_{1}

Calculation of the interactive force between two horizontally stacked circular uniformly charged rings placed along the common vertical axis conducive to nonlinear oscillations under gravity has been addressed [1]. Although challenging, nonetheless the scope of the study limited to uniform charge distributions of the rings. Here we extend the analysis considering a charged ellipse with a nonuniform, curvature-dependent elliptic charge distribution exerting a force on a point-like charge placed on the vertical symmetry axis. Nonuniform charge distribution and its impact on various practical scenarios are not a common theme addressed in literature. Applying Computer Algebra System (CAS) particularly
*Mathematica* [2], we analyze the issue on hand augmenting the traditional scope of interest. We overcome the CPU expensive symbolic computation following our newly developed numeric/symbolic method [1]. For comprehensive understanding, we simulate the nonlinear oscillations.

Calculating the electric field of a uniform charge distribution on common geometric objects such as circle, rectangle, square, rumbas, etc., among others along their vertical symmetry axis is trivial [

The shown scenario in

To begin with, the needed parameters are stored in the listed values1, units are all in SI, the values are feasibly practical.

values1={k→9.10^{9},q1→1.10^{−6},q2→2.10^{−6},a→2.,b→0.5,m→0.110^{−3},g→9.8};

The listing includes the electrostatic coupling constant, k = 1 4 π ∈ 0 = 9 × 10 9 with ∈ 0 being the permittivity of vacuum, q_{1} is the charge on the ellipse. The loose point-like particle has a mass m and charge q_{2}. The a and b are the semi major and minor lengths of the ellipse, respectively with implicit eccentricity e = 1 − ( b a ) 2 , e = 0.96 [

is practical. However, the mass of the particle ought to be determined compatibly to fulfilling the objective of the search: oscillations. An ill-posed mass would cause catastrophe; the particle falls through the ellipse with no chance of return more on this in follow-up paragraphs.

It is beneficiary quantifying the latter paragraph. The code below generates two ellipses and their associated curvatures.

plotellipse1=ParametricPlot[{a Cos[θ],b Sin[θ]}/.values1,{θ,0,2π},AxesLabel ->{"x","y"},GridLines->Automatic,PlotStyle->Black];

plotellipse2=ParametricPlot[{a Cos[θ],b Sin[θ]}/.values2,{θ,0,2π},AxesLabel ->{"x","y"},GridLines->Automatic,PlotStyle->{Dashing[{0.01}],Black}];

plotEllipses21=Show[{plotellipse2,plotellipse1}];

plotcurature1=Plot[(a b)/((a Sin[θ])^2+(b Cos[θ])^2)^{3/2} 1/2 (a+b)/.values1, {θ,0,2π},PlotRange->All,GridLines->{{0,1/2 π,π,3/2 π,2π},Automatic}, AxesOrigin->{0,0},Ticks->{{0,1/2 π,π,3/2 π,2π},Automatic},AxesLabel ->{"θ(rad)","Curvature"},PlotStyle->Black];

plotcurature2=Plot[(a b)/((a Sin[θ])^2+(b Cos[θ])^2)^{3/2} 1/2 (a+b)/.values2, {θ,0,2π},PlotRange->All,GridLines->{{0,1/2 π,π,3/2 π,2π},Automatic}, AxesOrigin->{0,0},Ticks->{{0,1/2 π,π,3/2 π,2π},Automatic},AxesLabel ->{"θ(rad)","Curvature"},PlotStyle->{Dashing[{0.01}],Black}];

plotCurvature21=Show[{plotcurature2,plotcurature1}];

As shown in

As the second step we focus characterizing the linear-charge density of the ellipse. Accordingly, we modify the basic definition of density incorporating the elliptic curvature,

λ ( θ ) = q 1 π [ 3 ( a + b ) − ( 3 a + b ) ( a + 3 b ) ] a b [ ( a sin [ θ ] ) 2 + ( b cos [ θ ] ) 2 ] 3 2 1 2 ( a + b ) (1)

The denominator of the first term in (1) is the approximate length of the ellipse’s circumference [_{1} i.e. ∫ λ ( θ ) d l = q 1 with d l = ( a sin [ θ ] ) 2 + ( b cos [ θ ] ) 2 d θ we introduce a normalization factor enforcing this characteristic. This factor for listed values1 is,

normalizationFactor1=NIntegrate[1/(π(3(a+b)-√((3a+b)(a+3b)))) 1/2 (a+b) (a b)/((a Sin[θ])^2+(b Cos[θ])^2)^{3/2} √((aSin[θ])^2+(bCos[θ])^2))/.values1, {θ,0,2π}];

So that the normalized λ ( θ ) is,

λ1[θ_]=1/normalizationFactor1 q1/(π(3(a+b)-√((3a+b)(a+3b))))

1/2 (a+b) (a b)/((a Sin[θ])^2+(b Cos[θ])^2)^{3/2};

To evaluate the electric field of the charged ellipse along its vertical symmetry axis through the center at a height z we calculate first the electrostatic potential V(z) applying,

V ( z ) = k ∫ λ ( θ ) 1 distance ( z ) d l , (2)

where in (2) the distance ( z ) = z 2 + ( a cos [ θ ] ) 2 + ( b sin [ θ ] ) 2 . Substituting the latter in (2) and the rest of the terms i.e. the above noted, λ(θ) and dℓ results an integral that Mathematica is incapable evaluating symbolically; this is a CPU expensive procedure. To overcome this issue, we apply our method introduced and outlined in [

potentialintegrand1[z_,θ_]=λ1[θ]1/√(z^2+(aCos[θ])^2+(bSin[θ])^2)√((aSin[θ])^2+(bCos[θ])^2)/.values1;

V1[z_]=Table[{z,NIntegrate[Evaluate[kpotentialintegrand1[z,θ]/.values1],{θ,0,2π}]},{z,0,10,1}];

As shown, the potentials, V(z) beyond z = 4 m irrespective of the size of the ellipse, {a, b} are indistinguishable. There are differences at shorter heights. The

differences are not ignorable yet have common similarities. The lowest set describes the ring with the least potential. This sounds, because intuitively the ring has the largest size a = b = 2.0 corresponding to the largest distance(z). On the contrary the top data set has the strongest potential associated with the pinched ellipse with the shortest distance(z). The observed similarities lead to exploring one unique analytic model function fitting the data. It appears at smaller z’s the potential behaves as Gaussian and at far distances fall off exponentially. With these observed featured and with some trial and error we build the model accordingly as:

modelz1[z_]:=c1+d1z+e1e^(-f1z^2)

fitV1z=FindFit[V1[z],modelz1[z],{c1,d1,e1,f1},z];

We then plot the data and the fitted modeled functions.

plotfitdataz1=Plot[modelz1[z]/.fitV1z,{z,0,10},PlotStyle→Black,AxesOrigin →{0,0}];

Next, we apply the fundamental relationship, E = − ∇ V ( z ) to calculate the electric field assisting to calculate the force, F = q 2 E . These are coded accordingly,

Efield1=-D[Evaluate[modelz1[z]/.fitV1z],{z,1}];

Eforce1=(q2/.values1)Efield1//Simplify;

As shown all three modeled functions have the same force value at the shorted height and do overlap indistinguishably at high heights. In the mediocre heights they reach a local maximum. The weakest of the three is the dashed curve and is the one associated with the circular ring as commented previously.

Force axis in

Solve[Eforce1==mg/.values1,z]

{{z→0.23},{z→3.4}}

As shown, for the selected mass there are two associated equilibrium positions. This is meaningful because the force shown on

Now we set up the equation of motion. Applying Newton’s force law,

F n e t = m z ¨ ( t ) with the standard notation that the supper double dots are the acceleration. The F n e t = F e l e c t r i c − m g . The needed code is,

EquationOfMotion1=z1"[t]-(Eforce1/.z→z1[t])(1/m/.values1)+g/.values1;

Numeric solutions of this equation with appropriately chosen initial conditions are coded below. The initial position of the charged particle is set higher than the equilibrium position and is dropped freely with initial zero speed.

solEquationOfMotion1=NDSolve[{EquationOfMotion1==0,z1[

The output is oscillatory as expected. Plots of oscillations are shown in

Having this information in hand we craft a code putting the charged particle in motion. Since the format of the journal does not allow showing the animation, we embed its snapshot. The interested reader capable of running Mathematic code may request for a copy of the code. The animation gives a fell how a nonlinear oscillation behaves and what the impact of the nonuniform charge distribution is (

With a few objectives, we proposed this investigating research project. The objectives stem from the interest in the nonlinear oscillations of physical phenomena in conjunction with electrostatic related physics issues. To begin with we considered a charged ellipse. An ellipse has less geometric symmetries vs. a circle. And, naturally, the charge distribution on an ellipse inherits the same broken symmetries. In general, “the less the symmetry, the more the challenges.” Distribution of charge on an ellipse is a curvature-dependent, uneven function. We have overcome the challenges caused by this unevenness conducive calculating quantities such as electrostatic potential and electric field; these are seldomly discussed in scientific literature. Having calculated these quantities, we furthered the investigation by applying them to scenarios where a loose point-like charged particle oscillates. The uneven elliptic charge distribution creates peculiar electric field and consequently makes the oscillations nonlinear. The entire calculations in this work are carried out applying Computer Algebra System (CAS for short), specifically Mathematica. In occasions, the numeric aspect of the computation surpassed the symbolic. For instance, evaluation of the electrostatic potential due to the unevenness of the charge distribution is concluded applying semi symbolic-numeric approach, similar to what we reported [

The author gracefully acknowledges the John T. and Page S. Smith Professorship funds for completing and publishing this work.

The author declares no conflicts of interest regarding the publication of this paper.

Sarafian, H. (2020) Impact of Eccentricity on Nonlinear Oscillations of a Point-Like Charge in the Electric Field of a Curvature-Dependent Elliptic Charged Ellipse. American Journal of Computational Mathematics, 10, 603-611. https://doi.org/10.4236/ajcm.2020.104035