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Two electrons and a positive charge q ar...

Two electrons and a positive charge q are hold along a straight line. At what position and for what value of q will the system be in equilibrium.

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Let two electrons of charges - e each be held at A and B. The third charge + q must be placed at the center O of AB. The forces on + q, due to two electrons and it is in equilibrium.
For the charge (-e) at A to be in equilibrium , Fig force on charge at A due to - e charge at A due to + q charge at 0 = Zero

`(1)/(4pi in_(0)) ((-e) (-e))/(x^(2)) + (1)/(4pi in_(0)) (q(-e))/((x//2)^(2)) = 0`
or `(1)/(4pi in_(0)) (e^(2))/(x^(2)) = (1)/(4pi in_(0)) (q(e)xx4)/(x^(2))`
or e = `4q or q = e//4`
If charge at 0 is moved slightly towards A, it would not return to 0 on its own and shall continue to move towards A. Hence equilibrium is unstable.
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