A thin semi-circular ring of radius r has a positive charge q distributed uniformly over it. The net field ⃗E at the centre O is [the ring lies in the upper half plane, ̂i along the axis of the diameter and ̂j pointing from O towards the arc]

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(4) -q/(2π²ε₀r²) ̂j. Linear charge density λ=q/(π r). An element at angle φ contributes dE=(λ dφ)/(4πε₀r) directed from the element towards O. By symmetry about the vertical axis, the components along ̂i from the two halves cancel; only the components along the vertical survive. E=∫₀^π(λ sin φ dφ)/(4πε₀r)=λ/(4πε₀r)×2=λ/(2πε₀r). Substituting λ=q/(π r): E=q/(2π²ε₀r²). The ring is positive and sits in the upper half, so the field at O points downward, away from the arc: ⃗E=-q/(2π²ε₀r²) ̂j.

A thin semi-circular ring of radius r has a positive charge q distributed uniformly over it. The net field ⃗E at the centre O is [the ring lies in the upper half plane, ̂i along the axis of the diameter and ̂j pointing from O towards the arc]

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