Consider a long thin wire of uniform linear charge density λ Let us find electric field due to the wire at the point P at a normal distance. PN = r from the wire Let AB = dx be a small elementary portion of the wire at a distance x from point N.
Then charge on the elementary portion AB, dq = λ dn
Therefore, electric field due to the small portion AB at the point P,
The component dE sin θ due to the different small portions, such as AB, will cancel each other. Therefore, effective electric field due to the small portion AB,
The electric field due to the whole long wire can be found by integrating the above between proper limits i.e.,
