Visualisation of the gradient fields of a multifocal (k = 1..n) curve with n foci for a generalised potential function of the form φ(r)=Σqₖ*|rₖ –r|ᵐ, where m∈ℝ

A multifocal oval is a curve defined as the locus of a point A(r) that moves in such a way that the sum of the distances φ(r) to n points remains constant. It is also known as a Multifocal Ellipse (n-ellipse) with n foci (or n poles). The potential function φ(r) is generalized by this applet to generate the corresponding multifocal curves. The sum of the m-th powers of the distances to n points is given here by φ(r)=Σqₖ*|rₖ–r|ᵐ, where k=1..n and m∈ℝ. qₖ - weight factors, which here, by analogy with electromagnetism, we will call "charges". ●The purpose of the applet is to visualise double orthogonal systems for scalar functions of the above type. In particular: for m = -1, this is the well-known result in electromagnetism that electric field lines and equipotential lines form a double orthogonal system. For m=1 -n-ellipse , m=2 - case. ●The applet constructs loci. For a given scalar field φ(r), it creates -equipotential lines, and for the corresponding gradient fields E(r), it creates -iso-magnitude lines, which are lines of constant field magnitude (but not their direction!), denoted by modE. The vector field E(r) is obtained by taking the gradient of the potential function φ(r). ● For a set of point "charges", these loci are always closed, and the "charges" are their foci. In other words, these lines are multifocal curves. ● Equipotential lines are always perpendicular to the field lines of the conservative vector field E(r), which are calculated numerically in the applet. Description of Services of this applet you will find in 1, images of examples in 2, 3, 4, 5, 6, 7, 8, 9.

[size=85] The Heatmap, created by scanning outside parameter [b][color=#980000]c∈[c01,c02][/color][/b], draws black and white curves.[/size] — The Heatmap, created by scanning outside parameter **c∈[c01,c02]**, draws black and white curves.

Examples of five charges

[size=85][b][color=#333333]Figure 1:[/color][/b][color=#333333]The [/color][i][b][color=#ff00ff]equipotential lines[/color][/b][/i][color=#333333] of a scalar field [/color][b][color=#ff00ff]φ [/color][/b][color=#333333]are always perpendicular to the [/color][i][color=#980000][b]lines of force[/b] [/color][color=#333333]of the corresponding gradient field [/color][/i][b]E[/b][color=#333333]. [/color]
[/size] — **Figure 1:**The **equipotential lines** of a scalar field φ are always perpendicular to the **lines of force** of the corresponding gradient field E.

[size=85][b]Figure 2: [/b][color=#333333]The [/color][b]Heatmap [/b][color=#333333]showing the distribution of the scalar field [/color][b][color=#ff00ff]φ[/color][/b][color=#333333] around the charges, along with the directions of the corresponding gradient vector field [/color][b]E[/b][color=#333333], is depicted as [/color][b][color=#ea9999]arrows[/color][/b][color=#333333]. Here the [/color][b][color=#ff00ff]Equipotential lines[/color][/b][color=#333333] of the scalar field [/color][b][color=#ff00ff]φ[/color][/b][color=#333333] are closed curves whose [/color][b][color=#ff0000]foci[/color][/b][color=#333333] are located on the charges.[/color][/size] — **Figure 2:** The **Heatmap** showing the distribution of the scalar field φ around the charges, along with the directions of the corresponding gradient vector field E, is depicted as **arrows**. Here the **Equipotential lines** of the scalar field φ are closed curves whose **foci** are located on the charges.

[size=85][b][color=#333333]Figure 3[/color][color=#980000]: [/color][/b]The [b]Heatmap[/b] and [b]Contour lines[/b] shows the distribution of the [i]constant value of the vector field strength modulus[/i] [b]E[/b] for five point charges (denoted [b]modE[/b]). In this case, the [i]lines representing the constant value of the vector field strength[/i] modulus [b]E[/b] (but not their directions!) always form closed curves, the [b]foci[/b] of which are[i] located at the charges[/i].
[/size] [size=85]The table shows the values of the potential [b][color=#ff00ff]φ[/color][/b] and [b]modE[/b] at test points 1–6, which are located on the locus [b]modE[/b]=[color=#980000][b]c[/b][/color]. Test points 7 and 8 are at points where the field [b]E[/b] is almost zero.[/size] — **Figure 3:** The **Heatmap** and **Contour lines** shows the distribution of the *constant value of the vector field strength modulus* E for five point charges (denoted **modE**). In this case, the *lines representing the constant value of the vector field strength* modulus E (but not their directions!) always form closed curves, the **foci** of which are located at the charges. The table shows the values of the potential φ and **modE** at test points 1–6, which are located on the locus **modE**=c. Test points 7 and 8 are at points where the field E is almost zero.

Visualisation of the gradient fields of a multifocal (k = 1..n) curve with n foci for a generalised potential function of the form φ(r)=Σqₖ*|rₖ –r|ᵐ, where m∈ℝ

Neue Materialien

Entdecke Materialien

Entdecke weitere Themen