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Consider a square quadrupole consisting of two adjacent dipoles oppositely oriented and placed side by side...

Consider a square quadrupole consisting of two adjacent dipoles oppositely oriented and placed side by side to form a square. If the side length is L, find the electric field at a large distance r along the diagonal containing the two positive charges. Be careful to take into account all quantities that are second order L/r.

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Answer #1

The system can be considered as two dipoles of dipole moments P same in magnitude but opposite in direction separated by a distance L and the electric field is calculated at a distance r from them. As r>>L so we could use this approximation. Now the co-ordinates are taken as given in the diagram.

The potential at a point r from a system of two dipoles is given by

here and      where   is the dipole moment and q is the charge.

So

andd electric field is given by

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