Fig 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 gtgt 1, and contract your results with that due to an electric dipole and an electric monopole (i.e, a single charge). <br> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/PR_XII_V01_C01_S01_591_Q01.png" width="80%">

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Answer Text

Solution :
Four charges of same magnitude are placed at points X, Y, Y and Z respectively, as shown in the following figure. <br> <img src="https://d10lpgp6xz60nq.cloudfront.net/physics_images/NCERT_PHY_V01_XII_C02_E01_022_S01.png" width="80%"> <br> A point is located at P, Which is r distance away from point Y. <br> The system of charges forms an electric quadrupole. <br> It can be considered that the system of the electric quadrupole has three charges. <br> Charge +q placed at point X <br> Charge `-2q` placed at point Y <br> Charge `+q` placed at point Z <br> `XY = YZ = a` <br> `YP= r` <br> `PX = r+a` <br> Electrostatic potential caused by the system of three charges at point P is given by, <br> `V = (1)/(4pi in_(0))[(q)/(XP) - (2q)/(YP)+(q)/(ZP)]` <br> `= (1)/(4pi in_(0))[(q)/(r+a) -(2q)/(r)+(a)/(r-a)]` <br> `= (Q)/(4pi in_(0))[(r(r-a)-2(r+a)(r-a)+r(r+a))/(r(r+a)(r-a))]` <br> `= (q)/(4pi in_(0))[(r^(2)-ra-2r^(2)+2a^(2)+r^(2)+ra)/(r(r^(2)-a^(2)))] = (q)/(4pi in_(0))[(2a^(2))/(r(r^(2)-a^(2)))]` <br> `= (2qa^(2))/(4pi in_(0) r^(3)(1-(a^(2))/(r^(2))))` <br> Since `(r)/(a) gt gt 1` <br> `:. (a)/(r) lt lt 1` <br> `(a^(2))/(r^(2))` is taken as negligible. <br> `:. V = (2qa^(2))/(4pi in_(0) r^(3))` <br> It can be inferred that potential, `V prop (1)/(r^(3))` <br> However, it is known that for a diople, `V prop (1)/(r^(3))` <br> And , for a monopole, `V prop (1)/(r)`

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