AP Physics C: E&M FRQ Practice

💡AP Physics C: E&M

Practice FRQ 1 of 201/20

1. An insulating sphere of radius $R = 0.20\ \text{m}$ is centered at the origin. The sphere has a spherical cavity of radius $a = 0.050\ \text{m}$ that is concentric with the sphere, as shown in Figure 1. The insulating material (the region $a < r < R$ ) contains a uniform volume charge density $\rho = +6.0\times10^{-6}\ \text{C/m}^3$ . The cavity (the region $r < a$ ) is empty. Ignore edge effects and assume electrostatic equilibrium.

Figure 1. Insulating sphere with concentric spherical cavity and uniform volume charge density in the insulating material.

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

Using Gauss's law, derive an expression for the magnitude $E(r)$ of the electric field as a function of the radial distance $r$ from the center for each of the three regions $0 ≤ r < a$ , $a < r < R$ , and $r > R$ . Express your answers in terms of $\rho$ , $a$ , $R$ , $r$ , and physical constants, as appropriate.

ii.

Derive an expression for the potential difference $\Delta V = V(a) - V(R)$ . Begin your derivation by writing a fundamental physics principle or an equation from the reference information, and use your result from part A(i). Express your answer in terms of $\rho$ , $a$ , $R$ , and physical constants, as appropriate.

iii.

On the axes shown in Figure 2, sketch a graph of the magnitude of the electric field $E$ as a function of $r$ from $r = 0$ to a position that is outside the sphere (past $r = R$ ). Clearly indicate any changes in functional form at $r = a$ and $r = R$ .

Figure 3. Same geometry as Figure 1, with the cavity filled by a linear dielectric of dielectric constant κ.

Derive an expression for the electric flux $\Phi_E$ through a spherical Gaussian surface of radius $r_g = 0.030\ \text{m}$ (which lies entirely within the dielectric-filled cavity). Express your answer in terms of $\rho$ , $a$ , $r_g$ , $\kappa$ , and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. A linear dielectric material with dielectric constant $\kappa = 4.0$ is inserted so that it completely fills the cavity region $0 ≤ r < a$ , as shown in Figure 3. The charge density $\rho$ in the insulating material remains uniform and unchanged.