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Question

Two point charges 2 q and 8 q are placed at a distance r apart. Where should a third charge q be placed between them so that the electrical potential energy of the system is a minimum
  1. at a distance of r3 from 2 q
  2. at a distance of 2 r3 from 2 q 
  3. None of the above
  4. at a distance of r16 from 2 q 


Solution

The correct option is C None of the above
Let the third charge q is placed at a distance of x from the charge 2 q. Then potential energy of the system is
U=k[(2 q)(8 q)r(2 q)(q)x(8 q)(q)rx]
Here, k=14πϵ0
U is minimum, where
2x+8rx is maximum.
Let 2x+8rx=y
For y to be maximum dydx=0
or  2x2+8(rx)2=0
or  xrx=28=12
or  x=r3
But at x=r3,d2ydx2= positive
i.e., at x=r3, y is minimum or U is maximum. So, there can't be any point between them where potential energy is minimum.

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