BASIC CONCEPTSBefore going into the detail of maxwell’s equations it is important to view some important concepts.ELECTRIC FIELD INTENSITY:

If we consider one charge fixed in position, say Q1, and move a second chargeslowly around, we note that there exists everywhere a force on this second charge;in other words, this second charge is displaying the existence of a force field that isassociated with charge, Q1. Call this second charge a test charge Qt . The force on itis given by Coulomb’s law,

Writing this force as a force per unit charge gives the electric field intensity, E1 arisingfrom Q1:

E1 is interpreted as the vector force, arising from charge Q1 , that acts on a unit positivetest charge. More generally, we write the defining expression:

in which E, a vector function, is the electric field intensity evaluated at the test chargelocation that arises from all other charges in the vicinity—meaning the electric fieldarising from the test charge itself is not included in E.

ELECTRIC FLUX AND FLUX DENSITY:About 1837 Michael Faraday performed an experiment.He had a pair of concentricmetallic spheres constructed, the outer one consisting of two hemispheres that could befirmly clamped together. He also prepared shells of insulating material (or dielectricmaterial, or simply dielectric) that would occupy the entire volume between theconcentric spheres. He found that:

1. With the equipment dismantled, the inner sphere was given a known positivecharge.2. The hemispheres were then clamped together around the charged sphere withabout 2 cm of dielectric material between them.3. The outer sphere was discharged by connecting it momentarily to ground.4. The outer space was separated carefully, using tools made of insulating material

in order not to disturb the induced charge on it, and the negative induced chargeon each hemisphere was measured.Faraday found that the total charge on the outer sphere was equal in magnitude tothe original charge placed on the inner sphere and that this was true regardless of thedielectric material separating the two spheres. He concluded that there was some sortof “displacement” from the inner sphere to the outer which was independent of themedium, and we now refer to this flux as displacement, displacement flux, or simplyelectric flux.Faraday’s experiments also showed, of course, that a larger positive charge on theinner sphere induced a correspondingly larger negative charge on the outer sphere,leading to a direct proportionality between the electric flux and the charge on the innersphere. The constant of proportionality is dependent on the system of units involved,and we are fortunate in our use of SI units, because the constant is unity. If electricflux is denoted by _ (psi) and the total charge on the inner sphere by Q, then forFaraday’s experiment

and the electric flux _ is measured in coulombs.

Electric flux density, measured in coulombs per square meter (sometimes describedas “lines per square meter,” for each line is due to one coulomb), is giventhe letter D, which was originally chosen because of the alternate names of displacementflux density or displacement density. The electric flux density D is a vector field and is a member of the “flux density”class of vector fields, as opposed to the “force fields” class, which includes the electric field intensity E. The direction of D at a point is the direction of the flux lines at thatpoint, and the magnitude is given by the number of flux lines crossing a surface normalto the lines divided by the surface area. Consider, the following figure:

The electric flux density is in the radial direction and has a value of:

and at a radial distance r , where a ≤ r ≤ b,

If we now let the inner sphere become smaller and smaller, while still retaining acharge of Q, it becomes a point charge in the limit, but the electric flux density at apoint r meters from the point charge is still given by

But according to above equation, the “ELECTRIC INTENSITY” is given as:

So,

CURRENT DENSITY:Electric charges in motion constitute a current. The unit of current is the ampere (A), defined as a rate of movement of charge passing a given reference point (or crossing a given reference plane) of one coulomb per second. Current is symbolized by I , and therefore

Current is thus defined as the motion of positive charges, even though conduction in metals takes place through the motion of electrons..In field theory, we are usually interested in events occurring at a point rather than within a large region, and we find the concept of current density, measured in amperes per square meter (A/m2), more useful. Current density is a vector represented by J.

DIVERGENCE:The divergence of A is defined as

and is usually abbreviated div A. The physical interpretation of the divergence of avector is obtained by describing carefully the operations implied by the right-handside of above equation.

“The divergence of the vector flux density A is the outflow of flux from a small closed surfaceper unit volume as the volume shrinks to zero.Divergence at a point (x,y,z) is the measure of the vector flow out of a surface surrounding that point”.

 That is, imagine a vector field represents water flow. Then if the divergence is a positive number, this means water is flowing out of the point (like a water spout - this location is considered a source). If the divergence is a negative number, then water is flowing into the point (like a water drain - this location is known as a sink).

First, imagine we have a vector field as shown in Figure, and we want to know what the divergence is at the point P:

We also draw an imaginary surface (S) surrounding the point P. Now imagine the vector A represents water flow. Then, if we add up the amount of water flowing out of the surface, would the amount be positive? The answer, is yes: water is flowing out of the surface at every location along the surface S. Hence, we can say that the divergence at P is positive.

Now consider the following figure. We have a new vector field B surrounding the point P:

If we imagine the water flowing, we would see the point P acting like a drain or water sink. In this case, the flow out of the surface is negative - hence, the divergence of the field B at P is negative.

Following figure shows a vector field  surrounding the point:

In the top of Figure , water flows out of the surface, but at the bottom it flows in. Since the field has equal flow into and out of the surface S, the divergence is zero.

Consider the following figure:

Here we have a vector field D that wraps around the point P. At each point along the surface S, the field is flowing tangentially along the surface. Therefore the field is not flowing into or out of the surface at each point. Hence, again, we have the divergence of D equal to 0 at P.

Consider the following figure:

The vector field E has a large vector above the point P indicating a strong field there - a lot of water flowing out of the surface. The vector to the left of P is small and tangential to the surface, so there is no flow into or out of S at that point. The same is true for the vector to the right of P. And the vector below P is small, indicating a smaller amount of water flowing into the surface. Hence, we can guess the divergence is positive - more water is flowing out of the surface than into it.

Mathematically,

CURL:“The curl of any vector is a vector, and any component of the curl is given bythe limit of the quotient of the closed line integral of the vector about a small path ina plane normal to that component desired and the area enclosed, as the path shrinksto zero.” It should be noted that this definition of curl does not refer specifically to aparticular coordinate system. The mathematical form of the definition is:

where _SN is the planar area enclosed by the closed line integral. The N subscriptindicates that the component of the curl is that component which is normal to thesurface enclosed by the closed path. It may represent any component in any coordinatesystem.

We may describe curl as circulation per unit area. The closed path is vanishinglysmall, and curl is defined at a point.

To understand this, we will use the analogy of flowing water to represent a vector function (or vector field). In the following figure , we have a vector function (V) and we want to know if the field is rotating at the point D (that is, we want to know if the curl is zero).

To determine if the field is rotating, imagine a water wheel at the point D. If the vector field representing water flow would rotate the water wheel, then the curl is not zero:

We can see that the water wheel would be rotating in the counter-clockwise direction. Hence, this vector field would have a curl at the point D.

As the curl rotates the water wheel in the x-y plane, the direction of the curl is taken to be the z-axis (perpendicular to plane of the water wheel). In addition, the curl follows the right-hand rule: if the thumb points in the +z-direction, then right hand will curl around the axis in the direction of positive curl.

In the figure , the water wheel rotates in the clockwise direction. Hence, the z-component of the curl for the vector field in the figure is negative.

Imagine that the vector field F  has z-directed fields. Let the symbol   represent a vector in the +z-direction and the symbol   represent a vector in the -z direction:

The wheel will not rotate if the water is flowing up or down around it. Only x- and y- directed vectors can cause the wheel to rotate when the wheel is in the x-y plane. Hence, the z-directed vector fields can be ignored for determining the z-component of the curl.

Now, let's take more examples to make sure we understand the curl.

Since the water wheel is in the y-z plane, the direction of the curl (if it is not zero) will be along the x-axis. The red vector in the figure is in the +y-direction. However, it will not rotate the water wheel, because it is directed directly at the center of the wheel and won't produce rotation. The green vector in Figure 4 will try to rotate the water wheel in the clockwise direction, but the black vector will try to rotate the water wheel in the counter-clockwise direction - therefore the green vector and the black vector cancel out and produce no rotation. However, the brown vector will rotate the water wheel in the counter clockwise direction. Hence, the net effect of all the vectors in the figure is a counter-clockwise rotation. The result is that the curl in the figure 4is positive and in the +x-direction.

We can use a very small paddle wheel as a “curl meter.” Our vector quantity, then, must be thought of as capable of applying a force to each bladeof the paddle wheel, the force being proportional to the component of the field normalto the surface of that blade. To test a field for curl, we dip our paddle wheel into thefield, with the axis of the paddle wheel lined up with the direction of the component of

curl desired, and note the action of the field on the paddle. No rotation means no curl;larger angular velocities mean greater values of the curl; a reversal in the direction ofspin means a reversal in the sign of the curl. To find the direction of the vector curl andnot merely to establish the presence of any particular component, we should placeour paddle wheel in the field and hunt around for the orientation which produces thegreatest torque. The direction of the curl is then along the axis of the paddle wheel,as given by the right-hand rule.

As an example, consider the flow of water in a river. Following figure shows thelongitudinal section of a wide river taken at the middle of the river. The water velocityis zero at the bottom and increases linearly as the surface is approached. A paddlewheel placed in the position shown, with its axis perpendicular to the paper, will turnin a clockwise direction, showing the presence of a component of curl in the direction of an inward normal to the surface of the page. If the velocity of water does not changeas we go up- or downstream and also shows no variation as we go across the river(or even if it decreases in the same fashion toward either bank), then this componentis the only component present at the center of the stream, and the curl of the watervelocity has a direction into the page.

In the following figure , the streamlines of the magnetic field intensity about an infinitelylong filamentary conductor are shown. The curl meter placed in this field of curvedlines shows that a larger number of blades have a clockwise force exerted on thembut that this force is in general smaller than the counterclockwise force exerted onthe smaller number of blades closer to the wire. It seems possible that if the curvatureof the streamlines is correct and also if the variation of the field strength is just right,the net torque on the paddle wheel may be zero. Actually, the paddle wheel does notrotate in this case, since

So,

Mathematically:

MAXEWELL’S EQUATIONSFollowing are the MAXEWELL EQUATIONS in differential form:

MAXWELL’S FIRST EQUATION

Gauss' Law is the first of Maxwell's Equations which dictates how the Electric Field behaves around electric charges. Gauss' Law can be written in terms of the Electric Flux Density and the Electric Charge Density as:

[Equation 1]

In Equation [1], the symbol   is the divergence operator.

Equation [1] is known as Gauss' Law in point form. That is, Equation [1] is true at any point in space. That is, if there exists electric charge somewhere, then the divergence of D at that point is nonzero, otherwise it is equal to zero.

We have already explained that divergence states that what is left behind when a vector passes a certain point. This equation states that the source of electric flux is electric charge.Whenever electric charge is there ,the divergence has non zero value which is equal to the volume charge density.

According to this total electric flux through any closed surface is times the total chargeenclosed by the closed surfaces, representing Gauss's law of electrostatics, As this does notdepend on time, it is a steady state equation. Here for positive , divergence of electric field is

positive and for negative , divergence is negative. It indicates that is scalar quantity

To get some more intuition on Gauss' Law, let's look at Gauss' Law in integral form. To do this, we assume some arbitrary volume (we'll call it V) which has a boundary (which is written S). Then integrating Equation [1] over the volume V gives Gauss' Law in integral form:

[Equation 2]

As an example, look at following figure. We have a volume V, which is the cube. The surface S is the boundary of the cube (i.e. the 6 flat faces that form the boundary of the volume).

Equation [2] states that the amount of charge inside a volume V (= ) is equal to the total amount of Electric Flux (D) exiting the surface S. That is, to determine the Electric Flux leaving the region V, we only need to know how much electric charge is within the volume. We rewrite Equation [2] with more of the terms defined in Equation [3]:

[Equation 3]

Look at the point P in following Figure , where we have drawn the D field vector:

We can rewrite any field in terms of its tangential and normal components.From Equation [3], we are only interested in the component of D normal (orthogonal or perpendicular) to the surface S. We write this as Dn. The tangential component Dt  flows along the surface. If we imagine the D field as a water flow, then only the component Dn would contribute to water actually leaving the volume - Dt is just water flowing around the surface.

Hence, Gauss' law is a mathematical statement that the total Electric Flux exiting any volume is equal to the total charge inside. Hence, if the volume in question has no charge within it, the net flow of Electric Flux out of that region is zero. If there is positive charge within a volume, then there exists a positive amount of Electric Flux exiting any volume that surrounds the charge. If there is negative charge within a volume, then there exists a negative amount of Electric Flux exiting (i.e. the Electric Flux enters the volume).

Gauss' Law states that electric charge acts as sources or sinks for Electric Fields.

If you use the water analogy again, positive charge gives rise to flow out of a volume - this means positive electric charge is like a source (a faucet - pumping water into a region). Conversely, negative charge gives rise to flow into a volume - this means negative charge acts like a sink (fields flow into a region and terminate on the charge).

This gives us a lot of intuition about the way fields can physically act in any scenario. For instance, here are possible and impossible situations for the Electric Field, as decided by the universe in the Law of Gauss it setup:

If you observe the way the D field must behave around charge, you may notice that Gauss' Law then is equivalent to the Force Equation for charges, which gives rise to the E field equation for point charges

[Equation 4]

Equation [4] shows that charges exert a force on them, which means there exists E-fields that are away from positive charge and towards negative charge. This means opposite charges attract and negative charges repel. And since D and E are related by permittivity, we see that Gauss' Law is a more formal statement of the force equation for electric charges.

In summary, Gauss' Law means the following is true:

D and E field lines diverge away from positive charges D and E field lines diverge towards negative charges D and E field lines start and stop on Electric Charges Opposite charges attract and negative charges repel

The divergence of the D field over any region (volume) of space is exactly equal to the net amount of charge in that region

MAXEWELL’S SECOND EQUATION

Second equation is the statement of Gauss law in magnetic field. It states that:

As the divergence indicates the source, so this equation states that isolated magnetic polesor magnetic monopoles cannot exist as they appear only in pairs and there is no source or sink

for magnetic lines of forces. It is also independent of time i.e. steady state equation.This is illustrated in following figure:

Magnets attract other magnets similar to how electric charges repel or attract like electric charges. However, there is something special about these magnets - they always have a positive and negative end. This means every magnetic object is a magnetic dipole, with a north and south pole. No matter how many times you break the magnetic in half, it will just form more magnetic dipoles. Gauss' Law for Magnetism states that magnetic monopoles do not exist - or at least we haven't found them yet.

Because we know that the divergence of the Magnetic Flux Density is always zero, we now know a little bit about how these fields behave. We will present a couple of examples of legal and illegal Magnetic Fields, which are a consequence of Gauss' Law for Magnetism:

In summary, the second of Maxwell's Equations - Gauss' Law For Magnetism - means that:

1. Magnetic Monopoles Do Not Exist2. The Divergence of the B or H Fields is Always Zero Through Any Volume

3. Away from Magnetic Dipoles, Magnetic Fields flow in a closed loop. This is true even for plane waves, which just so happen to have an infinite radius loop.

MAXWELL’S THIRD EQUATION:

Third equation is simply the statement of Faraday law. It states that:

According to this equation, a magnetic field that is changing in time will give rise to a circulating E-field. This means we have two ways of generating E-fields - from Electric Charges (or flowing electric charge, current) or from a magnetic field that is changing.

Faraday was a scientist experimenting with circuits and magnetic coils way back in the 1830s. His experiment setup, which led to Farday's Law, is shown in figure:

The experiment itself is somewhat simple. When the battery is disconnected, we have no electric current flowing through the wire. Hence there is no magnetic flux induced within the Iron (Magnetic Core). The Iron is like a highway for Magnetic Fields - they flow very easily through magnetic material. So the purpose of the core is to create a path for the Magnetic Flux to flow.

When the switch is closed, the electric current will flow within the wire attached to the battery. When this current flows, it has an associated magnetic field (or magnetic flux) with it. When the wire wraps around the left side of the magnetic cor, a magnetic field (magnetic flux) is induced within the core. This flux travels around the core. So the Magnetic Flux produced by the wired coil on the left exists within the wired coil on the right, which is connected to the ammeter.

Now, a funny thing happens, which Faraday observed. When he closed the switch, then current would begin flowing and the ammeter would spike one way. But this was very brief, and the current on the right coil would go to zero. When the switch was opened, the measured current

would spike to the other side , and then the measured current on the right side would again be zero.

Faraday figured out what was happening. When the switch was initially changed from open to closed, the magnetic flux within the magnetic core increased from zero to some maximum number (which was a constant value, versus time). When the flux was increasing, there existed an induced current on the opposite side.

Similarly, when the switch was opened, the magnetic flux in the core would decrease from its constant value back to zero. Hence, a decreasing flux within the core induced an opposite current on the right side.

Faraday figured out that a changing Magnetic Flux within a circuit (or closed loop of wire) produced an induced EMF, or voltage within the circuit. He wrote this as:

[Equation 2]

In Equation [2],   is the Magnetic Flux within a circuit, and EMF is the electro-motive force, which is basically a voltage source. Equation [2] then says that the induced voltage in a circuit is the opposite of the time-rate-of-change of the magnetic flux. Equation [2] is known as Lenz's Law. Lenz was the guy who figured out the minus sign. We know that an electric current gives rise to a magnetic field - but thanks to Farady we also know that a magnetic field within a loop gives rise to an electric current. 

Deriving Faraday's Law

Let's imagine a simple loop, with a time varying B field within it:

We know that the rate of change of the total magnetic flux is equal to the opposite of the EMF, or the electric force within the wire. The total magnetic flux is simply the integral (or sum) of the B field over the area enclosed by the wire:

[Equation 3]

To find the total EMF induced around the whole circuit, we sum up over the length of the wire the EMF produced at each point. This is known as a line integral. This is written as:

[Equation 4]

Now, recall that the Electric Field is directly related to force from electric charges. And Voltage is also defined as the sum (integral) of the Electric Field across a path [recall that the E-field is measured in Volts/meter]. Hence, the E-field is actually the spatial-derivative of voltage (E-field

is equal to the rate of change of the voltage with respect to distance). These facts are summed up in the following:

[Equation 5]

Hence, Equations [4] and [5] tell us that the differential amount of EMF at any point along the circuit is equal to the E field at that location. Therefore:

[Equation 6]

Stokes figured out that integrating (averaging) of a field around a loop is exactly equivalent to integrating the curl of the field within the loop.The curl is the measure of the rotation of a field, so the curl of a vector field within a surface should be related to the integral of a field around a loop that encloses the surface.

[Equation 7]

If we replace Farday's Law of Equation [2], with the terms we found in Equation [3] and Equation [7], then we get:

[Equation 8]

In Equation [8], we note that if we have two integrals over surfaces, and the surfaces can be however we choose, then the quantities we integrate must also be the same.

Interpretation of Farday's Law

Faraday's law shows that a changing magnetic field within a loop gives rise to an induced current, which is due to a force or voltage within that circuit. We can then say the following about Farday's Law:

Electric Current gives rise to magnetic fields. Magnetic Fields around a circuit gives rise to electric current.

A Magnetic Field Changing in Time gives rise to an E-field circulating around it.

A circulating E-field in time gives rise to a Magnetic Field Changing in time.

Farday's Law is very powerful as it shows that if a current gives rise to a Magnetic Field then a Magnetic Field can give rise to an electric current. And a changing E-field in space gives rise to a changing B-field in time

MAXWELL’S FOURTH EQUATION:

This equation is the statement of Ampere’s circuital law with the extension of a concept called DISPLACEMENT CURRENT”. This was devised by Maxwell to account for the varying fields. This states that:

[Equation 1]

Ampere was a scientist experimenting with forces on wires carrying electric current. He was doing these experiments back in the 1820s, about the same time that Farday was working on Faraday's Law. Ampere and Farday didn't know that there work would be unified by Maxwell himself, about 4 decades later.

Ampere's Law relates an electric current flowing and a magnetic field wrapping around it:

[Equation 2]

Equation [2] can be explained: Suppose you have a conductor (wire) carrying a current, I. Then this current produces a Magnetic Field which circles the wire.

The left side of Equation [2] means: If you take any imaginary path that encircles the wire, and you add up the Magnetic Field at each point along that path, then it will numerically equal the

amount of current that is encircled by this path (which is why we write   for encircled or enclosed current).

Let's do an example for fun. Suppose we have a long wire carrying a constant electric current, I[Amps]. What is the magnetic field around the wire, for any distance r [meters] from the wire?

Consider the following figure. We have a long wire carrying a current of I Amps. We want to know what the Magnetic Field is at a distance r from the wire. So we draw an imaginary path around the wire, which is the dotted blue line .

Ampere's Law states that if we add up (integrate) the Magnetic Field along this blue path, then numerically this should be equal to the enclosed current I.

Now, due to symmetry, the magnetic field will be uniform (not varying) at a distance r from the wire. The path length of the blue path is equal to the circumference of a circle of

radius r:  .

If we are adding up a constant value for the magnetic field , then the left side of Equation [2] becomes simple:

[Equation 3]

Hence, we have figured out what the magnitude of the H field is. And since r was arbitrary, we know what the H-field is everywhere. Equation [3] states that the Magnetic Field decreases in magnitude as you move farther from the wire (due to the 1/r term).

So we've used Ampere's Law (Equation [2]) to find the magnitude of the Magnetic Field around a wire. However, the H field is a Vector Field, which means at every location is has both a magnitude and a direction. The direction of the H-field is everywhere tangential to the imaginary loops. The right hand rule determines the sense of direction of the magnetic field:

We can rewrite Ampere's Law in Equation [2]:

[Equation 4]

On the right side equality in Equation [4], we have used Stokes' Theorem to change a line integral around a closed loop into the curl of the same field through the surface enclosed by the loop (S).

We can also rewrite the total current ( ) as the surface integral of the Current Density (J):

[Equation 5]

So now we have the original Ampere's Law (Equation [2]) rewritten in terms of surface integrals (Equations [4] and [5]). Hence, we can substitute them together and get a new form for Ampere's Law:

[Equation 6]

Now, we have a new form of Ampere's Law: the curl of the magnetic field is equal to the Electric Current Density.

There is a problem with Equation [6], and Maxwell figured out this problem.

Displacement Current Density

Ampere's Law was written as in Equation [6] up until Maxwell. So let's look at what is wrong with it. First, we have to know another vector identity - the divergence of the curl of any vector field is always zero:

[Equation 7]

So let's take the divergence of Ampere's Law as written in Equation [6]:

[Equation 8]

So Equation [8] follows from Equations [6] and [7]. But it says that the divergence of the current density J is always zero.

If the divergence of J is always zero, this means that the electric current flowing into any region is always equal to the electric current flowing out of the region (no divergence). This seems somewhat reasonable, as electric current in circuits flows in a loop. But let's look what happens if we put a capacitor in the circuit:

Now, we know from electric circuit theory that if the voltage is not constant (for example, any periodic wave, such as the 60 Hz voltage ) then current will flow through the capacitor. That is, we have I not equal to zero .

However, a capacitor is basically two parallel conductive plates separated by air. Hence, there is no conductive path for the current to flow through. This means that no electric current can flow through the air of the capacitor. This is a problem if we think about Equation [8]. To show it more clearly, let's take a volume that goes through the capacitor, and see if the divergence of J is zero:

In the figure , we have drawn an imaginary volume , and we want to check if the divergence of the current density is zero. The volume we've chosen, has one end (labeled side 1) where the current enters the volume via the black wire. The other end of our volume (labeled side 2) splits the capacitor in half.

We know that the current flows in the loop. So current enters through Side 1 of our red volume. However, there is no electric current that exits side 2. No current flows within the air of the capacitor. This means that current enters the volume, but nothing leaves it - so the divergence of J is not zero. We have just violated our Equation [8], which means the theory does not hold.

Maxwell knew that the Electric Field (and Electric Flux Density (D) was changing within the capacitor. And he knew that a time-varying magnetic field gave rise to a solenoidal Electric Field (i.e. this is Farday's Law - the curl of E equals the time derivative of B). So, why is not that a time varying D field would give rise to a solenoidal H field (i.e. gives rise to the curl of H). The

universe loves symmetry, so why not introduce this term? And so Maxwell did, and he called this term the displacement current density:

[Equation 9]

This term would "fix" the circuit problem we have , and would make Farday's Law and Ampere's Law more symmetric. But the existance of this term unified the equations and led to understanding the propagation of electromagnetic waves, and the proof that all waves travel at the same speed (the speed of light)! And it was this unification of the equations that Maxwell presented, that led the collective set to be known as Maxwell's Equations. So, if we add the displacement current to Ampere's Law as written in Equation [6], then we have the final form of Ampere's Law:

[Equation 10]

Intrepretation of Ampere's Law

The following are consequences of this law:

A flowing electric current (J) gives rise to a Magnetic Field that circles the current

A time-changing Electric Flux Density (D) gives rise to a Magnetic Field that circles the D field

We know that a time varying D gives rise to an H field, but from Farday's Law we know that a varying H field gives rise to an E field and so on and so forth and the electromagnetic waves propagate .