Obtaining an expression for electric field intensity Showing behaviour at large distance Net Electric Field at point $P=\int\limits_{0}^{2\pi a} dE \cos\theta$ $dE=$ Electric field due to a small element having charge $dq$ $=\frac{1}{4\pi\varepsilon_0}\frac{dq}{r^2}$Let $\lambda = \text{Linear charge density}$ $=\frac{dq}{dl}$ $dq=\lambda dl$ Hence $E=\int\limits_{0}^{2\pi a}\frac{1}{4\pi\varepsilon_0}\cdot\frac{\lambda dl}{r^2}\times\frac{x}{r}$ , where $\cos\theta=\frac{x}{r}$ $=\frac{\lambda x}{4\pi\varepsilon_0 r^3}(2\pi a)$ $=\frac{1}{4\pi\varepsilon_0}\frac{Qx}{(x^2+a^2)^{3/2}},$ where total charge $Q=\lambda \times 2\pi a$At large distance i.e., $x\gg a$ $E\simeq \frac{1}{4\pi\varepsilon_0}\cdot\frac{Q}{x^2}$ This is the electric field due to a point charge at distance $x$ .