AP Physics C: E&M FRQ Practice

💡AP Physics C: E&M

Practice FRQ 1 of 201/20

1. A nonconducting solid sphere of radius $R = 0.30\ \text{m}$ is centered at the origin. The sphere contains a spherical cavity of radius $a = 0.10\ \text{m}$ that is also centered at the origin. The material of the sphere (for $a < r < R$ ) has a uniform volume charge density $\rho = +6.0\times10^{-6}\ \text{C/m}^3$ . The cavity region $r < a$ is vacuum. A thin, conducting spherical shell of inner radius $R$ and outer radius $R_\text{out} = 0.32\ \text{m}$ is concentric with the sphere. The conducting shell has a net charge $Q_\text{shell} = -1.0\times10^{-7}\ \text{C}$ . The entire configuration is electrostatic and isolated, as shown in Figure 1.

Figure 1. Concentric spherical cavity, uniformly charged insulating material, and conducting spherical shell (cross-sectional view through a diameter).

Figure 2. Axes for sketching electric field magnitude E as a function of radial distance r.

Using Gauss's law, derive an expression for the magnitude $E$ of the electric field as a function of the radial distance $r$ for the region $a < r < R$ . Express your answer in terms of $\rho$ , $a$ , $r$ , and physical constants, as appropriate.

ii.

Derive an expression for the magnitude $E$ of the electric field as a function of the radial distance $r$ for the region $r > R_\text{out}$ . Your expression must be in terms of $\rho$ , $a$ , $R$ , $Q_\text{shell}$ , $r$ , and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

iii.

On the axes shown in Figure 2, sketch a graph of $E$ as a function of $r$ from $r = 0$ to a position that is outside the conducting shell.

Figure 3. Same geometry as Figure 1, but the cavity (r < a) is filled with a linear dielectric of relative permittivity κ.

Derive an expression for the magnitude of the electric field $E$ in the dielectric-filled cavity region $r < a$ . Express your answer in terms of $\rho$ , $a$ , $\kappa$ , and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. The cavity region $r < a$ is filled with a linear dielectric material of relative permittivity $\kappa = 4.0$ , as shown in Figure 3. The charge density in the insulating material for $a < r < R$ remains $\rho = +6.0\times10^{-6}\ \text{C/m}^3$ , and the conducting shell still has net charge $Q_\text{shell} = -1.0\times10^{-7}\ \text{C}$ .