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Surface Charge Distributions in Equilibrium

Surface Charge Distributions in Equilibrium
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2018-11-01 10:08
Solid GeometryCalculusIntegral
This example demonstrates equilibrium surface charge distributions. It involves core calculus concepts such as integration and differentiation. Using GeoGebra commands like CurveCartesian, Point, Union, Vector, etc., complex 3D shapes can be constructed. Suitable for university physics and mathematics courses.

GeoGebra Commands & Parameters

  1. CurveCartesian((a * cos(q)), (b * sin(q)), q, skip, π / 2 - skip) → d
  2. Point(d) → A
  3. CurveCartesian((a * cos(q)), (b * sin(q)), q, acos(x(A) / a) + skip, π / 2 - skip) → e
  4. Point(e) → B
  5. CurveCartesian((a * cos(q)), (b * sin(q)), q, acos(x(B) / a) + skip, π / 2 - skip) → f
  6. Point(f) → C
  7. CurveCartesian((a * cos(q)), (b * sin(q)), q, acos(x(C) / a) + skip, π / 2 - skip) → g
  8. Point(g) → D
  9. CurveCartesian((a * cos(q)), (b * sin(q)), q, acos(x(D) / a) + skip, π / 2 - skip) → h
  10. Point(h) → E
  11. Union(Union[Mirror[orig, yAxis], Mirror[orig, (0, 0)]], Mirror[orig, xAxis]) → reflections
  12. Vector(A, A + Sum[Sequence[(kqsq * Vector[Element[reflections, k], A] / Distance[A, Element[reflections, k]]^(3)), k, 1, 15]] + (kqsq * Vector[B, A] / Distance[A, B]^(3)) + (kqsq * Vector[C, A] / Distance[A, C]^(3)) + (kqsq * Vector[D, A] / Distance[A, D]^(3)) + (kqsq * Vector[E, A] / Distance[A, E]^(3))) → F_A
  13. Vector(B, B + Sum[Sequence[(kqsq * Vector[Element[reflections, k], B] / Distance[B, Element[reflections, k]]^(3)), k, 1, 15]] + (kqsq * Vector[A, B] / Distance[B, A]^(3)) + (kqsq * Vector[C, B] / Distance[B, C]^(3)) + (kqsq * Vector[D, B] / Distance[B, D]^(3)) + (kqsq * Vector[E, B] / Distance[B, E]^(3))) → F_B
  14. Vector(C, C + Sum[Sequence[(kqsq * Vector[Element[reflections, k], C] / Distance[C, Element[reflections, k]]^(3)), k, 1, 15]] + (kqsq * Vector[B, C] / Distance[C, B]^(3)) + (kqsq * Vector[A, C] / Distance[C, A]^(3)) + (kqsq * Vector[D, C] / Distance[C, D]^(3)) + (kqsq * Vector[E, C] / Distance[C, E]^(3))) → F_C
  15. Vector(D, D + Sum[Sequence[(kqsq * Vector[Element[reflections, k], D] / Distance[D, Element[reflections, k]]^(3)), k, 1, 15]] + (kqsq * Vector[B, D] / Distance[D, B]^(3)) + (kqsq * Vector[C, D] / Distance[D, C]^(3)) + (kqsq * Vector[A, D] / Distance[D, A]^(3)) + (kqsq * Vector[E, D] / Distance[D, E]^(3))) → F_D
  16. Vector(E, E + Sum[Sequence[(kqsq * Vector[Element[reflections, k], E] / Distance[E, Element[reflections, k]]^(3)), k, 1, 15]] + (kqsq * Vector[B, E] / Distance[E, B]^(3)) + (kqsq * Vector[C, E] / Distance[E, C]^(3)) + (kqsq * Vector[D, E] / Distance[E, D]^(3)) + (kqsq * Vector[A, E] / Distance[E, A]^(3))) → F_E
  17. Tangent(E, c) → i,
  18. Tangent(D, c) → j,
  19. Tangent(C, c) → k,
  20. Tangent(B, c) → l,
  21. Tangent(A, c) → m,
  22. OrthogonalLine(E, i) → n
  23. OrthogonalLine(D, j) → p
  24. OrthogonalLine(C, k) → q
  25. OrthogonalLine(B, l) → r
  26. OrthogonalLine(A, m) → s
  27. Point(n) → F
  28. Point(p) → G
  29. Point(q) → H
  30. Point(r) → I
  31. Point(s) → J
  32. Ray(A, J) → t
  33. Ray(B, I) → f_1
  34. Ray(C, H) → g_1
  35. Ray(D, G) → h_1
  36. Ray(E, F) → i_1
  37. Integral(sqrt(((a * sin(x)))^(2) + ((b * cos(x)))^(2)), acosd(x(A) / a), acosd(x(B) / a)) → l2
  38. Integral(sqrt(((a * sin(x)))^(2) + ((b * cos(x)))^(2)), acosd(x(B) / a), acosd(x(C) / a)) → l3
  39. Integral(sqrt(((a * sin(x)))^(2) + ((b * cos(x)))^(2)), acosd(x(C) / a), acosd(x(D) / a)) → l4
  40. Integral(sqrt(((a * sin(x)))^(2) + ((b * cos(x)))^(2)), acosd(x(D) / a), acosd(x(E) / a)) → l5
  41. CurveCartesian((a * cos(q)), (b * sin(q)), q, (-acos(x(A) / a)), acos(x(A) / a)) → o
  42. CurveCartesian((a * cos(q)), (b * sin(q)), q, acosd(x(A) / a), acosd(x(B) / a)) → u
  43. CurveCartesian((a * cos(q)), (b * sin(q)), q, acosd(x(B) / a), acosd(x(C) / a)) → v
  44. CurveCartesian((a * cos(q)), (b * sin(q)), q, acosd(x(C) / a), acosd(x(D) / a)) → w
  45. CurveCartesian((a * cos(q)), (b * sin(q)), q, acosd(x(D) / a), acosd(x(E) / a)) → a_1
  46. CurveCartesian((a * cos(q)), (b * sin(q)), q, acosd(x(E) / a), π - acosd(x(E) / a)) → b_1