An electron orbits a nucleus in a circular path with a radius R$=51011$mR $=5\backslash$times $10^\{-11\}\backslash ,\backslash$text$\{$m$\}$R$=51011$m and a velocity v$=2106$m$/$sv $=2\backslash$times $10^6\backslash ,\backslash$text$\{$m$/$s$\}$v$=2106$m$/$s$.$ A static, non$-$uniform magnetic field is applied along the zzz$-$axis, varying spatially as:

B$($r$)=$B$0(1$r$2$R$2)$z$^,\backslash$vec$\{$B$\}($r$)=$ B$\_0\backslash$left$(1-\backslash$frac$\{$r$^2\}\{$R$^2\}\backslash$right$)\backslash$hat$\{$z$\},$B$($r$)=$B$0(1$R$2$r$2)$z$^,$

where B$0=1$TB$\_0=1\backslash ,\backslash$text$\{$T$\}$B$0=1$T$.$

$($a$)$ Derive the expression for the Lorentz force acting on the electron due to the spatially varying magnetic field. $(13$ points$)$

$($b$)$ Calculate how the force affects the stability of the electron's circular orbit and determine if a radial drift is induced. $($