Calculating surface charge density as a function of time

Question

An infinitely large conducting sheet of thickness $d$ and of conductivity $eta$ is uncharged. At an instant $t=0$, an external uniform and electric field $E$ is switched on. Due to this, the conducting sheet starts developing surface charge density $sigma$. Now how do we find $sigma(t)$?
For $sigma(t)$, I think first we have to find the elctric field which is induced inside the sheet as a function of time. So how do we find it? Also, I wonder whether will the electric field just outside the sheet vary with time or will it remain constant?
The answer given is $sigma(t)=epsilon_{o}Eleft(1-e^{dfrac{-eta t}{epsilon_{o}}}right)$

Pranay · Accepted Answer

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, $dsigma_{t}=frac{JAdt}{A}=Jdt$
$$implies dsigma_{t}=etabigg(E-frac{sigma_{t}}{epsilon_{o}}bigg)dt$$
$$implies intlimits_{0}^{sigma_{t}}dfrac{dsigma_{t}}{bigg(E-frac{sigma_{t}}{epsilon_{o}}bigg)}=intlimits_{o}^{t}eta dt$$
$$implies sigma_{t}=epsilon_{o}Eleft(1-e^{dfrac{-eta t}{epsilon_{o}}}right)$$

maverick · Answer

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 $t$.

Calculating surface charge density as a function of time

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