📢Why internal charge instantly vanishes

These 2 sources are the mandatory and rapid exponential decay of net electrical charge within a conducting material. Mathematically, the combination of the continuity equation, Ohm's law ($\vec{\jmath}=\sigma \vec{E}$), and Gauss's law demonstrates that if the charge density ($\rho$) is non-zero, it must decrease exponentially over time, described by the relationship $\frac{\partial \rho}{\partial t}=-\frac{\sigma}{\varepsilon_0} \rho$. This mathematical necessity proves that the only stationary solution for charge inside a conductor with non-zero conductivity ($\sigma$) is zero charge density ($\rho=0$), with the decay rate determined by the constant $\sigma / \varepsilon_0$. Conceptually, this highly efficient self-correction means that any net electrical charge introduced inside the material must instantaneously dissipate and move to the surface. This unavoidable flow of charge is driven by the internal electric field and is governed by the charge relaxation time, which is extremely short in metals due to their typically enormous conductivity, making the dissipation effectively immediate.

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Why internal charge instantly vanishes | Audiosviadean on Notion