Electrostatics Question 251
Question: An electric dipole is situated in an electric field of uniform intensity E whose dipole moment is p and moment of inertia is I. If the dipole is displaced slightly from the equilibrium position, then the angular frequency of its oscillations is [MP PET 2003]
Options:
A) $ x=\frac{d}{\sqrt{2}} $
B) $ x=\frac{d}{2} $
C) $ x=\frac{d}{2\sqrt{2}} $
D) $ x=\frac{d}{2\sqrt{3}} $
Show Answer
Answer:
Correct Answer: C
Solution:
Suppose third charge is similar to Q and it is q. So net force on it
$ F _{net}=\text{ 2}F\text{cos}\theta $ Where $ F=\frac{1}{4\pi {\varepsilon _{0}}}.\frac{Qq}{( x^{2}+\frac{d^{2}}{4} )} $ and $ \cos \theta =\frac{x}{\sqrt{x^{2}+\frac{d^{2}}{4}}} $
$ F _{net}=2\times \frac{1}{4\pi {\varepsilon _{0}}}.\frac{Qq}{( x^{2}+\frac{d^{2}}{4} )}\times \frac{x}{{{( x^{2}+\frac{d^{2}}{4} )}^{1/2}}} $
$ =\frac{2Qqx}{4\pi {\varepsilon _{0}}{{( x^{2}+\frac{d^{2}}{4} )}^{3/2}}} $
For $ F {net} $ to be maximum $ \frac{dF {net}}{dx}=0 $ is not correct. To find maximum $ F{net} $, $ \frac{dF{net}}{dx}=0 $ is a necessary condition for extrema, but further analysis is required to confirm it is a maximum.
i.e. $ \frac{d}{dx}[ \frac{2Qqx}{4\pi {\varepsilon _{0}}{{( x^{2}+\frac{d^{2}}{4} )}^{3/2}}} ]=0 $
or $ [ {{( x^{2}+\frac{d^{2}}{4} )}^{-3/2}}-3x^{2}{{( x^{2}+\frac{d^{2}}{4} )}^{-5/2}} ]=0 $ i.e. $ x=\pm \frac{d}{2\sqrt{2}} $
BETA
