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In the figure shown, an electric dipole is placed at a distance `x` from an infinitely long rod of linear charge density `lambda`. (`a`) Find the net force acting on the dipole ? (`b`) What is the work done in rotating the dipole through `180^(@)` ? (`c`) If the dipole is slightly rotated about its equilibrium position, find the time period of oscillation. Assume that the dipole is linearly restrained. |
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Answer» (`a`) The electric field at a distance `x` is `E=(lambda)/(2piepsilon_(0)x)` The net force on the dipole is `F=p(dE)/(dx)=(2a)q[(-lambda)/(2piepsilon_(0)x^(2))]=(-lambdaqa)/(piepsilon_(0)x^(2))` The negative sign indicates that the force is attractive. (`b`) `W=DeltaU` `U_(i)=-vecp*vecE=-pEcos0^(@)=-pE` `U_(f)=-vecp*vecE=-pEcos180^(@)=+pE` `W=2pE=2(2a)q(lambda)/(2piepsilon_(0)x)=(2lambdaqa)/(piepsilon_(0)x)` (`c`) Restoring torque `tau=-pEsintheta` `tau=I(d^(2)theta)/(dt^(2))=(2ma^(2))(d^(2)theta)/(dt^(2))` `:. 2ma^(2)(d^(2)theta)/(dt^(2))+[(lambdaqa)/(piepsilon_(0)x)]theta=0` or `(d^(2)theta)/(dt^(2))+[(lambdaq)/(2piepsilon_(0)max)]theta=0` `T=2pisqrt((2piepsilon_(0)max)/(lambdaq))=sqrt((8pi^(2)epsilon_(0)max)/(lambdaq))` |
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