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The figure given below 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. Contrast your result with that due to an electric dipole and an electric monopole (i.e. a single charge). |
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Answer» Solution :As we know that the electric potential at a point due to a system of n charges is given by: `V=(1)/(4pi epsilon_(0)) sum (q_(n))/(r_(n))` Electric potential at point P due to the given charge array can be calculated as: `V=(1)/(4pi epsilon_(0))((q)/(r+a)+2xx(-q)/(r )+(q)/(r-a))=(1)/(4pi epsilon_(0)) ((2qa^(2))/(r(r^(2)-a^(2))))` `RARR V=(1)/(4piepsilon_(0))((2qa^(2))/(r(r^(2)-a^(2))))=(1)/4pi epsilon_(0)((2qa^(2))/(r^(3)(1-(a^(2))/(r^(2)))))` As, `(r )/(a ) gt gt 1, rArr V=(1)/(4piepsilon_(0))((2qa^(2))/(r^(3)))` Thus, `V prop (1)/(r^(3))` For a monopole (SINGLE charge) electric potential varies as: `V prop (1)/(r )` For large VALUES of r electric potential due to an electric dipole varies as `V prop (1)/(r^(2))`and that due to quadrapole as `V prop (1)/(r^(3))` |
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