A total charge Q is distributed uniformly throughout a...

A total charge $Q$ is distributed uniformly throughout a spherical shell of inner and outer radii $r_{1}$ and $r_{2}$ respectively. Show that the electric field due to the charge is $$\begin{array}{ll} \overrightarrow{\mathbf{E}}=\overrightarrow{0} & \left(r \leq r_{1}\right) \\ \overrightarrow{\mathbf{E}} & =\frac{Q}{4 \pi \varepsilon_{0} r^{2}}\left(\frac{r^{3}-r_{1}^{3}}{r_{2}^{3}-r_{1}^{3}}\right) \hat{\mathbf{r}} \\ \overrightarrow{\mathbf{E}} & =\frac{Q}{4 \pi \varepsilon_{0} r^{2}} \hat{\mathbf{r}} \end{array}$$

A total charge $Q$ is distributed uniformly throughout a spherical shell of inner and outer radii $r_{1}$ and $r_{2}$ respectively. Show that the electric field due to the charge is $$\begin{array}{ll}
\overrightarrow{\mathbf{E}}=\overrightarrow{0} & \left(r \leq r_{1}\right) \\
\overrightarrow{\mathbf{E}} & =\frac{Q}{4 \pi \varepsilon_{0} r^{2}}\left(\frac{r^{3}-r_{1}^{3}}{r_{2}^{3}-r_{1}^{3}}\right) \hat{\mathbf{r}} \\
\overrightarrow{\mathbf{E}} & =\frac{Q}{4 \pi \varepsilon_{0} r^{2}} \hat{\mathbf{r}}
\end{array}$$

Key Concepts

Gauss's Law

Gauss's Law is a fundamental principle linking the net electric flux through a closed surface to the charge enclosed by that surface. It simplifies the calculation of electric fields, especially in cases with high symmetry, by allowing one to relate the field to the total charge within a chosen Gaussian surface.

Spherical Symmetry

Spherical symmetry implies that the physical properties of a charge distribution are invariant under rotation about the center. This property ensures that the electric field at any point depends solely on the radial distance from the center, meaning the field is radial and its magnitude is the same on any spherical surface centered at the nucleus of the symmetry.

Gaussian Surface Selection

Selecting an appropriate Gaussian surface leverages the symmetry of the problem to simplify the evaluation of the electric flux. In a spherically symmetric charge distribution, choosing a concentric spherical Gaussian surface allows the electric field magnitude to be taken as constant over the surface, facilitating the integration of the flux and the application of Gauss's Law.

Volume Charge Density

Volume charge density characterizes how charge is distributed within a volume and is defined as the charge per unit volume. In uniformly charged bodies, it provides a means to relate the total charge of the object to the dimensions of the charge distribution, enabling the calculation of the enclosed charge when applying Gauss's Law.

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