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 minimum?

Step-by-step solution

Let the third charge -q is placed at a distance of x from the charge 2 q . Then, the potential energy of the system is U=k [ 2 q(8 q) r - (2 q)(q) x - (8 q)(q) r-x ] Here, k= 1 4 ₀ U is minimum, where 2 x + 8 r-x is maximum Let 2 x + 8 r-x =y For y to be maximum, d y d x =0- 2 x^2 + 8 (r-x)^2 =0 x r-x = 2 8 = 1 2 x= r 3 But at x= r 3 , d^2 y d x^2 = positive i.e. at x= r 3 , y is minimum or U is maximum. So, there cannot be any point between them where the potential energy is minimum.