A thread with uniform linear charge density λ is bent into a circular arc of radius R subtending angle θ at the centre. The arc is symmetric about its angular bisector. The electric field magnitude at the centre is:
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Question

A thread with uniform linear charge density λ is bent into a circular arc of radius R subtending angle θ at the centre. The arc is symmetric about its angular bisector. The electric field magnitude at the centre is:

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G1 Physics · Accepted Answer

For an element subtending dφ, dq = λR dφ and dE = k dq/R² = (kλ/R)dφ. Components perpendicular to the angular bisector cancel. Along the bisector, E = (kλ/R) ∫[-θ/2 to θ/2] cosφ dφ = (2kλ/R) sin(θ/2).