A valid model of a cell membrane is as a parallel$-$plate capacitor whose plate area is the surface area of the cell and whose thickness is the thickness of the lipid bilayer.

a$)$ You place a probe across the membrane of a spherical cell with a radius of $10$ um and find that there is a voltage of $5$ mV across it and a charge density of $2$ nC per square centimeter of the outer surface. Let$$s say you know that the thickness of the lipid bilayer is $4\mathrm{.}83$ nm$.$ What is the dielectric constant of the cell membrane, and what is its capacitance?

b$)$ You bring the cell$$s voltage to $8$ mV and it immediately begins to discharge. If the charge on the cell drops to $10^-5$ nC in $10$ nanoseconds, what is the equivalent resistance of the membrane? $($Hint: do not worry about the geometry or resistivity of the cell.$)$