Question

What is electric field intensity at any point on the axis of a charged rod of length '$L$' and linear charge density '$\lambda$'? The point is separated from the nearer end by a.

Answer 1

Consider an element, $dx$ at a distance, $x$ from the point $P$, where we have to find the electric field. the elemental charge, $dq=\lambda dx$
Then, $dE=k.\frac{\lambda dx}{x^2}$ or $E=k\lambda \stackrel{a+L}{\underset{a}{\int }}\frac{1}{x^2}dx=k\lambda {[-\frac{1}{x}]}_a^{a+L}=[\frac{-1}{a+L}+\frac{1}{a}]$
Thus, $E=\frac{\lambda }{4\pi {\in }_0}[\frac{1}{a}-\frac{1}{L+a}]$ $[k=\frac{1}{4\pi {\epsilon }_0}]$