8.5 Fields

8.5.1 Scalar Field and Vector Fields

scalar-field

In practical situations the simple case, that a single force vector acts on an object, will occur relatively seldom. An approximate example for this would be the collision of two bodies in outer space sufficiently far away from other bodies, such that their influence can be neglected. Then one could consider one of the bodies to be at rest and characterize the other one with a vector whose absolute value and direction correspond to its momentum p = mv.

Much more common is the situation, that there are influences at every r = (x,y,z) in space on the object of interest. They can either be described by vectors, having length and direction, for example the gravitational force in the vicinity of a planet, or by scalars, that have no direction, as the density of an atmosphere or the temperature. Both quantities, force and density, influence the movement of a test body in the vicinity of the planet. The gravitation has the effect of a directed acceleration, while the density causes a deceleration independent of the direction of movement but dependent on the position.

In the first case we call it a vector field and in the second a scalar field, In both cases the characteristic quantities depend on the space coordinates, thus for the vector field absolute value and direction and for scalar field the value. For the case of a non-stationary field they also depend on the time.

We want to visualize for both visualizations distributions, where you will have the opportunity to edit formulas for the position dependence of absolute value and direction or to design them yourself. That will give you a feeling for the characteristics of typical fields.

8.5.2 Visualization Possibilities for Scalar-and Vector fields

To visualize a scalar field in all generality one would need four variables, three for the position coordinates and one for the position dependent scalar itself. This information can obviously not represented with a 3D-simulation that is projected on a plane. In addition some fields will change with time as a fifth variable. Thus one has to work with certain restrictions for the visualization.

For stationary problems the time does not play a role as variable.

For problems with rotational symmetry, for example the gas density distribution ρ(r around a planet with rotational symmetry around an axis, one can restrict the presentation to a cross section through the center of the planet at a right angle to this axis and plot the gas density ρr as third coordinate over the cross section xy. Thus one obtains a 3D-surface in the space xyρ. The field distribution in space then shows rotational symmetry with respect to the distribution on the cross section.

A second possibility would be for this example, to ask where the curves of equal density are located and to create a family of such curves as contour plot. This task can be solved computationally by intersecting the planes ρ = constant for values on a equidistant ρ-grid with the 3D-surface xyρ and finding the intersection curves. Then this contour plot has the familiar appearance of a geographical contour lines display.

In the general case one would have to produce a family of such presentations for the different values of those variables that have been neglected so far. Fortunately however ,the cases of practical interest are mostly stationary and possess high symmetry, such that the methods described above can visualize the important characteristics quite well.

For vector fields one has to show in addition the direction of the vectors localized in space and their absolute value. This requires further restrictions for the visualization.

One is mostly interested in the general structure of the field, which can be shown by putting arrows on a regular grid, that show directions and absolute values at the respective positions. If one only wants to show only the direction of the vectors, one can use the same length for all arrows, which makes the presentation clearer. To indicate the absolute value one then can use different shades of colour.

For the presentation of a three-dimensional vector field one can stack several such cross sections over each other. As a static picture such a 3D-projection is often quite confusing. However, if one moves the projection direction interactively, either with the mouse or automatically around an axis, one obtains a rather good idea of the distribution.

All these tools are provided by the common numerical programs and we will show examples for these in the following.

8.5.3 Basic Formalism of Vector Analysis

vector-field

Pure scalar fields without relation to a vector field, (the density distribution is such an example), are not very interesting. Much more attractive are scalar fields, from which vector fields can be deduced, since they simplify their description in extraordinary ways and reduce them to a cause, that can be described by one position dependent parameter.

We refer to a scalar potential field P, if the components of a vector field V are obtained via taking partial derivatives with respect to x,y,z of P 2, i.e. via differentiation after one variable at a time, while the other variables are considered as constant. The underlying questions is then, how the scalar value changes, if one moves from a space point r = (x,y,z) to a neighboring space point r + dr = (z + dx,y + dy,z + dz). Thus one can take for each variable, i.e. partially, the first term of the Taylor expansion, if dr is small enough. One then obtains

dP = P(x + dx,y + dy,z + dz) - P(x,y,z) = P x dx + P y dy + P z dz = P x , P y , P z dx dy dz = gradP dr

The vector called grad P denotes the change of the scalar P in the three space directions. For a given space point its direction depends on the change of potential in the three directions; it points in the direction of maximum change. Its absolute value depends on the absolute values of these changes.

V = gradP = P x P y P z ; V = P x 2 + P y 2 + P z 2

As shorthand for the partial differentation with respect too all three coordinates, that is applied to the scalar potential, one uses the symbol Nabla (),an overturned greek letter Δ. This symbol reminds one of the form of an antique harp (ναβλα in greek, nablium in latin. Nabla symbolizes a vector operator which is therefore written as a matrix with one column or one line. To stress the vector character of this operator, one usually puts an arrow on top of it.

= x y z ;V = P = x y z P = P x P y P z ;P = gradP

Using the nabla-notation has the advantage, that it also allows a unified notation for other differential operators, for which different symbols are traditionally used, that hide their common origin. For example the vector field characterized by P is traditionally denoted by gradP (gradient of P) and referred to as gradient field, because if characterizes the steepness of the potential field.

We will now show some further applications of the Nabla-symbol and its traditional synonyms. In the first two examples the operator will not be applied to a scalar field but to a vector field. In analogy to the gradient of a scalar field we now deal with the change of a vector field from a space point r to a neighboring point r + dr.

a = x y z ax ay az = ax x +ay y +az z

a =  diva divergence(scalar field)ofa

×a = x y z ×ax ay az = az y -ay z ax z -az x ay x -ax y ;

×a = curla;curl (vector field)  of a

2 = x y z x y z = 2 x2 + 2 y2 + 2 z2 2 = ΔLaplace-Operator 2P = ΔP = divgradP, LaplaceP(read ”Laplace”P)scalar field.

The meaning of the symbols and operations is in short the following:

The divergence of a vector field in Cartesian coordinates is obtained computationally as the (symbolic) scalar multiplication of the nabla-operator with the vector and therefore it is a scalar field. It describes the source strength of the vector field. Where it does not vanish field lines either originate or converge.

An example from the Maxwell equations: divD = ρ; the charges are the sources of the electrical vector field D.

The curl of a vector field is obtained computationally as (symbolic) vectorial multiplication of the nabla-operator with the vector field, thus at every space point it is a vector, that is orthogonal to the input vector field. The curl of a vector field describes the vorticity of a vector field, which has closed field lines, unless curl a = 0 everywhere.

Another example from the Maxwell equations: curlH = j; the current density j determines the vector field H that is orthogonal the current density and has closed field lines everywhere.

The Laplace operator is obtained via (symbolic)scalar multiplication of the nabla-operator with itself and therefore yields a scalar field.

An example for its application: under the assumption, that the electrical field strength is the gradient field of an electrostatic potential, i.e. E = -gradΦ, we obtain from one of the Maxwell equations, namely divD = ρ the Poisson equation ΔΦ = -ρε0. Using this equation the electrostatic potential due to a given charge density ρ can be calculated, and from this the electrical vector field. In a portion of space without charges the potential equation ΔΦ = 0 applies.

Between the different operators the following general relations apply:

For every scalar field V we have: curlgradV = ×V = 0, i.e. for a gradient field the (local) curl is zero, there are no vortices.

divrota = 0. The (local) divergence of the curl field of a vector field is zero, because a pure vortex field does not have any sources.

8.5.4 Potential Fields of point sources as 3D-Surface

Especially elementary. simple and at the same time important are the potential fields, that are caused by point sources in space. They describe both the gravitational attraction between masses mi as well as the attraction or repulsion between charges ei, that can be positive or negative. The common property of these forces is, that with growing distance the effect of the point source is spread over the surface of a sphere and therefore decreases like 14πr2. The potential field then has apart from an additive constant as integral of the vector field in polar coordinates the form

-mi r ,or -ei r .

The effect on a similar object increases with decreasing distance, because 1r becomes larger, when the distance becomes smaller. The minus sign has the effect that the force F = -gradP = mi r2 r0 is positive if the mi are positive. In the case of gravitation this is always the always the case. In the electrostatic example the increasing repulsion of equally charged objects turns into an increasing attraction, if they are of opposite charge.

Die nachfolgende interaktive Simulation von Skalarfeldern zeigt als Beispiele: The following interactive simulation of scalar fields shows as examples:

The first object is normalized to 1; the distance r can be changed continuously with a slider.

The potential distribution (the value P can be chosen with the left slider), is calculated for planes in an adjustable distance z to the xy-plane, which can be set with the right slider. The potential distribution in the z-plane is shown by the intersection curve between the potential surface and the z-plane.

The fields diverge at the respective point sources, since lim 1 r = . In the simulation this is prevented, by excluding the plane z = 0. For a realistic field distribution one would have to work with extended charged objects instead of with point sources.

This simulation provides many possibilities, whose detailed description cannot be given here. The formulas are editable, such that in addition to the given fields additional fields can be calculated. In the description pages we discuss this further and useful experiments are proposed.

In Fig.8.3 you can see the whole interactive appearance for the three body problem with three equal objects. We show a potential cross section in the plane z = 0 in the “remote field”, where the potential surface has not yet decayed into partial surfaces.


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Figure 8.3: 3D-plot of the potential field of point sources lying in a plane. The figure shows the field of three similar point sources. With the right slider the mass or charge can be fixed, with the left slider the yellow intersection plane can be shifted and with the bottom sliders b,c the mass or charge ratios can be set. In addition different projections of the 3D-field can be chosen.

8.5.5 Potential Fields of Point sources as Contour Diagram

The visualization of potential curves in cross sectional planes shown above is very flexible. However it takes some careful thought to understand what is actually being calculated and what one sees. The extensive details in the description pages will assist in this regard.

This is the advantage of a presentation as contour diagram. It immediately shows a family of potential curves of equal potential distance in a plane.

For the computation the same algorithm as above is used. We now however only show the intersection of the yellow plane from Fig.8.3 with the potential surface and at the same time we show a number of potential lines (here 35, they cannot all be separated with the eye). The following interactive diagram Fig.8.4 shows in the xy-plane equipotential lines for a large mass with two smaller masses in its vicinity. One recognizes at the same time the near field, where the equipotential lines encircle the individual objects, and the far field, where the equipotential lines encircle all objects, as well as the neutral points, with gradP = 0, in which an object with out its own momentum would not know where to turn (the force as gradient acts in the direction of the largest potential change).


Figure 8.4 has not been translated yet!!!


Figure 8.4: Contour representation of a potential field, in the picture the field of a point source with two smaller satellites (mass ratio 0.2) is shown. The scanning plane can be moved with the right slider. 35 potential lines are calculated. That these appear to have corners at small distances is an artifact of the calculation.

The three following static pictures show next to each other the three-body potential of three equal bodies in a plane with distance z = 0.42 to the xy-plane, as well as dipole- and a quadrupole field which have very instructive appearance in this presentation. One must however take into account, that the equipotential lines in the individual z-planes do not represent identical potentials. The 35 potential lines are determined separately for each layer. Thus one obtains a qualitative picture, while the 3D-presentation is quantitative.

Via moving the z-plane one quickly obtains in this simulation an idea about the spatial distribution of the potential.


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Figure 8.5: Examples: contour diagrams for three point masses, for a dipole of a positive and a negative charge and quadrupole formed from two neighboring dipoles.

8.5.6 Plane Vector Fields

In reality vector fields have many components, derivatives and variables: three for the position, three for the components of the direction, the absolute value and the time. For the numerical calculation this does not really pose a problem, one simply has to do the calculation with the required number of dimensions.

One however has to accept many limitations for the visualization, since the desired relationships have to be shown as a projection on a plane. It becomes relatively easy if one assumes that the vector field is in a stationary state (no time dependence of the direction or absolute value) and if we restrict ourselves to vectors in a plane, as we will do in this subsection.

The local distribution of the vector direction can be shown quite clearly with arrows, whose origins are placed on a regular grid in the plane. The pictorial presentation of the absolute value of the vector is however less convincing. If one chooses the length of the arrows as measure the arrangement easily becomes unclear, since the dependence on the position can be quite strong, for example for a quadratic dependence from the distance to the source. If one chooses shades of colours, the achievable range is small and the presentation is only of a qualitative nature.

We have chosen a uniform arrow length. Colour shading gives a qualitative hint about the absolute value of the vector field. For the quantitative presentation of the absolute value of the vector we use the velocity of a test object, that moves in the vector field (ax,ax). Thus the time is used as another dimension of the presentation.

The red test object moves after the start along the field lines with a velocity that is determined from the components of the vector field ax,ay according to very simple, coupled ordinary differential equations (see chapter 9):

vx = dax dt ;vy = day dt .

In two dimensions especially the formula for the curl becomes clearer:

curla = ×a = x y ×ax ay = 0 0 ay x -ax y

diva = a = x y ax ay = ax x +ay y .

The curl has only got one component, namely in z-direction, since it has to be orthogonal to the xy-plane of the vector field. The source strength only depends on the changes in x-and y-direction.

In general the components of the vectors will will be functions of both variables x and y, ax = ax(x,y);ay = ay(x,y), such that the scalar divergence and the absolute value of the curl depend on the position. The direction of the curl is for a plane field always the orthogonal axis, normally called the z-axis.

In two dimensions the curl can be calculated easily for given formulas of the components. Remember, that one treats the respective second variable as constant for the partial differentiation with respect to the first one.

Vortices in the vector field can be easily recognized in the chosen presentation.

A vector field is vortex free, if its curl is vanishes everywhere:

rota = 0foray x -ax y = 0 ay x = ax y

This is for example the case, if ax = ax(x);ay = ay(y) i.e. if the components only depend on their own coordinate. In this case the partial derivatives vanish identically (further details are given in the simulation description).

Sources in a vector field can be recognized visually by sequences of vectors, i.e. field lines that start or end at them. A field is free of sources and sinks (negative sources), if the divergence vanishes everywhere.

diva = 0forax x + ay y = 0 ;ax x = -ay y

This is for example the case, when the vector components are independent of the coordinates, i.e. if the derivatives vanish identically.

In other cases one needs to examine critically if the formally satisfied condition provides an answer that make sense in all points of the vector field. This is for example no the case, if the limiting process when calculating the derivative results in a undetermined expression (00). A secure statement is obtained, if one surrounds the suspected source with a circle (a sphere in three dimensions) and sums up the number of field lines that cross the curve while taking the signs into account. Is the number of field lines entering the circle the same as the number of field lines leaving it, the corresponding point is free of sources. This statement becomes exact when integrating over the volume and taking the limit of vanishing radius.


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Figure 8.6: Animated movement of a test body in a given vector field. The topmost, not editable text field shows preset field types and their vector components ax,ay. The components can be edited separately in the small text fields, such that arbitrary fields can be created. The right slider controls the zoom factor, the lower slider the arrow length that is constant over the whole plane. For the preset fields divergence and rotation are displayed.

Fig.8.6, that shows at the start a vector field with two vortices, leads to the interactive simulation. The test object, that is initially at rest (initial velocity 0), can be moved to an arbitrary position of the field with the mouse prior to the time simulation in order to investigate the total field in detail.

With the selection field one can choose one of a number of typical fields. The formulas for the components as well as the respective divergence and curl are then given in text fields. The formulas field can be edited, such that arbitrary component formulas can be entered to study the corresponding fields.

The scale slider zoom on the right, allows to investigate the field on larger or smaller scales. This variation possibility is important, because the number of the arrows shown is constant for clarity, but at a larger scale details, such as vortices and sources, can be lost.

The arrow length, that is constant for the whole plane, can be adjusted with the second slider.

Further details and suggestions for experiments are given in the description pages of the simulation.

8.5.7 3D-Field due to Point Charges

In Fig.8.7 the vector field of a quadrupole is shown. It leads to the simulation of the general electric vector field of point charges. The distribution of its direction is visualized through a periodic grid of arrows having constant length, that show the local direction of the electric field strength vector in every point. The vector length is adjustable, but constant everywhere within the picture.

The absolute value of the field strength is indicated by colour shading. In addition a threshold value for the lowest absolute value for which vectors are shown can be selected. This provides a field strength dependent envelope surface for the whole field.

The opaque yellow plane can be shifted parallel to the z-axis, such that a spatial cross section through the vector field is shown.

The space orientation of the presentation can be adjusted with the mouse; in addition defined projections can be selected.

The number of objects can be chosen freely, and one can switch between particles of the same or opposite polarity. This shows the large difference of the far fields between multipoles of opposite polarity particle configurations of uniform polarity.

In the initial state all particles are positioned uniformly on a circle around the origin in the xy-plane. They can be individually moved with the mouse, such that arbitrary configurations are possible.

A convincing visualization of a situation that depends on so many parameters via projection to the observation plane requires a careful coordination of point distance, arrow length, threshold level and observation angle. The spatial impression becomes quite vivid if one changes the orientation of the projection slowly by pulling with the mouse.

The description contains further details and suggestions for experiments.


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Figure 8.7: 3D field of point sources that are located at arbitrary positions, in the figure a quadrupole consisting of two positive and and two negative point charges. With the sliders the arrow length, the threshold of the field strength level that are displayed and the position of the yellow scanning plane can be adjusted. The number of charges is entered by hand in the number field while the switch determines, whether the charges have all the same or alternating opposite signs. On the top different projections can be selected. All individual particles and the 3D-projection can be moved or adjusted with the mouse.

8.5.8 3D Movement of a Point Charge in a Homogeneous Electromagnetic Field

The movement of a charge in an electric field is quite easy to understand. It follows the direction of the electric field and the charge is accelerated proportionally to the absolute value of the electric field vector.

The movement in a magnetic field is much more complicated. In this case the vector product of magnetic field and velocity determines the acceleration of the charged test mass. Thus the charge is deflected at a right angle both to the magnetic field and the direction of its velocity and the strength of the deflection depends on the angle between magnetic field and current direction of movement, namely F ~v ×B. The effect of this force is, that the orbit moves in spirals around the direction of the magnetic field lines.


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Figure 8.8: Movement of a charge with given initial velocity vector in a homogeneous electromagnetic field. The components of the homogeneous fields and of the velocity vector are shown in the sliders at the right and can be set as initial values before starting the simulation. The velocity vector for the simulation is set with the lower slider. The simulation is started with the Start button.

Through the combined effect of magnetic and electric field the accelerations are added and very different movement patterns can come about. We want to visualize this for the simple example of a homogeneous field, for which the electric and magnetic field are constant in absolute value and direction in the whole space.

The interactive Figure 8.8 shows after opening the movement of a charge, whose initial velocity vector has components in positive y-direction and negative z-direction and that is subject to the accelerating effect of the green electric field vector and the “rolling up” effect of the red magnetic field. The spiral drawn in magenta is stretched in the direction of the E field. The axis of the spiral is parallel to the red vector of the magnetic field, as becomes apparent if turning the 3D-simulation appropriately.

Arbitrary initial conditions can be set with three sliders each for the homogeneous vector components of electric field, magnetic field and for the initial velocity of the point charge. After pushing the Start button, which triggers the calculation of the orbit using the respective differential equations, the sliders for the velocity components move according to their changes as function of time. The velocity of the calculation can be adjusted.

The start point of the charge can be adjusted with the mouse. resetObjects moves the start point to the origin, while leaving all other settings unchanged. reset changes all parameters back to their default settings. clear erases all orbits. Thus one can also superimpose orbits with different settings.

The simulation provides a wealth of possibilities, of which a few limiting cases are named below:

The description of the simulation contains further details and suggestions for experiments.

End of chapter 8