We recently finished studying about multipole expansion in electrodynamics class, so i just wanted to summarize what i learned in class. The monopole moment which is the net charge of the system vanishes, when the magnitude of positive charge is equal to the magnitude of negetive charge. In this case, the potential far away from the system of charges is dominated by the dipole moment if present. If not present, then the higher order momen will dominate the potential.
A simplest example of quadrapole would be charges of equal magnitude in the corners of a square, where positive charges are kept diagonally opposite to each other so are the negative charges. The electric potential due to the electric quadrapole is given by
Here, 
is the electric permittivity, Q is the charge, factors ni, nj are the components of the unit vector from teh point of interest to the location of the quadrupole moment.
A simplest example of quadrapole would be charges of equal magnitude in the corners of a square, where positive charges are kept diagonally opposite to each other so are the negative charges. The electric potential due to the electric quadrapole is given by
is the electric permittivity, Q is the charge, factors ni, nj are the components of the unit vector from teh point of interest to the location of the quadrupole moment.
Evident from formula, potential due to quadrapole fall off by 1/r^3, that's why when a gravitational wave from the distant pulsar arrives on earth, its too weak 
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I made a simple comsol model to see the electric field line of a quadrapole, and the potential in the near space.
electric field line originating from the conductors in red representing positive charge, and ending in conductors in blue representing negetive charge.
The contour plot of the potential . Yellow is the positive region, and blue is the negetive potential region.
