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Irodov Solutions : Problem 3.7

Point charges q and -q are located at the vertices of a square with diagonals 2l as shown in the figure. Find the magnitude of electric field intensity at a point lying symmetrically with respect to the vertices of the square at a distance x from its centre.

We know that electric field intensity vector due to a point charge q at a location r is given by E = (kq/r³)r where k = 1/4πϵₒ. Using this expression the individual field produced (at a point say P) by each individual charge can be written and then by the superposition principle, the resultant field at point P can be obtained by adding those individual fields vectorially. So let's first assign a coordinate system to the given system of charges as shown below in the figure.
We have to find resultant electric field strength vector at point P(0,0,√(l²+x²)). Let E₁, E₂, E₃ and E₄ be the individual fields (at point P) due to the charges placed at vertices 1, 2, 3 and 4 respectively.  Then,

Alternative Method

Observe the diagram shown below.

Here, net electric field due to the charges placed at vertices 2 and 4 is