Optical Tweezer Manipulation of a Polar Nanoparticle: Interaction with Mirror Dipole of Grounded Conducting Sphere

Problem Statement

In vacuum, a perfectly grounded conducting sphere of radius $a$a is placed with its center at the origin $O(0,0,0)$O(0,0,0).

At a point $A(0,0,r_0)$A(0,0,r0​) on the $z$z-axis at distance $r_0$r0​ ($r_0 > a$r0​>a) from the sphere, a polar nanoparticle of negligible size is fixed. This particle possesses a permanent electric dipole moment of magnitude $p$p (referred to simply as the "dipole").

Initially, the dipole direction points toward the origin $O$O (i.e., in the $-z$−z direction).

While the particle's center of mass position is kept at point $A$A, an external manipulation (such as an optical tweezer) quasi-statically rotates the dipole until it is finally parallel to the positive $x$x-axis direction.

Find the work $W$W done by the external force on the particle during this rotation.

The spatial extent of the dipole itself is sufficiently small compared to $r_0 - a$r0​−a, so it can be treated as an ideal point dipole. Let the proportionality constant in Coulomb's law be $k$k. Gravity and all other interactions are neglected.

Constraints

• Coulomb's law constant: $k = 9.0\times10^9 \text{ N}\cdot\text{m}^2/\text{C}^2$k=9.0×109 N⋅m2/C2
• Radius of conducting sphere: $a = 1.0\times10^{-8} \text{ m}$a=1.0×10−8 m
• Position of particle: $r_0 = 2.0\times10^{-8} \text{ m}$r0​=2.0×10−8 m
• Magnitude of dipole moment: $p = 3.0\times10^{-25} \text{ C}\cdot\text{m}$p=3.0×10−25 C⋅m

Input Format

The work $W \text{ [J]}$W [J] is expressed as $W = X\times10^{-18}$W=X×10−18. Give the value of positive integer $X$X.