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Trilinear Coordinates in Triangle of Line Charges

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Suppose three finite line charges with positive linear charge densities , , are located on the sides opposite the vertices of the triangle . It can be shown that the electric field is zero at the point with trilinear coordinates that are in the same ratios as .

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You can drag the vertices , , and vary the charge densities with the controls. The graphic shows a stream plot of the resulting electric field within the triangle.

Check "show trilinear coordinates" to see the point where the three contributions to the field cancel (where ); it has trilinear coordinates , , . The ratios of the densities to the trilinear coordinates then become equal: , as shown in the inset table.

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Contributed by: S. M. Blinder (January 2023)
Open content licensed under CC BY-NC-SA

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Details

The starting point for computation of the electric field is the potential for a finite uniformly charged rod of length and linear charge density , parallel to the axis:

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The electric field is then given by . The total field is a superposition of three such terms, each rotated to coincide with a side of the triangle.

Reference

[1] Suren. "A New Electric Field Interpretation of Barycentric and Trilinear Coordinates." (Aug 2, 2022) https://faculty.evansville.edu/ck6/encyclopedia/ANewElectricFieldInterpretationSuren.pdf

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