A particle with charge Q is on the y axis a distance a from the origin and a particle with charge q is on the x axis a distance d from the origin. The value of d for which the x component of the force on the second particle is the greatest is?

Force on q due to Q

= k Qq / (a² + d²)

x component

= k Qq / (a² + d²) x d / √ (a² + d²)

F = kQq d / (a² + d²) ³/²

differentiating F with respect to d

dF / Dd = kQq [ d. - 3/2 (a² + d²) ⁻⁵/² 2d + (a² + d²) ⁻³/²]=0 for maximum F

- 3d² / (a² + d²) + 1 = 0

a² + d² = 3 d²

a² = 2d²

d = a / √2