Three charges -q,+2q, and -q are arranged on a line as shown in fig. Calculate the field at a distance rgtgta on the line.

Explanation

The field at point P is superposition of fields `vecE_(1),vecE_(2)`, and `vecE_(3)` due to each charge. Given by
`vecE_(1)=-(q)/(4 pi epsilon_(0)(r-a)^(2))hat i`
`vecE_(1)= +(2q)/(4 pi epsilon_(0)r^(2))hat i`
`vecE_(3)=-(q)/(4 pi epsilon_(0)(r+a)^(2))hat i`
Now,
`vec(E)=vecE_(1)+vecE_(2)+vecE_(3)=(q)/(4 pi epsilon_(0))[-(1)/((r-a)^(2))+(2)/(r^(2))-(1)/((r+a)^(2))]hat i`
`E=(q)/(4 pi epsilon_(0)r^(2))[-{1-((a)/(r))}^(-2)+2-{1+((a)/(r))}^(-2)]`
if `r gtgt a`, we can use the binomial approximation
`(1+alpha)^(n)~=1+n alpha+(n+(n-1))/(|__ 2)alpha^(2)+` ....for `alphaltlt1`
Therefore, `E=(q)/(4 pi epsilon_(0)r^(2))[-{(1-2((a)/(r)))+(-2(-2-1))/(2)(-(a)/(r))^(2)}`
`+2-{1-2(a)/(r)+(-2(-2-1))/(2)((a)/(r))^(2)}]`
`= (6a^(2)q)/(4pi epsilon_(0)r^(4))`
The charge in this problem may be consider two dipoles placed close together. Such an arrangement of charge is called an electric quadruploe.