Four particles each of mass m are placed at the vertices of a square of side l. the potential at the centre of square is

Explanation

From the figure above,
`PQ = QR = RS = SP = l`
`PR = QS = sqrt(l^(2) + l^(2)) = sqrt(2)l`
`OP = OQ = OR = OS = sqrt(2)l//2 = l//sqrt(2)`
Using the principle of superposition, total potential energy of the system of particles is
`U = U_(PQ) + (U_(RQ) + U_(RP)) + (U_(SP) + U_(SQ) + U_(SR))`
`= 4R_(PQ) + 2U_(SQ)`
`[because U_(PQ) = U_(SP) = U_(SR) = U_(RQ) and U_(RP) = U_(SQ)]`
`U = 4(-(Gm m)/(l)) + 2(-(Gm m)/(l//sqrt(2)))`
`= -(2Gm^(2))/(l)(2+(1)/(sqrt(2)))`
`= (-2Gm^(2))/(l) [2+0.707]`
`= (-5.41 Gm^(2))/(l)`
Gravitational potential at the centre O is
`V = V_(P) + V_(Q) + V_(R ) + V_(S) = 4V_(P) = 4((-Gm)/(OP))`
`= 4((-Gm)/(l//sqrt(2)))`
`= (-4sqrt(2) Gm)/(l)`