Consider the two small, equal$-$mass, charged spheres shown in the figure. The top sphere is suspended from the ceiling by a filament, and has a charge of $q_1=31\mathrm{.}7nC.$ The bottom sphere has a charge of $q_2=-58\mathrm{.}0nC,$ and is directly below the top sphere. Assume $d=2\mathrm{.}00cm$ and $m=7\mathrm{.}60g.$

$($a$)$ Calculate the tension $($in $N)$ in the filament.

Draw a free$-$body diagram of the upper sphere. Note that there are essentially three forces acting on the sphere: you can negiect the gravitational force from the lower sphere because its magnitude is so small. Write Newton's second law in the vertical direction and solve for the tension T$.$N

$($b$)$ If the filament can withstand a maximum tension of $0\mathrm{.}180N,$ what is the smallest value $d$ can have before the filament breaks? $($Give your answer in cm$.)$

Apply Newton's second law to the top sphere, and solve your equation for the distance $d.$ Set $T=0\mathrm{.}180N,cm$