Undergrad Electric field of ring at any point

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Calculating the electric field due to a uniformly charged ring at points not on the perpendicular axis involves complex integrals. Users have noted that applying Coulomb's law and the divergence of electric potential often leads to complicated calculations. The integral for this scenario does not yield a simple closed-form expression, which can be a source of frustration. It's essential to approach the problem with an understanding of the complexities involved. The discussion emphasizes the challenges of finding a straightforward solution in this context.
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How to calculate the field due to a uniformly charged ring at a point not on the perpendicular axis of the plane of the ring
I have tried using coulomb's law and divergennce of the electric potential but I always stumble with a complicated integeral
 
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That’s because the corresponding integral is complicated. You should worry if you found a simple closed form expression.
 
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Thread 'What determines if wave components are independent vs coherent?'

I'm working through something and want to make sure I understand the physics. In a system with three wave components at 120° phase separation, the total energy calculation depends on how we treat them: If coherent (add amplitudes first, then square): E = (A₁ + A₂ + A₃)² = 0 If independent (square each, then add): E = A₁² + A₂² + A₃² = 3/2 = constant In three-phase electrical systems, we treat the phases as independent — total power is sum of individual powers. In light interference...
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