Show that a surface density of charge given by \(\varepsilon_{0}\left(\varepsilon_{1}-1\right) E_{m}\) is alone sufficient to account for the appearance of \(\varepsilon_{\text {, }}\) in the denominators of Eqs \((11.3)\). \(E_{\mathrm{m}}\) is the E-field in the medium.

Imagine electric field lines streaming through space - electric flux quantifies this flow of electric field through a surface. Interpret it like water flow through a net; the more water or denser the stream, the greater the flow. In mathematical terms, electric flux (denoted by \(\Phi\)) is the dot product of the electric field \(\vec{E}\) and the vector area \(\vec{A}\), integrated over a closed surface. \[\Phi = \oint_{S} \vec{E} \cdot d\vec{A}\]\]

Surface charge density \(\sigma\), akin to a crowded beach, measures how densely packed charges are, but on a two-dimensional surface. It's defined as the charge per unit area and can have varying distributions across different materials. In our exercise, surface charge density modifies the behavior of the electric field in a material. The equation relating it to the electric field in a medium is given by \(\sigma = \varepsilon_{0}(\varepsilon_{1}-1) E_{m}\), where \(\varepsilon_{0}\) is the permittivity of free space, \(\varepsilon_{1}\) is the relative permittivity of the medium, and \(E_{m}\) is the magntiude of the electric field in the medium. This equation is key to understanding how charges influence electric fields in various materials.

Selecting the right Gaussian surface is like picking the right tool for a job - it simplifies your work. In Gauss's Law applications, a Gaussian surface is an imaginary closed surface chosen to exploit symmetries in the problem. The strategic choice of surface (sphere, cylinder, etc.) can make calculations of electric flux much more manageable due to the alignment of the electric field with the surface's orientation. An example from our exercise is the cylindrical Gaussian surface inside a medium; here, only the top and bottom contribute to the flux!

Moving from air into water, swimmers feel resistance - similarly, electric fields change when entering different media. This change is governed by the medium's relative permittivity \(\varepsilon_{1}\). The electric field within a medium \(E_{m}\), as seen in our exercise, is different from what would be in a vacuum due to this property. It helps us understand and calculate how materials influence the behavior of electric fields, vital for designing capacitors and insulators.

Gauss's Law is like a bookkeeper, meticulously accounting for all electric charges within a closed surface. In its integral form, it mathematically links electric flux \(\Phi\) to the total charge \(Q_{\text{enclosed}}\) contained within a Gaussian surface. It's expressed as \[\oint_{S} \vec{E} \cdot d\vec{A}=\frac{Q_{\text {enclosed }}}{\varepsilon_{0}}\]\], where \(\varepsilon_{0}\) is the permittivity of free space. This fundamental equation is paramount in electrostatics, explaining how the presence of charges creates electric fields that permeate space.