Figure shows an electric quadrupole,
with quadrupole moment (Q=2qt”)
The electric field at a distance from
its centre at the axis of the
quadrupole is given by
Answers
V∝ 1/r³ for a dipole and V∝ 1/r for a monopole.
Explanation:
Four charges of same magnitude are placed at points X, Y, Y, and Z respectively. A point is located at P, which is r distance away from point Y. Given that the system of charges forms an electric quadrupole.
So the system of the electric quadrupole has three charges as follows:
Charge +q placed at point X
Charge −2q 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 = 1/4π∈₀ [ q/XP - 2q/YP + q/ZP]
= 1/4π∈₀ [ 1/r+a - 2q/r + q/r-a]
= q/4π∈₀ [ r(r-a) - 2(r+a)(r-a) - r(r+a) / r(r+a)(r-a) ]
= q/4π∈₀ [ r²-ra - 2r² + 2a² + r² + ra] / [r(r² - a²)]
= q/4π∈₀ [ 2a² / r(r² - a²) ]
= 2qa² /4π∈₀. r³(1 - a²/r²) ]
We know that r/a >> 1, so a/r << 1.
Hence a²/r² will be negligible.
V = 2qa² /4π∈₀r³
V∝ 1/r³ for a dipole and V∝ 1/r for a monopole.