#### Question

Figure 2.34 shows a charge array known as an *electric quadrupole*. For a point on the axis of the quadrupole, obtain the dependence of potential on *r *for *r*/*a *>> 1, and contrast your results with that due to an electric dipole, and an electric monopole (i.e., a single charge).

#### Solution

Four charges of same magnitude are placed at points X, Y, Y, and Z respectively, as shown in the following figure.

A point is located at P, which is *r* distance away from point Y.

The system of charges forms an electric quadrupole.

It can be considered that the system of the electric quadrupole has three charges.

Charge +*q *placed at point X

Charge −2*q* placed at point Y

Charge +*q* placed at point Z

XY = YZ = *a*

YP =* r*

PX = *r* + *a*

PZ = *r* − *a*

Electrostatic potential caused by the system of three charges at point P is given by,

`V=1/(4piin_0)[q/(XP)-(2q)/(YP)+q/(ZP)]`

=`1/(4piin_0)[q/(r+a)-(2q)/r+q/(r-a)]`

`=q/(4piin_0)[(r^2-ra-2r^2+2a^2+r^2+ra)/(r(r^2-a^2))]=q/(4piin_0)[(2a^2)/(r(r^2-a^2))]`

`=(2qa^2)/(4piin_0r^3(1-a^2/r^2))`

Since`r/a`>>1,

`therefore a/r `<<1

`a^2/r^2`is taken as negligible.

`therefore V=(2qa^2)/(4piin_0r^3)`

It can be inferred that potential, `Vprop1/r^3`

However, it is known that for a dipole, `Vprop1/r^3`

And, for a monopole, `Vprop1/r^3`