Question

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 r/a >> 1, and contrast your results with that due to an electric dipole, and an electric monopole (i.e., a single charge).

Answer 1

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 −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/(4piin_0)["q"/("XP") - (2"q")/("YP") + "q"/("ZP")]`

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

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

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

= `(2"qa"^2)/(4piin_0"r"^3(1 - "a"^2/"r"^2))`

Since `"r"/"a"` >> 1,

∴ `"a"/"r" ` << 1

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

∴ `"V" = (2"qa"^2)/(4piin_0"r"^3)`

It can be inferred that potential, `"V" prop 1/"r"^3`

However, it is known that for a dipole, `"V" prop 1/"r"^2`

And, for a monopole, `"V" prop 1/"r"`