AP Physics C: Electricity and Magnetism FRQ Practice

💡AP Physics C: Electricity and Magnetism

Practice FRQ 1 of 171/17

2. A solid insulating sphere of radius $R = 0.10\ \text{m}$ is centered at the origin. The sphere has a uniform volume charge density $\rho = +6.0\times10^{-6}\ \text{C/m}^3$ . Concentric with the sphere is a thin conducting spherical shell with inner radius $a = 0.20\ \text{m}$ and outer radius $b = 0.22\ \text{m}$ . The conducting shell has a net charge $Q_{\text{shell}} = -3.0\times10^{-8}\ \text{C}$ . The space between the sphere and the shell and the space outside the shell are vacuum with permittivity $\varepsilon_0 = 8.85\times10^{-12}\ \text{F/m}$ . Assume electrostatic equilibrium has been reached. Figure 1 shows the experimental setup.

Figure 1. Concentric charged insulating sphere and conducting spherical shell (cross-section).

Figure 2. Bar chart template for |E| at four radii, with |E| at r = 0.30 m as the reference.

In Figure 2, draw bars to represent $|\vec{E}|$ at $r = 0.05\ \text{m}$ , $0.15\ \text{m}$ , and $0.21\ \text{m}$ relative to $|\vec{E}|$ shown at $r = 0.30\ \text{m}$ . If $|\vec{E}| = 0$ , write a "0" in that column. The electric field is radial. The partially completed bar chart in Figure 2 shows a bar that represents the magnitude of the electric field $|\vec{E}|$ at $r = 0.30\ \text{m}$ .

Derive an expression for the magnitude of the electric field $|\vec{E}(r)|$ for the region $0 < r < R$ in terms of $\rho$ , $r$ , $\varepsilon_0$ , and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Figure 3. Axes for sketching |E| versus r from 0 to 0.40 m with region boundaries at R, a, and b.

On the axes shown in Figure 3, sketch a graph of $|\vec{E}|$ as a function of $r$ for $0 ≤ r ≤ 0.40\ \text{m}$ . Your graph must clearly indicate the behavior in each region: $0 ≤ r ≤ R$ , $R < r < a$ , $a < r < b$ , and $r > b$ .

Indicate whether the sketch you drew in part C is or is not consistent with Gauss's law for the Gaussian surface at $r_G = 0.21\ \text{m}$ . Briefly justify your answer by referencing the functional relationship between electric flux $\Phi_E$ , enclosed charge $Q_{\text{enc}}$ , and the electric field in a conductor in electrostatic equilibrium. A spherical Gaussian surface of radius $r_G = 0.21\ \text{m}$ (which lies within the conducting material of the shell) is considered. The shell is in electrostatic equilibrium. The following values may be used: $R = 0.10\ \text{m}$ , $\rho = +6.0\times10^{-6}\ \text{C/m}^3$ , $Q_{\text{shell}} = -3.0\times10^{-8}\ \text{C}$ , $\varepsilon_0 = 8.85\times10^{-12}\ \text{F/m}$ .