A conductor carries its charge in a thin sheet on its surface, and the E-field just inside and just outside the sheet should be \(\sigma /\left(2 \varepsilon_{0}\right)\) as in Fig. \(3.5 a\). How do you reconcile this with the zero field which in fact occurs inside conducting material and the field of \(\sigma / \varepsilon_{0}\) which occurs just outside?

A conductor is a material that allows the free movement of electrons. When an electric field is applied to a conductor, the free electrons redistribute themselves throughout the material, creating an induced electric field. The induced electric field opposes the external electric field, and since the electrons are free to move, they continue to redistribute until the net electric field inside the conductor is zero. This is a fundamental property of conductors.

Surface charge density (\(\sigma\)) represents the amount of charge per unit area on the surface of a conductor. It is an important parameter in determining how an external electric field interacts with a conductor. The electric field just outside the conductor is proportional to this surface charge density, and is given by \(\frac{\sigma}{\varepsilon_0}\), where \(\varepsilon_0\) is the permittivity of free space. This formula shows how more surface charge results in a stronger electric field just outside the conductor.

As we mentioned earlier, the net electric field inside a conductor is zero. However, just inside the surface, there might be a non-zero electric field due to the surface charge distribution. This E-field is perpendicular to the surface and is given by \(\frac{\sigma}{2\varepsilon_0}\). It represents the field that would exist if we considered only half of the surface charge density (hence the factor of \(\frac{1}{2}\)). This formula provides an approximation of the electric field at points very close to the surface of the conductor, within the conductor itself.

Now that we have an understanding of electric field behavior inside and outside conductors and the roles of surface charge density and permittivity, we can reconcile the given formula with the zero field inside and the field of \(\frac{\sigma}{\varepsilon_0}\) just outside the conductor. The formula \(\frac{\sigma}{2 \varepsilon_0}\) provides an approximation for the electric field very close to the surface of the conductor, within the conductor itself, where the \(\frac{1}{2}\) factor accounts for considering only half of the surface charge density. However, as we move deeper into the conductor, the electric field quickly becomes zero due to the opposing induced electric field, satisfying the condition that the net electric field inside a conductor is always zero. Meanwhile, the electric field just outside the conductor is given by \(\frac{\sigma}{\varepsilon_0}\), which is twice the given formula, accounting for the full surface charge density. This shows that both conditions (zero field inside conductor and non-zero field just outside conductor) can coexist with the given formula as it describes electric field very close to the surface but still within the conductor.