When a shower is turned on in a closed bathroom, the splashing of the water on the bare tub can fill the room's air with negatively charged ions and produce an electric field in the air as great as $1000\frac{N}{C}.$ Consider a bathroom with dimensions $2\mathrm{.}8m3\mathrm{.}1m2\mathrm{.}0m.$ Along the ceiling, floor, and four walls, approximate the electric field in the air as being directed perpendicular to the surface and as having a uniform magnitude of $512\frac{N}{C}.$ Also, treat those surfaces as forming a closed Gaussian surface around the room's air. What are $($a$)$ the volume charge density $$ and $($b$)$ the number of excess elementary charges e per cubic meter in the room's air?

$($a$)$ Number

Units

$($b$)$ Number

Units