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Two charges -q each are separated by dis...

Two charges `-q` each are separated by distance `2d`. A third charge `+q` is kept at mid-point O. Find potential energy of `+q` as a function of small distance `x` from O due to `-q` charges. Prove that the charge at O is in an unstable equilibrium.

Text Solution

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Let the distribution be as shown
Let `P_(1),P_(2)` and `P_(3)` be the three points
Inside the plates
`therefore E_(P_(1))=E_(P_(2))=E_(P_(3))=0`
Now `E_(P_(1))=(1)/(2epsilon_(@)A)(Q-q_(1)-q_(1)+(5Q+q_(2))-q_(2)-10Q+q_(3)-q_(3)=0`

`impliesQ-q_(1)-q_(1)+5Q+q_(2)-q_(2)-10Q+q_(3)-q_(3)=0`
`impliesq_(1)=-2Q`
Similarly, `E_(P_(2))=0`
`implies q_(2)=-7Q`
And `E_(P_(3))=0`
`impliesq_(3)=3Q`
`therefore` The final distribution is :
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