AP Physics C: E&M FRQ Practice

💡AP Physics C: E&M

Practice FRQ 1 of 171/17

2. A solid insulating sphere of radius $R = 0.060\ \text{m}$ is centered at the origin. The sphere carries a total charge $Q = +8.0\ \mu\text{C}$ that is uniformly distributed throughout its volume. The sphere is embedded in a large, homogeneous insulating material with relative permittivity $\kappa = 3.0$ (so $\varepsilon = \kappa\varepsilon_0$ ). A point $P$ lies on the +x-axis at $x = 0.120\ \text{m}$ . A spherical Gaussian surface of radius $r = 0.040\ \text{m}$ (surface $S_1$ ) and another of radius $r = 0.120\ \text{m}$ (surface $S_2$ ) are centered on the origin, as shown in Figure 1.

Figure 1. Uniformly charged insulating sphere with two concentric spherical Gaussian surfaces and a point on the +x-axis.

Figure 2. Bar chart template for electric-field magnitude E at three radii (two to be completed by students).

In Figure 2, draw bars to represent $E$ at $r = 0.040\ \text{m}$ and $r = 0.120\ \text{m}$ relative to the given bar at $r = 0.060\ \text{m}$ . If $E = 0$ , write a "0" in that column. The magnitude of the electric field at distance r from the center is E(r). The partially completed bar chart in Figure 2 shows a bar that represents E at $r = 0.060\ \text{m}$ .

Derive an expression for the electric flux $\Phi_{S_1}$ through the spherical Gaussian surface $S_1$ of radius $r = 0.040\ \text{m}$ in terms of $Q$ , $R$ , $r$ , $\kappa$ , 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 electric-field magnitude E versus distance r from the center (0 to 0.150 m).

On the axes shown in Figure 3, sketch a graph of the electric field magnitude $E$ as a function of distance $r$ from the center for $0 ≤ r ≤ 0.150\ \text{m}$ . Your graph should be consistent with the bars you drew in part A.

Indicate whether the net electric flux $\Phi_{S_3}$ through $S_3$ is positive, negative, or zero. Briefly justify your answer by referencing conservation of charge and the relationship between electric flux and enclosed charge. A conducting spherical shell of inner radius $a = 0.080\ \text{m}$ and outer radius $b = 0.100\ \text{m}$ is placed concentrically around the charged insulating sphere, still embedded in the same dielectric (relative permittivity $\kappa = 3.0$ ). The shell is initially neutral and isolated. Consider a spherical Gaussian surface $S_3$ of radius $r = 0.090\ \text{m}$ (located within the conducting material).