$(2)$ The Lennard$-$Jones potential $W(z)$ for sphere$-$planar surface contact describes the energy

for a particle close to a wall, i$.$e$.$ at a distance $z.$ This expression differs from that of the

interaction of two atoms $($or uncharged apolar molecules$).$ It can be written as follows:

$W_{\text{t}\text{o}\text{t}\text{a}\text{l}}=\frac{2}{3}{}^{2}lon{}_1{}_2{}^{4}R[\frac{{}^7}{210z^7}-\frac{}{2}]$

where $R$ is the radius of the particle and $,{}_1,{}_2$ and $$ are constants.

$($a$)$ Which term is attractive and which one is repulsive? $5P$

$($b$)$ What is the expression for the force $F$ as a function of separation distance $z$

between a spherical particle and the wall? Derive the expression for $F(z)$ step by step

$($explicitly$)!40P$

$($c$)$ Plot both $W(z)$ and $F(z)$ in one single graph! $20P$