magneto static fields

Engineering Electromagnetics

Magnetostatic Fields

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Objectives:1. To define Biot-Savart’s Law2. To define Ampere’s Circuit Law3. To apply Biot Savart’s Law and Ampere’s Law in

the analysis of electromagnetic systems4. To determine Magnetic Flux Density5. To determine Magnetic Scalar and Vector

Potentials

IntroductionWe now focus our attention on static magnetic fields, which are characterized by H or B. There are similarities and dissimilarities between electric and magnetic fields. As E and D are related according to D = εE for linear material space, H and B are related according to B = μH.

MagnetostaticFields

Objectives:1. To define Biot-Savart’s Law2. To define Ampere’s Circuit Law3. To apply Biot Savart’s Law and Ampere’s Law in the

analysis of electromagnetic systems4. To determine Magnetic Flux Density5. To determine Magnetic Scalar and Vector Potentials

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A definite link between electric and magnetic fields was established by Oersted in 1820. As we have noticed, an electrostatic field is produced by static or stationary charges. If the charges are moving with constant velocity, a static magnetic (or magnetostatic) field is produced. A magnetostaticfield is produced by a constant current flow (or direct current). This current flow may be due to magnetization currents as in permanent magnets, electron-beam currents as in vacuum tubes, or conduction currents as in current-carrying wires.

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The development of the motors, transformers, microphones, compasses, telephone bell ringers, television focusing controls, advertising displays, magnetically levitated high-speed vehicles, memory stores, magnetic separators, and so on, involve magnetic phenomena and play an important role in our everyday life.

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There are two major laws governing magnetostaticfields: (1) Biot-Savart's law, and (2) Ampere's circuit law. Like Coulomb's law, Biot-Savart's law is the general law of magnetostatics. Just as Gauss's law is a special case of Coulomb's law, Ampere's law is a special case of Biot-Savart's law and is easily applied in problems involving symmetrical current distribution. The two laws of magnetostatics are stated and applied first; their derivation is provided later in the chapter.

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Biot-Savart’s LawBiot-Savart's law states that the magnetic field intensity dH produced at a point P, as shown in Figure, by the differential current element Idl is proportional to the product Idl and the sine of the angle α between the element and the line joining P to the element and is inversely proportional to the square of the distance R between P and the element.

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That is,

or

where k is the constant of proportionality. In SI units, k = 1/4π, so the equation becomes

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In vector form,

Thus the direction of dH can be determined by the right-hand rule with the right-hand thumb pointing in the direction of the current, the right-hand fingers encircling the wire in the direction of dH as shown in Figure(a).

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Alternatively, we can use the right-handed screw rule to determine the direction of dH: with the screw placed along the wire and pointed in the direction of current flow, the direction of advance of the screw is the direction of dH as in Figure(b).

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It is customary to represent the direction of the magnetic field intensity H (or current I) by a small circle with a dot or cross sign depending on whether H (or I) is out of, or into, the page as illustrated in Figure.

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Just as we can have different charge configurations, we can have different current distributions: line current, surface current, and volume current as shown in Figure.

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If we define K as the surface current density (in amperes/meter) and J as the volume current density (in amperes/meter square), the source elements are related as

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Thus in terms of the distributed current sources, the Biot-Savart law becomes

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As an example, let us apply the line current equation to determine the field due to a straight current carrying filamentary conductor of finite length AB as shown in Figure.

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We assume that the conductor is along the z-axis with its upper and lower ends respectively subtending angles α2 and α1 at P, the point at which H is to be determined. Particular note should be taken of this assumption as the formula to be derived will have to be applied accordingly. If we consider the contribution dH at P due to an element dl at (0, 0, z),

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But dl = dzaz and R = ρaρ - zaz, so

Hence,

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Letting z = ρcotα, dz = -ρcosec2αdα, the equation becomes

or

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The expression is generally applicable for any straight filamentary conductor of finite length. Notice that H is always along the unit vector aφ (i.e., along concentric circular paths) irrespective of the length of the wire or the point of interest P. As a special case, when the conductor is semi-infinite (with respect to P) so that point A is now at O(0, 0, 0) while B is at (0, 0, ∞); α1 = 90°, α2 = 0°, the equation becomes

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Another special case is when the conductor is infinite in length. For this case, point A is at (0, 0, -∞) while B is at (0, 0, ∞); α1 = 180°, α2 = 0°, so the equation reduces to

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To find unit vector aφ is to determine aφ from

where aℓ is a unit vector along the line current and aρis a unit vector along the perpendicular line from the line current to the field point.

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Ampere’s Circuit LawAmpere's circuit law states that the line integral of the tangential component of H around a dosed path is the same as the net current Ienc enclosed by the path.In other words, the circulation of H equals Ienc; that is,

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Ampere's law is similar to Gauss's law and it is easily applied to determine H when the current distribution is symmetrical. It should be noted that the equation always holds whether the current distribution is symmetrical or not but we can only use the equation to determine H when symmetrical current distribution exists. Ampere's law is a special case of Biot-Savart's law; the former may be derived from the latter.

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By applying Stoke's theorem to the left-hand side of the equation, we obtain

But

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Comparing the surface integrals clearly reveals that

This is the third Maxwell's equation to be derived; it is essentially Ampere's law in differential (or point) form whereas the other equation is the integral form. We should observe that ; that is, magnetostatic field is not conservative.

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Applications of Ampere’s LawWe now apply Ampere's circuit law to determine H for some symmetrical current distributions as we did for Gauss's law. We will consider an infinite line current, an infinite current sheet, and an infinitely long coaxial transmission line.

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A. Infinite Line CurrentConsider an infinitely long filamentary current I along the z-axis as in Figure.

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To determine H at an observation point P, we allow a closed path pass through P. This path, on which Ampere's law is to be applied, is known as an Amperian path (analogous to the term Gaussian surface). We choose a concentric circle as the Amperian path, which shows that H is constant provided p is constant.

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Since this path encloses the whole current I, according to Ampere's law

or

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B. Infinite Sheet of CurrentConsider an infinite current sheet in the z = 0 plane. If the sheet has a uniform current density K = KyayA/m as shown in Figure, applying Ampere's law to the rectangular closed path (Amperian path) gives

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To evaluate the integral, we first need to have an idea of what H is like. To achieve this, we regard the infinite sheet as comprising of filaments. As evident in Figure (b), the resultant dH has only an x-component. Also, H on one side of the sheet is the negative of that on the other side.

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Due to the infinite extent of the sheet, the sheet can be regarded as consisting of such filamentary pairs so that the characteristics of H for a pair are the same for the infinite current sheets, that is,

where Ho is yet to be determined.

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Evaluating the line integral of H along the closed path gives

From the equations, we obtain Ho = 1/2 Ky. Substituting Ho gives

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In general, for an infinite sheet of current density K A/m,

where an is a unit normal vector directed from the current sheet to the point of interest.

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C. Infinitely Long Coaxial Transmission LineConsider an infinitely long transmission line consisting of two concentric cylinders having their axes along the z-axis. The cross section of the line is shown in Figure, where the z-axis is out of the page.

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The inner conductor has radius a and carries current I while the outer conductor has inner radius b and thickness t and carries return current -I. We want to determine H everywhere assuming that current is uniformly distributed in both conductors. Since the current distribution is symmetrical, we apply Ampere's law along the Amperian path for each of the four possible regions: 0 ≤ ρ ≤ a, a ≤ ρ ≤ b, b ≤ ρ ≤b + t, and ρ ≥ b + t.

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For region 0 ≤ ρ ≤ a, we apply Ampere’s law to path L1, giving

Since the current is uniformly distributed over the cross section,

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Hence, the equation becomes

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For region a ≤ ρ ≤ b, we use path L2 as the Amperianpath,

since the whole current I is enclosed by L2.

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For region b ≤ ρ ≤ b + t, we use path L3, getting

where

and J in this case is the current density (current per unit area) of the outer conductor and is along -av that is,

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Thus

Substituting this, we have

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For region ρ > b + t, we use path L4, getting

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Putting the equations together gives

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The magnitude of H is sketched in the figure shown.

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Notice from these examples that the ability to take H from under the integral sign is the key to using Ampere's law to determine H. In other words, Ampere's law can only be used to find H due to symmetric current distributions for which it is possible to find a closed path over which H is constant in magnitude.

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Magnetic Flux DensityThe magnetic flux density B is similar to the electric flux density D. As D = εoE in free space, the magnetic flux density B is related to the magnetic field intensity H according to

where μo is a constant known as the permeability of free space. The constant is in henrys/meter (H/m) and has the value of

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The magnetic flux through a surface S is given by

where the magnetic flux Ψ is in webers (Wb) and the magnetic flux density is in webers/square meter (Wb/m2) or teslas.

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The magnetic flux line is the path to which B is tangential at every point in a magnetic field. It is the line along which the needle of a magnetic compass will orient itself if placed in the magnetic field. For example, the magnetic flux lines due to a straight long wire are shown in the Figure.

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The direction of B is taken as that indicated as "north" by the needle of the magnetic compass. Notice that each flux line is closed and has no beginning or end. It is generally true that magnetic flux lines are closed and do not cross each other regardless of the current distribution.

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In an electrostatic field, the flux passing through a closed surface is the same as the charge enclosed; that is,

Thus it is possible to have an isolated electric charge as shown in Figure (a), which also reveals that electric flux lines are not necessarily closed. Unlike electric flux lines, magnetic flux lines always close upon themselves as in Figure (b). This is due to the fact that it is not possible to have isolated magnetic poles (or magnetic charges).

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For example, if we desire to have an isolated magnetic pole by dividing a magnetic bar successively into two, we end up with pieces each having north and south poles as illustrated in Figure. We find it impossible to separate the north pole from the south pole.

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An isolated magnetic charge does not exist.Thus the total flux through a closed surface in a magnetic field must be zero; that is,

This equation is referred to as the law of conservation of magnetic flux or Gauss's law for magnetostatic fields just as § D • dS = Q is Gauss's law for electrostatic fields. Although the magnetostatic field is not conservative, magnetic flux is conserved.

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By applying the divergence theorem to the equation, we obtain

This equation is the fourth Maxwell's equation to be derived. The equations shows that magnetostaticfields have no sources or sinks. It suggests that magnetic field lines are always continuous.

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Maxwell’s Equations for Static EM FieldsHaving derived Maxwell's four equations for static electromagnetic fields, we may take a moment to put them together as in Table. From the table, we notice that the order in which the equations were derived has been changed for the sake of clarity.

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The choice between differential and integral forms of the equations depends on a given problem. It is evident from Table that a vector field is defined completely by specifying its curl and divergence. A field can only be electric or magnetic if it satisfies the corresponding Maxwell's equations. It should be noted that Maxwell's equations as in Table are only for static EM fields. As will be discussed in the next Chapter, the divergence equations will remain the same for time-varying EM fields but the curl equations will have to be modified.

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Magnetic Scalar and Vector PotentialsWe recall that some electrostatic field problems were simplified by relating the electric potential V to the electric field intensity E ( ). Similarly, we can define a potential associated with magnetostaticfield B. In fact, the magnetic potential could be scalar Vm or vector A.

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To define Vm and A involves recalling two important identities

which must always hold for any scalar field V and vector field A.Just as , we define the magnetic scalar potential Vm (in amperes) as related to H according to

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Combination of equations gives

Thus the magnetic scalar potential Vm is only defined in a region where J = 0. We should also note that Vmsatisfies Laplace's equation just as V does for electrostatic fields; hence,

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We know that for a magnetostatic field, V • B = 0. We can define the vector magnetic potential A (in Wb/m) such that

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Just as we defined

we can define

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To derive the line current equation, we have the equation

where R is the distance vector from the line element dl’ at the source point (x’, y’, z’) to the field point (x, y, z) as shown in the figure.

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and R = |R|, that is,

Hence,

where the differentiation is with respect to x, y, and z.

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Substituting the equation, we obtain

We apply the vector identity

where f is a scalar field and F is a vector field. Taking f = 1/R and F = dl’, we have

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Since operates with respect to (x, y, z) while dl’ is a function of (x’, y', z'),Hence,

and

It shows that

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By substituting the equations and applying Stokes's theorem, we obtain

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Thus the magnetic flux through a given area can be found using either the two equations. Also, the magnetic field can be determined using either Vm or A; the choice is dictated by the nature of the given problem except that Vm can only be used in a source-free region. The use of the magnetic vector potential provides a powerful, elegant approach to solving EM problems, particularly those relating to antennas.

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Derivation of Biot-Savart’s Law and Ampere’s LawBoth Biot-Savart's law and Ampere's law may be derived using the concept of magnetic vector potential. The derivation will involve the use of the vector identities

and

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Since Biot-Savart's law is basically on line current, we begin our derivation as

If the vector identity is applied by letting F = dl and f = 1/R, the equation becomes

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Since operates with respect to (x, y, z) while dl’ is a function of (x’, y', z'),Also

where aR is a unit vector from the source point to the field point.

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Thus equation (upon dropping the prime in dl') becomes

which is Biot-Savart’s Law.Using the identity with the equation, we obtain

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It can be shown that for a static magnetic field

so that upon replacing B with μoH, the equation becomes

which is called the vector Poisson's equation.

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In Cartesian coordinates, the equation may be decomposed into three scalar equations:

which may be regarded as the scalar Poisson's equations.

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It can also be shown that Ampere's circuit law is consistent with our definition of the magnetic vector potential. From Stokes's theorem

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From the previous equations,

Substituting this yields

which is Ampere's circuit law.

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QUESTIONS

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- maglev is short for magnetic levitation- it means that these trains will float over a guide way using the basic principles of magnets

TECHNOLOGY BRIEF: TECHNOLOGY BRIEF: MAGLEV TRAINSMAGLEV TRAINS

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- William Sturgeon developed the first practical electromagnet in the 1820s.- today, the principle of the electromagnet is used in motors, relay switches, in read/write heads of hard disks and tape drives, loud speakers, magnetic levitation, and many other applications

TECHNOLOGY BRIEF: TECHNOLOGY BRIEF: ELECTROMAGNETS AND MAGNETIC RELAYSELECTROMAGNETS AND MAGNETIC RELAYS

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- when an electric current generated by a power source flows through the wire coiled around the central core, it induces a magnetic field with field lines resembling those generated by a bar magnet- when subjected to a magnetic field, ferromagnetic materials, such as iron or nickel, get magnetized and act like magnets themselves


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Magnetic Relays- is a switch or circuit breaker that can be activated into the “ON” and “OFF” positions magnetically


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TECHNOLOGY BRIEF: TECHNOLOGY BRIEF: MAGNETIC RECORDINGMAGNETIC RECORDING

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- Valdemar Poulsen, a Danish engineer, invented magnetic recording by demonstrating in 1900 that speech can be recorded on a thin steel wire with a simple electromagnet- videotapes were introduced in the late 1950s for recording motion pictures for later replay on television


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- other types of magnetic recording media were developed since then, including the flexible plastic disks called “floppies”, the hard disks made of glass or aluminum, the magnetic drum, and the magnetic bubble memory- all take advantage of the same fundamental principle of being able to store electrical information through selective magnetization of a magnetic material, as well as the ability to retrieve it (playback) when so desired


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Record Process- sound waves incident on a microphone causes a diaphragm to vibrate, creating an electric current with corresponding time and amplitude variations as that of the original sound pattern- after amplification, the current signal drives a recording head consisting of an electromagnet to magnetize the tape as it is drawn past the head


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- the tape is made of a plastic base material with a coating of ferric oxide powder glued to its surface- when exposed to the magnetic field, the previously randomly oriented molecules of the ferromagnetic powder become permanently oriented along a specific direction, thereby establishing a magnetic imprint of the original sound signal on the tape


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Replay process- accomplished by reversing the process of read process- drawing the magnetized tape past a reproducing head induces a current having a vibration proportional to that on the tape, which is then amplified and converted back to sound waves through a loud speaker


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Homework:1. Find H at (-3 , 4, 0) due to the current filament

shown in Figure 7.7(a).

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2. A circular loop located on x2 + y2 = 9, z = 0 carries a direct current of 10 A along aφ. Determine H at (0, 0, 4) and (0, 0, -4) .

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3. Planes z = 0 and z = 4 carry current K = -10ax A/m and K = 10ax A/m, respectively. Determine H at(a) (1,1,1)(b) (0, -3 , 10)

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4. Given the magnetic vector potential A = -ρ2/4 azWb/m, calculate the total magnetic flux crossing the surface φ = -π/2, 1 ≤ ρ ≤ 2m, 0 ≤ z ≤ 5m.

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magneto static fields

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