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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 gt gt 1, and contrast your results with that due to an electric dipole, and an electric monopole (i.e., a single charge). |
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Answer» Solution :For the ELECTRIC QUADRUPOLE SHOWN in the above figure, net electric potential at point P is `V = 1/(4pi epsi_0)[q/((r+a)) -(2q)/r + q/((r-a))]= q/(4pi epsi_0)[1/(r + a) + 1/(r-a) - 2/r]` `=q/(4piepsi_0)[(r(r-a) + r(r + a) -2(r^2 - a^2))/(r(r^2 - a^2))]=q/(4piepsi_0).(2a^2)/(r(r^2-a^2))` `=(q.2a^2)/(4piepsi_0r^3(1- a^2/r^2)) = (2qa^2)/(4pi epsi_0 r^3(1 - a^2/r^2))` For` r/agt gt 1`i.e.,`a/r gt gt 1` we have : `V = (2qa^2)/(4piepsi_0r^3)` Thus, `V prop 1/r^3` i.e., electric potential due to a quadrupole is inversely proportional to the cube of the distance along its AXIS. However, for a dipole `V prop 1/r^2` and for a MONOPOLE `V prop 1/r`. |
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