is a consequence of Gauss's Law applied to an inverse square field. The flux for any cross-sectional surface of the tube will be the same. The total flux for any surface surrounding a charge q is q / ε 0 . In free space the electric displacement is given by the constitutive relation D = ε 0 E , so for any bounding surface

1961-442: The flux of the electric field E out of a closed surface is proportional to the electric charge Q A enclosed in the surface (independent of how that charge is distributed), the integral form is: where ε 0 is the permittivity of free space . If one considers the flux of the electric field vector, E , for a tube near a point charge in the field of the charge but not containing it with sides formed by lines tangent to

2014-422: The heat exchanger. For instance, this may be because it is air-cooled, giving it a lower heat transfer coefficient than if it were liquid-cooled. A high enough heat exchanger transfer coefficient is sufficient to avoid the need for a heat spreader. The use of a heat spreader is an important part of an economically optimal design for transferring heat from high to low heat flux media. Examples include: Diamond has

2067-535: The largest implementation of Rogue is a 6-cluster design, indicating that Apple has made customizations to the design in order to provide higher performance. This GPU is referred to as the GXA6850, with the "A" denoting the Apple customization. The A8X has video codec encoding support for H.264 . It has decoding support for H.264, MPEG‑4 , and Motion JPEG . The A8X's branch predictor has been claimed to infringe on

2120-405: The magnetic flux is time-dependent either because the boundary is time-dependent or magnetic field is time-dependent. In integral form: where d ℓ is an infinitesimal vector line element of the closed curve ∂ A {\displaystyle \partial A} , with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to

2173-412: The most common forms of flux from the transport phenomena literature are defined as follows: These fluxes are vectors at each point in space, and have a definite magnitude and direction. Also, one can take the divergence of any of these fluxes to determine the accumulation rate of the quantity in a control volume around a given point in space. For incompressible flow , the divergence of the volume flux

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2226-727: The probability of finding a particle in a differential volume element d r is d P = | ψ | 2 d 3 r . {\displaystyle dP=|\psi |^{2}\,d^{3}\mathbf {r} .} Then the number of particles passing perpendicularly through unit area of a cross-section per unit time is the probability flux; J = i ℏ 2 m ( ψ ∇ ψ ∗ − ψ ∗ ∇ ψ ) . {\displaystyle \mathbf {J} ={\frac {i\hbar }{2m}}\left(\psi \nabla \psi ^{*}-\psi ^{*}\nabla \psi \right).} This

2279-408: The red arrows denotes the boundary of the surface, and the orientation of the arrows with respect to the surface denotes the sign of the inner product of the vector field with the surface normals. If the surface encloses a 3D region, usually the surface is oriented such that the influx is counted positive; the opposite is the outflux . The divergence theorem states that the net outflux through

2332-627: The second factor is the mean free path and the square root (with the Boltzmann constant k ) is the mean velocity of the particles. In turbulent flows, the transport by eddy motion can be expressed as a grossly increased diffusion coefficient. In quantum mechanics , particles of mass m in the quantum state ψ ( r , t ) have a probability density defined as ρ = ψ ∗ ψ = | ψ | 2 . {\displaystyle \rho =\psi ^{*}\psi =|\psi |^{2}.} So

2385-484: The spreader will make it more effective to function as an air heat exchanger , as opposed to the original (presumably smaller) source. The low heat conduction of air in convection is matched by the higher surface area of the spreader, and heat is transferred more effectively. A heat spreader is generally used when the heat source tends to have a high heat- flux density , (high heat flow per unit area), and for whatever reason, heat can not be conducted away effectively by

2438-407: The surface does not fold back onto itself. Also, the surface has to be actually oriented, i.e. we use a convention as to flowing which way is counted positive; flowing backward is then counted negative. The surface normal is usually directed by the right-hand rule . Conversely, one can consider the flux the more fundamental quantity and call the vector field the flux density. Often a vector field

2491-497: The surface here need not be flat. Finally, we can integrate again over the time duration t 1 to t 2 , getting the total amount of the property flowing through the surface in that time ( t 2  −  t 1 ): q = ∫ t 1 t 2 ∬ S j ⋅ d A d t . {\displaystyle q=\int _{t_{1}}^{t_{2}}\iint _{S}\mathbf {j} \cdot d\mathbf {A} \,dt.} Eight of

2544-763: The surface in which flux is being measured is fixed and has area A . The surface is assumed to be flat, and the flow is assumed to be everywhere constant with respect to position and perpendicular to the surface. Second, flux as a scalar field defined along a surface, i.e. a function of points on the surface: j ( p ) = ∂ I ∂ A ( p ) , {\displaystyle j(\mathbf {p} )={\frac {\partial I}{\partial A}}(\mathbf {p} ),} I ( A , p ) = d q d t ( A , p ) . {\displaystyle I(A,\mathbf {p} )={\frac {\mathrm {d} q}{\mathrm {d} t}}(A,\mathbf {p} ).} As before,

2597-675: The surface is assumed to be flat, and the flow is assumed to be everywhere perpendicular to it. However the flow need not be constant. q is now a function of p , a point on the surface, and A , an area. Rather than measure the total flow through the surface, q measures the flow through the disk with area A centered at p along the surface. Finally, flux as a vector field : j ( p ) = ∂ I ∂ A ( p ) , {\displaystyle \mathbf {j} (\mathbf {p} )={\frac {\partial \mathbf {I} }{\partial A}}(\mathbf {p} ),} I ( A , p ) =

2650-429: The surface: d q d t = ∬ S j ⋅ n ^ d A = ∬ S j ⋅ d A , {\displaystyle {\frac {\mathrm {d} q}{\mathrm {d} t}}=\iint _{S}\mathbf {j} \cdot \mathbf {\hat {n}} \,dA=\iint _{S}\mathbf {j} \cdot d\mathbf {A} ,} where A (and its infinitesimal)

2703-407: The term corresponds to. In transport phenomena ( heat transfer , mass transfer and fluid dynamics ), flux is defined as the rate of flow of a property per unit area, which has the dimensions [quantity]·[time] ·[area] . The area is of the surface the property is flowing "through" or "across". For example, the amount of water that flows through a cross section of a river each second divided by

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2756-428: The transport definition—charge per time per area. Due to the conflicting definitions of flux , and the interchangeability of flux , flow , and current in nontechnical English, all of the terms used in this paragraph are sometimes used interchangeably and ambiguously. Concrete fluxes in the rest of this article will be used in accordance to their broad acceptance in the literature, regardless of which definition of flux

2809-502: Was one of the major developers of what we now call "electric flux" and "magnetic flux" according to the electromagnetism definition. Their names in accordance with the quote (and transport definition) would be "surface integral of electric flux" and "surface integral of magnetic flux", in which case "electric flux" would instead be defined as "electric field" and "magnetic flux" defined as "magnetic field". This implies that Maxwell conceived of these fields as flows/fluxes of some sort. Given

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