AP Physics C: Electricity and Magnetism FRQ Practice

💡AP Physics C: Electricity and Magnetism

Practice FRQ 1 of 21/2

4. A solid insulating sphere of radius 0.30 m is centered at the origin and has uniform volume charge density $\rho = +6.0\times10^{-6}\ \text{C/m}^3$ . A spherical cavity of radius 0.10 m is carved out of the sphere. The cavity's center is on the +x-axis at $x = 0.10\ \text{m}$ . The remaining charge in the insulating material is unchanged by the carving (charge is removed only from the cavity region). The object is surrounded by a uniform dielectric material with relative permittivity $\kappa = 2.0$ . Consider two points on the x-axis: Point 1 is at $x=+0.05\ \text{m}$ (inside the cavity), and Point 2 is at $x=-0.05\ \text{m}$ (inside the charged insulating material). Figure 1 shows this setup. Ignore any polarization surface-charge details and treat the dielectric effect using $\epsilon = \kappa\epsilon_0$ .

Figure 1. Charged insulating sphere with an off-center spherical cavity and two labeled points on the x-axis (dielectric with κ = 2.0).

$E_1$ is the magnitude of the electric field at Point 1, and $E_2$ is the magnitude of the electric field at Point 2.

Indicate whether $E_2$ is greater than, less than, or equal to $E_1$ by writing one of the following.

$E_2 > E_1$
$E_2 < E_1$
$E_2 = E_1$

Justify your answer.

Derive an expression for the electric flux $\Phi_E$ through a spherical Gaussian surface of radius $r = 0.25\ \text{m}$ centered at the origin. Express your answer in terms of $\rho$ , $\epsilon_0$ , $\kappa$ , and geometric constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Figure 2. Same charged sphere and points as Figure 1, but the cavity center is moved to x = +0.05 m (Point 1 coincides with the cavity center).

Indicate whether $E_{2,\text{new}}$ is greater than, less than, or equal to $E_2$ by writing one of the following. Later, the cavity is moved so that its center is at $x = +0.05\ \text{m}$ , as shown in Figure 2. The charge density of the insulating material remains $\rho = +6.0\times10^{-6}\ \text{C/m}^3$ and the surrounding medium remains a uniform dielectric with $\kappa = 2.0$ . The points are unchanged: Point 1 is at $x=+0.05\ \text{m}$ and Point 2 is at $x=-0.05\ \text{m}$ . Let $E_{1,\text{new}}$ and $E_{2,\text{new}}$ be the new magnitudes of the electric field at Points 1 and 2, respectively.

$E_{2,\text{new}} > E_2$
$E_{2,\text{new}} < E_2$
$E_{2,\text{new}} = E_2$

Briefly justify your answer by referencing your reasoning from part A and/or a superposition argument consistent with part B.

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