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An are of radius r subtends an angle θ at the center with x-axis in a Cartesian coordinate system. A charge is distributed over the arc such that the linear charge density is λ . Calculate the electric field at the origin.

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Consider small element on the arc as point charge and then proceed by integrating for all such points. The electric charge distributed on the portion of the arc making an angle d ϕ is dQ=λrdϕ.produced due to this portion at the origin will be

 

The electric field vector dE of this portion of the arc is indicated in the diagram.

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