In Figure 2, draw bars to represent $|E|$ at $r = 0.05\ \text{m}$ and $r = 0.15\ \text{m}$ relative to the bar shown at $r = 0.30\ \text{m}$. If $|E| = 0$, write a "0" in that column. The electric field magnitude at a distance r from the center is |E(r)|. The partially completed bar chart in Figure 2 shows a bar that represents |E| at r = 0.30 m.

Derive an expression for the electric field $E(r)$ (including sign/direction using the radial unit vector $\hat{\mathbf{r}}$) for the region $a < r < b$ in terms of $\rho$, $q$, $a$, $r$, $\varepsilon_0$, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

On the axes shown in Figure 3, sketch a graph of the electric flux $\Phi_E(r)$ through a spherical Gaussian surface of radius $r$, for $0 ≤ r ≤ 0.35\ \text{m}$. Your graph must indicate the sign of $\Phi_E$ and any changes in curvature or slope at $r=a$ and $r=b$.

Indicate whether the magnitude of the particle's initial acceleration is greater than, less than, or equal to $g$. Briefly justify your answer by comparing the magnitudes of the electric and gravitational forces at $r_0$ using $F_e = |Q|\,|E(r_0)|$ and $F_g = mg$. A small test particle of mass $m = 1.0\times10^{-6}\ \text{kg}$ and charge $Q = +1.0\times10^{-9}\ \text{C}$ is released from rest at $r_0 = 0.30\ \text{m}$ and moves radially under the influence of the electric force from the charge distribution and the gravitational force from Earth. Take $g = 9.8\ \text{m/s}^2$ downward. At the release point, the radial direction is horizontal, so the gravitational force is perpendicular to the radial electric force. Use the electric field at $r_0$ due to the total enclosed charge at that radius.