Solution:
Solution
Given,
$$k=\dfrac{1}{4\pi {{\varepsilon }_{o}}}\,\,and\,\,\lambda =\dfrac{q}{\pi R}\,\,\,\,\,(\,where,\,\lambda =\text{linear}\,\text{charge}\,\text{density})$$
$$\text{Small}\,\text{charge}\,\text{on}\,\text{small}\,\text{length(Rd}\theta )\,\,is\,,\,\,dq=\lambda Rd\theta $$
$$ \vec{E}=\int\limits_{-\pi /2}^{\pi /2}{dE}\cos \theta =2\int\limits_{0}^{\pi /2}{\dfrac{k\left( \lambda Rd\theta \right)}{{{R}^{2}}}}\cos \theta \,\left( -\hat{j} \right) $$
$$ \Rightarrow \vec{E}=2\times \left( \dfrac{1}{4\pi {{\varepsilon }_{0}}} \right)\,\,\,\left( \dfrac{q}{\pi R} \right)\dfrac{R}{{{R}^{2}}}[\sin \theta ]_{0}^{\pi /2}\,\left( -\hat{j} \right) $$
$$ \Rightarrow \dfrac{q}{2{{\pi }^{2}}{{\varepsilon }_{0}}{{R}^{2}}}[sin90-\sin 0]\left( -\hat{j} \right) $$
$$ \Rightarrow \vec{E}=\dfrac{q}{2{{\pi }^{2}}{{\varepsilon }_{0}}{{R}^{2}}}(-\hat{j}) $$
Hence, Electric field at point O is $$\dfrac{q}{2{{\pi }^{2}}{{\varepsilon }_{0}}{{R}^{2}}}(-\hat{j})$$