Answer to Question #86813 in Electricity and Magnetism for rahul

Question

Three charges q, 2q and q are to be placed on a straight wire of length 10 m.
Determine the positions where the charges should to be placed so that the
electrostatic potential energy of the system is a minimum.

Answer 1

$U=k(2\frac{q^2}{x}+\frac{q^2}{(10-x)}+2\frac{q^2}{10})$

$U=kq^2(\frac{2}{x}+\frac{1}{(10-x)}+\frac{1}{5})$

$\frac{dU}{dx}=0$

$-\frac{2}{x^2}+\frac{1}{(10-x)^2}=0$

$x^2-40x+200=0$

$x=5.86m$

Distance between charges 2q and q - 5.86 m; 2q and q - 10 m; q and q - 4.14 m