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(a) Derive an expression for the elecrtric field E due to a dipole of length '2aat a point distant r from the centre of the dipole on the axial line. (b) Drawa graph of E versus r for r gt gt a. (c)If the diploe were kept in a uniform external electric fields E_(0)Diagrammatically represent the position of the dipole in stable and unstable equilibrium and write the expressions for the torque acting on the dipole in both the cases. |
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Answer» Solution :(a) LET there be an ELECTRIC dipole consisting of charges -q and + q , separated by distance 2n and placed in vacuum . LetP be a point on the axial line at a distance r from centre O of the dipole on the dipole on the side of the charge `+ q . overset(to)(E)` due to charge- q at P is ` overset(to)(E)_(-q) = (-q)/(4piin_(0) (r + a)^(2)) hatP`(towards left) (along AB) Where, `hatP`unit vector along dipole AXIS from q to q ` overset(to)(E)_(-q) ` due to charge + q at P is ` overset(to)(E)_(-q) = (-q)/(4piin_(0) (r + a)^(2)) hatP`(along BP) ` overset(to)(E)_("axial")=overset(to)(E)_(+q) +overset(to)(E)_(-q)` ` overset(to)(E)_("axial")=(q)/(4piin_(0))= [(1)/((r- a)^(2)) - (1)/((r+ a)^(2)) ] hatP` ` overset(to)(E)_("axial")=(q)/(4piin_(0))= (4ar)/((r^(2) - a^(2)) )hatP = (1)/(4 piin _(0))XX (2Pr)/((r^(2) - a^(2))) hatp ` ( Directed towards `bar(BP)` product.) ` P = 2a xx q ` = Dipole moment For `r gt gt a , a^(2)` is neglected `rArroverset(to)(E)_("axial")=(1)/(4 pi in _(0)) (2P)/(r^(3))`(towardsright) (b)` E = (1) /(4 pi in _(0))xx (2p)/(r^(3))`  (c)Torque`tau = PE sin theta` In stable equillbrium ` theta = 0` (##SB_PHY_XII_17_OD_E01_026_S02.png" width="80%"> |
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