Calculating surface charge density as a function of time

Let at time $t$ charge on a area $A$ of the sheet be $Q$. charge density on one face will be $sigma_{t}$ and on the other face will be -$sigma_{t}$. the net field inside the conducting sheet will be $$E_{tot}=E-frac{Q}{Aepsilon_{o}}=E-frac{sigma_{t}}{epsilon_{o}}$$Current Density, $J=eta E_{tot}$

Due to this current a charge $JAdt$ flows from one surface to the other in a time interval dt. the increase in charge density, $dsigma_{t}=frac{JAdt}{A}=Jdt$ $$implies dsigma_{t}=etabigg(E-frac{sigma_{t}}{epsilon_{o}}bigg)dt$$ $$implies intlimits_{0}^{sigma_{t}}dfrac{dsigma_{t}}{bigg(E-frac{sigma_{t}}{epsilon_{o}}bigg)}=intlimits_{o}^{t}eta dt$$ $$implies sigma_{t}=epsilon_{o}Eleft(1-e^{dfrac{-eta t}{epsilon_{o}}}right)$$

For a conductor placed in electric field we have a vector form of Ohm's law-

$$vec{j} = etavec{E}$$

where $vec{j}$ is current density and $eta$ is conductivity of conductor (which is constant at given temperature), $vec{E}$ is net electric field inside conductor.

note that, for $vec{E}$, you need net electric field inside the conductor, which would be $E - cfrac{sigma}{epsilon_0}$($vec{sigma}$ is induced surface charge density at any instant), and since there would be a current inside the conductor due to this net field ($vec{E}$), for any elemental charge that would flow due to this $vec{E}$ will be $dq$, and the additional induced surface charge density would be $dsigma$, so $vec{j} = cfrac{dsigma}{dt}$. rest part is easy calculation using calculus , you can easily find induced surface charge density as a function of $t$.