A spherical region of space of radius \(a\) is filled with positive charge of volume density \(\rho\) which varies with distance \(r\) from the centre of the sphere but is constant for a given \(r\). Find the total charge within the sphere if (a) \(\rho=\rho_{0} r / a\), (b) \(\rho=\rho_{0}\left(1-r^{2} / a^{2}\right)\), (c) \(\rho=\rho_{0} \mathrm{e}^{-n / 4}\). Interpret the constant \(\rho_{0}\) in each case. (The last two have been suggested as approximate charge distributions in atomic nuclei.)

When we discuss electric charges in a continuous space—such as within a material or a defined volume—we often refer to the concept of volume charge density, symbolized as \(\rho\). Volume charge density is simply the amount of charge present in a unit volume of a material. Specifically, when looking at a problem of electric charge distributed throughout a spherical volume, the charge density \(\rho\) might vary from point to point within the sphere, depending on the distance \(r\) from the center.

The three cases outlined in the original exercise demonstrate different ways the charge density can be described. In case (a), the charge density increases linearly with the radius from the sphere's center, while in case (b), the charge density decreases with the square of the radius, implying a concentration of charge towards the center of the sphere. Case (c) introduces an exponential decay of the density away from the center—this sort of distribution is sometimes used to model atomic nuclei.

Understanding the distribution of charge helps us calculate not just the total charge within the sphere using integrals, but also inform us about the behavior of the electric field and potential around and inside the sphere, as we'll explore further with Gauss's law and electric potential concepts.

Moving from charge distributions to the resulting electric fields, Gauss's law is a powerful tool rooted in one of Maxwell's equations. It relates the electric field emanating from a volume to the charge enclosed within it. Formally, Gauss's law states that the total electric flux out of a closed surface (a Gaussian surface) is equal to the charge enclosed by that surface divided by the permittivity of the medium. Mathematically, it is expressed as \( \Phi_E = \frac{Q}{\varepsilon_0} \).

This law is especially useful because it simplifies what could be complex integrations to more simple ones, as long as there is some symmetry to exploit, as in the case of a spherically symmetric charge distribution. By choosing an appropriate Gaussian surface, which in this case would be concentric spheres with varying radii, we can easily calculate the electric field at any point within the sphere without actually solving complex equations.

The beauty of Gauss's law is that it doesn't matter how the charge is distributed within the Gaussian surface; the total flux depends only on the total charge inside. Thus, it is an indispensable tool in electrostatics, especially for problems with high degrees of symmetry.

Lastly, the electric potential inside a sphere is a scalar quantity that represents the electric potential energy per unit charge at a point within the spherical charge distribution. Unlike the electric field, which is a vector and gives information about the force experienced by a charge, the electric potential provides insight into the energy landscape of the system.

Inside a uniformly charged sphere, the electric potential is determined by both the charges outside the point of interest (which contribute a constant value because of the shell theorem) and the charges inside the radius of interest, which can be calculated by integrating the electric field stemming from those charges over the distance from the center of the sphere to the point in question. The result is a function that will illustrate how energy varies with respect to the position inside the sphere. In the case of non-uniform charge distributions, as highlighted in the original exercise, the electric potential must account for how the varying charge densities affect the potential at each point.