Charge Q is distributed uniformly on length l of a wire. It is bent in the form of a semicircular ring. Find the electric field at the centre of the ring.
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Correct option is D)
Consider two small elements of length dl each charge dq on each element. If λ is the line charge density, Q=λl=λ.πr. as wire is bent in the form of semicircular ring. Hence , dq=λdl=πrQ.(rdθ)=πQdθ. Here sine component of the field at center cancel out and only cosine will contribute the filed. thus, dEc=2dEcosθ=24πϵ0r2dqcosθ=2π2ϵ0r2Qcosθdθ or Ec=2ϵ0(πr)2Q∫0π/2cosθdθ=2ϵ0(πr)2Q(sin2π−sin0)=2ϵ0l2Q as (l=πr)