Testing Gauss’ Law for a Relativistically Oscillating

Point Charge in an Ellipsoid

G.R.Dixon, 7/20/2010

In a previous article it was demonstrated that Gauss’ law is obeyed for a relativistically oscillating point charge within a spherical surface. In this article it is demonstrated that the same may be true when the charge oscillates within an ellipsoid surface. The flux of E through this surface is computed at time t=0, which is when the particle passes through the origin. Once again the particle’s motion is

.

The formula for the ellipsoid cross section is

.

Fig. 1 plots the top half of the ellipsoid cross section.

Figure 1

Ellipsoid Cross Section

r is defined to be the chord from the origin to a point on the ellipse. Fig. 2 plots 1/r^{2} over the range of x. Fig. 3 plots the normal component of **E** at points on the ellipse. Note how, in this relativistic case, the two curves are significantly different.

Figure 2

1/r^{2} vs. x

Figure 3

E_{normal} vs x

The flux of E_{normal} through the ellipsoid surface is computed to be 112962897. On the other hand, q/e_{o}112994350. The percent difference (2.78E-2%) might be ascribable to numerical error, or it might denote a small but significant failure of Gauss’ law for the ellipsoid surface. The program used to compute the data in this article is included in the Appendix.

***Appendix***

Private Sub cmdEllipsoid_Click()

'*******************

'Compute the E Field Flux through an ellipsoid surface containing

'an oscillating charge.

'*******************

Const c As Double = 300000000# 'speed of light

Const eps0 As Double = 0.00000000000885 'permittivity constant

Const pi As Double = 3.141592654

Const q As Double = 0.001 'oscillating charge equals .001 coulomb

Const A As Double = 1 'amplitude of oscillation equals 1 meter

Const omega As Double = 0.99 * c / A 'max q speed is highly relativistic

Const steps As Long = 5000 'number of iterations

Const deltax As Double = 6 / steps

Dim i, j As Long 'loop counter

Dim dArea(steps) As Double 'Area increment of ring

Dim Px, Py As Double

Dim Ex(steps), Ey(steps) As Double 'electric field components

Dim Enormal(steps) As Double

Dim dtmin, dtmax, dt As Double

Dim ux, uy As Double

Dim drx, dry, dr As Double

Dim tr As Double 'retarded time

Dim x(steps), xr As Double

Dim y(steps) As Double

Dim dydx(steps) As Double

Dim r(steps) As Double

Dim ds(steps) As Double

Dim vr As Double

Dim ar As Double

Dim Eflux As Double

Dim Flux(steps) As Double 'Flux increments

Dim normx(steps) As Double

Dim normy(steps) As Double

Dim deltay(steps) As Double

Dim slopenormal(steps) As Double

Dim phi(steps) As Double

Eflux = 0

For j = 0 To steps - 1

x(j) = -3 + j * deltax

y(j) = 2 * Sqr(1 - x(j) ^ 2 / 9)

If Abs(x(j)) <> 3 Then

deltay(j) = -2 * x(j) / 3 / Sqr(9 - x(j) ^ 2)

Else

deltay(j) = deltax

End If

If Abs(x(j)) = 3 Then

ds(j) = 0

Else

ds(j) = Sqr((81 - 5 * x(j) ^ 2) / (81 - 9 * x(j) ^ 2)) * deltax

End If

dArea(j) = 2 * pi * y(j) * ds(j)

If Abs(x(j)) <> 3 Then

dydx(j) = -2 * x(j) / 3 / Sqr(9 - x(j) ^ 2)

End If

slopenormal(j) = 0

If dydx(j) <> 0 Then slopenormal(j) = Abs(-1 / dydx(j))

phi(j) = Atn(slopenormal(j))

If x(j) < 0 Then

normx(j) = -Cos(phi(j))

Else

normx(j) = Cos(phi(j))

End If

normy(j) = Abs(Sin(phi(j)))

r(j) = Sqr(4 + 5 * x(j) ^ 2 / 9)

Next j

Open "c:\\WINMCADC\Physics\GaussTest.PRN" For Output As #1

For j = 0 To steps - 1

Write #1, x(j), 1 / r(j) ^ 2

Next j

Close

MsgBox ("Ready to plot 1/rsquared")

'Compute electric field at points on the ellipse.

For j = 0 To steps - 1

Px = x(j)

Py = y(j)

dtmin = 0

dtmax = 10 * A

Do

dt = (dtmin + dtmax) / 2

tr = -dt

xr = A * Sin(omega * tr)

drx = Px - xr

dry = Py

dr = Sqr(drx ^ 2 + dry ^ 2)

If Abs(c * dt - dr) < 2 ^ (-30) Then Exit Do

If c * dt - dr > 0 Then

dtmax = dt

Else

dtmin = dt

End If

Loop

vr = omega * A * Cos(omega * tr)

ar = -(omega ^ 2) * A * Sin(omega * tr)

ux = c * drx / dr - vr

uy = c * dry / dr

Ex(j) = q / (4 * pi * eps0) * dr / (drx * ux + dry * uy) ^ 3 * (ux * (c ^ 2 - vr ^ 2) + dry * (-uy * ar))

Ey(j) = q / (4 * pi * eps0) * dr / (drx * ux + dry * uy) ^ 3 * (uy * (c ^ 2 - vr ^ 2) - drx * (-uy * ar))

Enormal(j) = Ex(j) * normx(j) + Ey(j) * normy(j)

Flux(j) = Enormal(j) * dArea(j)

Eflux = Eflux + Flux(j)

Next j

Open "c:\\WINMCADC\Physics\GaussTest.PRN" For Output As #1

For j = 0 To steps - 1

Write #1, x(j), Enormal(j)

Next j

MsgBox ("Ready to plot ENormal")

Close

Open "c:\\WINMCADC\Physics\GaussTest.PRN" For Output As #1

For j = 0 To steps - 1

Write #1, x(j), Flux(j)

Next j

Close

'MsgBox ("Ready for plotting dFluxes")

MsgBox ("E Flux = " & Eflux & "; Gauss = " & q / eps0)

MsgBox ("Percent Difference = " & Abs(Eflux - q / eps0) / Eflux * 100)

Stop

End Sub