Gauss's Law Practice Test

A very long, solid, nonconducting cylinder of radius $R=4.0\ \text{cm}$ has uniform volume charge density $\rho=+8.0\times10^{-7}\ \text{C/m}^3$. The cylinder is effectively infinite. A student chooses a coaxial cylindrical Gaussian surface of radius $r=2.0\ \text{cm}$ (inside the material) and length $L=0.60\ \text{m}$. Use $\varepsilon_0=8.85\times10^{-12}\ \text{C}^2/(\text{N}\cdot\text{m}^2)$. For $r<R$, $Q_{\text{enc}}=\rho(\pi r^2L)$ and Gauss’s Law gives $E(2\pi rL)=Q_{\text{enc}}/\varepsilon_0$. Based on the scenario, determine the electric field magnitude at $r=2.0\ \text{cm}$.

A conducting sphere of radius $R=0.050\ \text{m}$ is in electrostatic equilibrium and has total charge $Q=+1.2\times10^{-8}\ \text{C}$ on its surface. Inside a conductor in electrostatic equilibrium, the electric field is zero everywhere. A student considers a spherical Gaussian surface of radius $r=0.030\ \text{m}$ located entirely within the conducting material. Use $\varepsilon_0=8.85\times10^{-12}\ \text{C}^2/(\text{N}\cdot\text{m}^2)$. Although Gauss’s Law always holds, the enclosed charge for this interior Gaussian surface is $Q_{\text{enc}}=0$, so $\oint \vec{E}\cdot d\vec{A}=0$. Using the situation described, what is the magnitude of the electric field at $r=0.030\ \text{m}$ from the center?