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In the figure shown, find the ratio of the linear charge densities $λ_{1}$ (on semi-infinite straight wire) and $λ_{2}$ (on semi-circular part), that is $λ_{1} / λ_{2}$ so that the field at O is along y direction.

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Detailed Solution

Electric field due to semi-circular part: $E_{2} = \frac{λ_{2}}{2 {πε}_{0} R}$

Electric field due to straight part: $E_{1 x} = E_{1 y} = \frac{λ_{1}}{4 {πε}_{0} R}$

For net field to be along y-axis: $E_{2} = E_{1 x} \Rightarrow \frac{λ_{2}}{2 {πε}_{0} R} = \frac{λ_{1}}{4 {πε}_{0} R} \Rightarrow \frac{λ_{1}}{λ_{2}} = 2$

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