Long Answer Problems $(15$ pts each$)$ Show all work to receive full credit.

For each problem, find symbolic answers before numerical answers.

Written Problem $1$

Metal sphere $"$A$",$ with a radius $R_A=2\mathrm{.}00mm$ and mass $m_A=10$ kilogram, is fixed at the origin. A smaller metal sphere $"$B$",$ with $m_B=10\mathrm{.}0$ grams and radius $R_B=1\mathrm{.}00mm,$ is connected to the fixed sphere $"A"$ by a thin, taut conducting string that is $1\mathrm{.}00$ m long. A total charge of $9\mathrm{.}00C$ is placed on the connected spheres.

$($a$)$ Assuming no charge remains on the thin string, determine how the $9\mathrm{.}00C$ charge is distributed on the spheres, i$.$e$.,$ how much of this charge lies on sphere $"$A$"$ and how much lies on sphere $"$B$"?$

$($b$)$ Find the resulting electrical force between these spheres.

$($c$)$ The string between them is then cut, so sphere $"$ B $"$ begins accelerating away from sphere $"$A$".$ Determine the speed of sphere $"$B$"$ when it is very far from sphere $"$A$".$ Assume that only the electrical force between the spheres is acting on sphere $"$B$"$ as it accelerates away from sphere $"$A$".$

$($Show all work to receive full credit.$)$

Written Problem $2$ In the circuit shown below, after the switch $"$ S $"$ is closed,