One type of electric quadrupole is formed by four charges located at the vertices of a square of side 2a. Point P lies a distance x from the centre of the quadrupole on a line parallel to two sides of the square as shown in the figure. For , we have to show that the electric field at P is approximately given by

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A quadrupole is a combination of four charges of equal magnitude with two of opposite signs such that the total charge is zero. A quadrupole configuration is shown in the figure. It can be considered as a combination of two dipoles. We recall that the field at a distant point from a dipole at the location as shown in the figure below is

for

, and

The net electric field at P in the configuration of the quadrupole shown in the figure can be found as a vector sum of electric fields due to the two dipoles. It is given by

For we approximate

In this approximation

As , we find