Three charges each equal to q, are placed at the three corners of a square of side a . Find the electric field at the fourth corner.

Explanation

Let the charges be placed at the corners A, B and C (figur) Qe shall calculate the electric field at the fourth corner D. The field `E_1` due to the charge at A will have the magnitude `q/4piepsilon0 a^2 and will be along CD. The field E_3 due to the charge at B will have the magnitude `q / (4piepsilon0 sqrt(2a^2) and will be along BD. As E_1 and E_3 are equal in magnitude, their resultant willl be along the bisector of the angle between E_1 , E_2 and hence along E_3. The magnitude of this resultant is `sqrt E_1^2+ E_2^2 ` as the angle between`E_1 and E_ 2 is pi/2 ` .The resultant electric field at D is, therfore, along E_3 and has magnitude `sqrt (E_1^2 + E_2^2) + E_3 . = sqrt ((q/4piepsilon_0 a^2)^2 + (q/ 4piepsilon_0 a^2)^2 ))+ q /( 4piepsilon(sqrt 2a)^2 `
. = `q/ 4piepsilon_0 [ sqrt 2/a^2 + 1 / 2a^2] = (2sqrt 2+1) q /8piepsilon_0 a^2