Aime F. answered • 02/12/23
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PhD in Physics (Yale), have taught Methods of Engineering Analysis
It actually makes a big difference if the question is asked about 3 points in a 2D universe, or 3 points in a 2D plane z = 0 contained in a 3D universe. Let's assume the latter case, because it's simpler. In vector notation, we're asked for the variables Ex, Ey, E, η in E = (Ex,Ey,Ez) = (cosζcosη,cosζsinη,sinζ)E. We will see that Ez and ζ will both be zero because the 3 points lie in the same plane.
By Coulomb's Law, E = (q₁r₁/|r₁|³ + q₂r₂/|r₂|³)/4πε₀, where r₁ and r₂ are the displacement vectors from points P₁ and P₂ to P.
Aime F.
In the other case, "a 2D universe" is like a 3D universe in which nothing can depend on z. So the field E from a single charge must point transversely from the axis of a cylinder of radius r, and E must depend only on r (not z or azimuth). The Gauss Law for a cylinder segment of length L then reads 2πrLE = λL/ε₀ where λ is the charge linear density in C/m units. I don't see how to interpret your question this way, so I recommend my previous interpretation.
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02/13/23
Aime F.
If it were a 2D universe, Coulomb's law would have a different dependence on the displacement-vector magnitudes.02/12/23