# 2.22 Figure 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 $\frac{r}{a}>>1$, and contrast your results with that due to an electric dipole, and an electric monopole (i.e., a single charge).

Here, As we can see

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

$V = \frac{1}{4\pi \varepsilon _0}\left [ \frac{q}{r+a}-\frac{2q}{r}+\frac{q}{r-a} \right ]$

$V = \frac{1}{4\pi \varepsilon _0}\left [ \frac{r(r-a)-2(r+a)(r-a)+r(r+a)}{r(r+a)(r-a)}\right ]=\frac{q}{4\pi \epsilon _0}\left [ \frac{2a^2}{r(r^2-a^2)} \right ]$

$V =\frac{q}{4\pi \epsilon _0}\left [ \frac{2a^2}{r^3(1-\frac{a^2}{r^2})} \right ]$

Since

$\frac{r}{a}>>1$

$V=\frac{2qa^2}{4\pi \epsilon _0r^3}$

From here we conclude that

$V\propto \frac{1}{r^3}$

Whereas we know that for a dipole,

$V\propto \frac{1}{r^2}$

And for a monopole,

$V\propto \frac{1}{r}$

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