Find the E-field at a distance \(r\) from the centre of each of the spherically symmetrical charge distributions of Problem 1.14.

Gauss's Law is a fundamental principle of electromagnetism that relates the distribution of electric charge to the resulting electric field. The law is encapsulated in the mathematical statement:

\[\begin{equation}\oint \vec{E} \cdot d\vec{A} = \frac{q_\text{enclosed}}{\epsilon_0}\end{equation}\]
This equation tells us that the total electric flux through a closed surface, which is the product of the electric field (\(\vec{E}\)) and the area of the surface (\(d\vec{A}\)), is proportional to the charge enclosed within the surface (\(q_\text{enclosed}\)), divided by the electric constant (\(\epsilon_0\)). Essentially, this means that the total flux depends only on the charge within the surface, not on the shape or size of the surface itself. This law is particularly useful when dealing with symmetrical charge distributions because it allows for a simplified calculation of the electric field.

A spherical charge distribution is a model where the charge is distributed uniformly throughout a spherical volume or on its surface. In the context of Gauss's Law, this symmetry allows us to set up a Gaussian surface that coincides with the symmetry of the charge distribution, in this case a sphere. The radial symmetry of the spherical charge distribution means that the electric field at any point on the Gaussian surface will have the same magnitude and will be directed radially outward or inward, depending on the sign of the charge distribution. This uniformity significantly simplifies the analysis, allowing us to treat the electric field magnitude as a constant over the surface when applying Gauss's Law.

Volume charge density (represented by \(\rho\)) is a measure of how much electric charge is contained within a particular volume. It is defined as the charge (\(Q\)) per unit volume (\(V\)): <\br><\br>\[\begin{equation}\rho = \frac{Q}{V}\end{equation}\]<\br>In a homogenous spherical charge distribution, the density is constant. However, if the charge varies with radius, as in a sphere with a charge density that decreases linearly from the center to the surface, \(\rho(r) = \rho_0 (1 - \frac{r}{R})\), the volume charge density is no longer constant and must be integrated over the volume to find the enclosed charge for calculating the electric field using Gauss's Law.

Determining the electric field inside a sphere requires an understanding of the charge enclosed by a Gaussian surface within the sphere. The charge enclosed is proportional to the volume of the Gaussian surface when the charge distribution is uniform. As the Gaussian surface enlarges with increasing radius within the sphere, so does the enclosed charge. By applying Gauss's Law, we find that the electric field inside the sphere increases linearly with distance from the center when the charge density is constant. The general expression for the electric field inside a sphere with radius \(r\), for \(r \le R\), where \(R\) is the sphere's radius, is given by:<\br><\br>\[\begin{equation}E = \frac{Qr}{4 \pi \epsilon_0 R^3}\end{equation}\]<\br>When the charge density varies, as in the case of a linearly decreasing charge density, the enclosed charge must be determined by integrating the charge density over the volume up to radius \(r\). Only then can the electric field be calculated using Gauss's Law.

For the electric field outside a sphere, the Gaussian surface encloses the entire charge within the sphere, regardless of the Gaussian surface's radius as long as it is larger than the sphere ('for \(r > R\)'). According to Gauss's Law, outside a sphere with a homogenous charge distribution, the electric field behaves as if all the charge were concentrated at the center of the sphere. Therefore, the electric field decreases with the square of the distance from the center (\(\frac{1}{r^2}\)), resembling the field of a point charge. The expression for the electric field outside the sphere, for \(r > R\), is given by:<\br><\br>\[\begin{equation}E = \frac{Q}{4 \pi \epsilon_0} \cdot \frac{1}{r^2}\end{equation}\]<\br>This inverse square relationship holds true for any radial distance greater than the radius of the sphere, signifying that, from afar, the spherical charge distribution can be replaced by a point charge located at the center with the same magnitude of total charge.