Trilinear Coordinates in Triangle of Line Charges

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 .
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.

THINGS TO TRY

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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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