Lines of Force for Two Point Charges

In an earlier Demonstration by Stephen Wolfram, two-dimensional equipotential contours ϕ(x,y) , representing cross sections of equipotential surfaces, were plotted for two point charges of variable magnitude and location. It was also possible to make a vector plot of the corresponding electric field E(x,y). It is much trickier to plot the lines of force emanating from a positive point charge and converging on a negative point charge. These are everywhere tangent to the electric field vectors and form a set of contours ψ(x,y), shown as red curves, everywhere orthogonal to the equipotentials ϕ(x,y), shown in gray.

Derivation of ψ(x,y) is quite complicated in either Cartesian or spherical coordinates. It is, however, straightforward in prolate ellipsoidal coordinates defined by the variables ξ=((Subscript[r, 1])+(Subscript[r, 2]))/R and η=((Subscript[r, 1])-(Subscript[r, 2]))/R. Here (Subscript[r, 1]) and (Subscript[r, 2]) are the distances from the field point to charges (Subscript[q, 1]) and (Subscript[q, 2]), respectively, and R is the distance between the two charges. Expressed in the Cartesian coordinates of the graphic: (Subscript[r, 1])=(Sqrt[(((x-(Subscript[x, 1])))^2)+(((y-(Subscript[y, 1])))^2)]), (Subscript[r, 2])=(Sqrt[(((x-(Subscript[x, 2])))^2)+(((y-(Subscript[y, 2])))^2)]), R=(Sqrt[((((Subscript[x, 2])-(Subscript[x, 1])))^2)+((((Subscript[y, 2])-(Subscript[y, 1])))^2)]). The equipotentials are given by ϕ(ξ,η)=((Subscript[q, 1])/(2R(ξ+η)))+((Subscript[q, 2])/(2R(ξ-η))). The function ψ(ξ,η) representing the orthogonal network of lines of force must then satisfy the equation ∇ϕ(ξ,η)·∇ψ(ξ,η)=0. The solution ψ(ξ,η)=(1/((ξ^2)-(η^2)))[(Subscript[q, 1]) (ξ-η+(ξ^2)η-ξ (η^2))+(Subscript[q, 2]) (-η+(ξ^2) η+-ξ+ξ (η^2))], transformed back to Cartesian coordinates, is represented by the red contour plot shown in the graphic.

THINGS TO TRY

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DETAILS

Snapshot 1: lines of force for two unequal charges of opposite sign

Snapshot 2: repulsion between like charges

Snapshot 3: lines of force for a point dipole, obtained for two infinitesimally separated opposite charges

RELATED LINKS

Contributed by: S. M. Blinder

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