AP Physics C: E&M FRQ Practice

💡AP Physics C: E&M

Practice FRQ 1 of 201/20

1. A solid insulating sphere of radius $R_1 = 3.0\ \text{cm}$ is centered inside a thin, conducting spherical shell with inner radius $R_2 = 7.0\ \text{cm}$ and outer radius $R_3 = 9.0\ \text{cm}$ , as shown in Figure 1. The insulating sphere has a uniform volume charge density $\rho = +6.0\times10^{-6}\ \text{C/m}^3$ . The conducting shell has a net charge $Q_\text{shell} = -2.0\times10^{-9}\ \text{C}$ . The system is in electrostatic equilibrium and is surrounded by air. The permittivity of air may be taken as $\varepsilon_0 = 8.85\times10^{-12}\ \text{F/m}$ .

Figure 1. Insulating sphere centered inside a conducting spherical shell (electrostatic equilibrium).

A clean black-and-white cross-sectional (cutaway) diagram of two concentric spheres, centered on the same point.

Overall layout and alignment:
- The diagram is centered on the page with a single common center point placed at the exact visual center of the figure.
- Show a horizontal diameter line passing through the center point from left to right (a thin line) to emphasize the radial geometry.
- Place a small dot at the center and label it "center".
- Add a radial arrow labeled "r" pointing directly to the right from the center dot.

Objects (concentric layers, drawn as circles in cross-section):
1) Inner solid insulating sphere:
- Draw a filled (solid) circle centered on the center dot.
- Its boundary is the circle of radius R1.
- Label the boundary circle with the text "R1 = 3.0 cm" placed just outside the sphere boundary on the right side, with a short leader line pointing to the boundary.
- Inside the solid sphere region, place the text "uniform volume charge density" on one line and "ρ = +6.0×10⁻⁶ C/m³" on the next line.

2) Conducting spherical shell (shown as an annulus in cross-section):
- Draw two concentric circles centered on the same center dot representing the inner and outer surfaces of the conductor.
- The inner surface radius is R2 and the outer surface radius is R3, creating a visible annular thickness.
- The region between the R2 circle and the R3 circle is the conducting material; depict it with light gray shading (uniform) to distinguish it from the empty cavity and the insulating sphere.
- Label the inner boundary of the conductor with "R2 = 7.0 cm" placed just inside the cavity region near the right side, with a leader line to the inner circular boundary.
- Label the outer boundary of the conductor with "R3 = 9.0 cm" placed just outside the outermost circle near the right side, with a leader line to the outer circular boundary.
- On the shaded conductor region, place the text "conducting shell" and beneath it "net charge Q_shell = −2.0×10⁻⁹ C".
- Add the text "electrostatic equilibrium" adjacent to the conductor region (outside the annulus, near the top-right), indicating the conductor is in equilibrium.

Region identification (explicit spatial relationships):
- Clearly show that the insulating sphere of radius R1 is completely inside the empty cavity region that extends from r = R1 to r = R2.
- Do NOT draw any off-center placement: all circles must be perfectly concentric.

Style constraints:
- Use medium-thickness black outlines for all circular boundaries.
- Use no grid, no background texture.
- No numerical values other than: R1 = 3.0 cm, R2 = 7.0 cm, R3 = 9.0 cm, ρ = +6.0×10⁻⁶ C/m³, Q_shell = −2.0×10⁻⁹ C.
- No extra symbols for charge distribution on surfaces; only the text labels above.

Figure 2. Axes for graphing electric-field magnitude E versus radial distance r.

A blank set of Cartesian axes intended for a hand-drawn sketch (no curve pre-drawn).

Axes (all required labels and exact numeric markings):
- Horizontal axis: labeled "r (cm)" centered below the axis.
- Horizontal axis numeric range: starts at 0 at the origin and ends at 12 at the far right.
- Horizontal tick marks and labels: ticks every 1 cm from 0 to 12, with the numbers "0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12" shown.
- Emphasized special ticks: the tick at 3 is labeled additionally with "R1" directly above the number 3; the tick at 7 is labeled additionally with "R2" above the number 7; the tick at 9 is labeled additionally with "R3" above the number 9. (Keep the numeric labels 3, 7, 9 visible as well.)

- Vertical axis: labeled "E (N/C)" centered along the vertical axis.
- Vertical axis numeric scale: show tick marks but no numbers (to allow any appropriate vertical scaling by students). The ticks are evenly spaced from the origin to the top edge.
- The origin is explicitly labeled "0" at the intersection of the axes.
- Put arrowheads on the positive end of the r-axis (to the right) and on the positive end of the E-axis (upward).

Curve shape description (required, but this figure contains no curve):
- No curve is drawn in this figure. The only marks are the axes, tick marks, and the R1/R2/R3 indicators.

Style constraints:
- Black axes with medium line thickness.
- No grid lines.
- No title other than the caption outside the plotting area.

Using Gauss's law, derive an expression for the magnitude $E$ of the electric field as a function of radial distance $r$ for the region $R_1 < r < R_2$ . Express your answer in terms of $\rho$ , $R_1$ , $r$ , and physical constants, as appropriate.

ii.

Derive an expression for the net charge $Q_\text{inner}$ on the inner surface of the conducting shell and the net charge $Q_\text{outer}$ on the outer surface of the conducting shell. Express your answers in terms of $\rho$ , $R_1$ , $Q_\text{shell}$ , 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 outer surface of the shell.

Figure 3. Same geometry as Figure 1, but the inner sphere region is a linear dielectric (relative permittivity κ).

A black-and-white cross-sectional (cutaway) diagram with the same concentric geometry and label placement rules as Figure 1, plus a clear dielectric indication.

Overall layout and alignment:
- Use the exact same centered concentric-circle layout: a center dot at the visual center, a thin horizontal diameter line, and a radial arrow labeled "r" pointing right.

Objects (concentric layers):
1) Inner sphere region (dielectric-filled):
- Draw the inner sphere boundary as a circle of radius R1.
- Fill the entire region from the center out to R1 with a distinct diagonal hatch pattern (e.g., 45° lines) to indicate dielectric material.
- Place text inside this region reading "linear dielectric" on one line and "relative permittivity κ" on the next line.
- Label the boundary with "R1 = 3.0 cm" just outside the R1 boundary on the right side with a leader line.

2) Conducting spherical shell:
- Draw the inner conductor surface at radius R2 and the outer conductor surface at radius R3, both concentric with the center.
- Shade the annular conductor region (between R2 and R3) with uniform light gray (different from the dielectric hatching).
- Label inner conductor boundary "R2 = 7.0 cm" with a leader line.
- Label outer conductor boundary "R3 = 9.0 cm" with a leader line.
- On the conductor region, include the same text: "conducting shell" and "net charge Q_shell = −2.0×10⁻⁹ C" and also the note "electrostatic equilibrium".

Spatial relationships (explicit):
- The dielectric occupies only the region r < R1.
- The cavity region between R1 and R2 is unshaded/blank (air).
- The conducting metal occupies only the region between R2 and R3.

Style constraints:
- Medium black outlines for all boundaries.
- No extra numeric values besides R1, R2, R3, and Q_shell.
- Do not show any off-center placement; all circles must be perfectly concentric.

Derive an expression for the magnitude $E$ of the electric field as a function of $r$ for the region $0 < r < R_1$ with the dielectric present. Express your answer in terms of $\rho$ , $\kappa$ , $r$ , and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. A dielectric material with relative permittivity $\kappa = 4.0$ completely fills the region $0 < r < R_1$ , as shown in Figure 3. The free charge density in the sphere remains uniform and equal to $\rho = +6.0\times10^{-6}\ \text{C/m}^3$ , and the conducting shell still has net charge $Q_\text{shell} = -2.0\times10^{-9}\ \text{C}$ .