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 $R_1 = 4.0\ \text{cm}$ is centered inside a hollow conducting spherical shell with inner radius $R_2 = 10.0\ \text{cm}$ and outer radius $R_3 = 12.0\ \text{cm}$ , as shown in Figure 1. The insulating sphere has a uniform volume charge density $\rho = +2.0× 10^{-6}\ \text{C/m}^3$ . The conducting shell has a net charge $Q_{\text{shell}} = -6.0× 10^{-10}\ \text{C}$ . The region between the sphere and the shell is initially air, which may be treated as vacuum with permittivity $\varepsilon_0$ . The system is in electrostatic equilibrium.

Figure 1. Cross-sectional view of a uniformly charged insulating sphere centered within a conducting spherical shell (electrostatic equilibrium).

Draw a clean, black-and-white 2D cross-sectional diagram (a diameter cut) of concentric spherical objects centered on a single point.

Overall layout:
- The diagram is centered on a single common center point. Mark the exact center with a small solid dot.
- Show three concentric circles (cross-sections of spheres) with clearly different radii. The smallest circle is the insulating sphere of radius R1. The next circle is the inner surface of the conducting shell of radius R2. The largest circle is the outer surface of the conducting shell of radius R3.
- Ensure the spacing reflects the numerical radii exactly in proportion: R1 is much smaller than R2, and R3 is only slightly larger than R2. Specifically, the thickness of the conducting material (R3−R2) is much smaller than the vacuum gap (R2−R1).

Circle radii and dimension labels (must be explicitly printed on the figure):
- Along a horizontal radius line extending to the right from the center, draw three short radial tick marks where it intersects each circle.
- Add three radial dimension arrows from the center to each tick mark, stacked with slight vertical offsets so the arrows do not overlap.
- Label these arrows exactly as:
- "R1 = 4.0 cm" pointing to the smallest circle.
- "R2 = 10.0 cm" pointing to the middle circle (inner conductor surface).
- "R3 = 12.0 cm" pointing to the largest circle (outer conductor surface).

Region identification (must be explicit text on the figure):
- Label the interior region inside the smallest circle as: "Insulator (uniform ρ)".
- Place the charge-density label inside the insulating sphere: "ρ = +2.0×10^-6 C/m^3".
- Label the annular region between the R1 and R2 circles (the empty gap) with the text: "Vacuum (ε0)" and also include the inequality "R1 < r < R2" directly beneath it.
- Label the conducting material region between the R2 and R3 circles with: "Conductor" and the inequality "R2 < r < R3".
- Label the exterior region outside the largest circle with: "Outside (r > R3)".

Conductor net charge label:
- Place a label near the conducting shell (between the R2 and R3 circles, or just outside with a leader line pointing to the shell) reading exactly: "Net charge on shell: Q_shell = −6.0×10^-10 C".

Electrostatic-equilibrium cue (visual, not an equation):
- Do not draw field lines.
- Optionally, include a small note near the conductor reading: "Electrostatic equilibrium".

Line styling:
- All three circles are solid black lines.
- The conductor region (between R2 and R3) is lightly shaded with uniform diagonal hatching to indicate metal.
- The vacuum gap (R1 to R2) is left unshaded (white).
- The insulating sphere (r < R1) is filled with a very light gray (different from the conductor hatching) to distinguish it from vacuum.

Text placement constraints:
- All text must be outside the circles or in open areas inside regions so it does not overlap the circle boundaries.
- Use leader lines (thin arrows) for labels that would otherwise collide with circles.

Figure 2. Axes for sketching electric-field magnitude E versus radial distance r for the concentric-sphere system.

Create a blank Cartesian graph with clearly labeled axes and tick marks.

Axes (all items required):
- Horizontal axis label: "r (cm)".
- Horizontal axis range: from 0 to 18 cm.
- Horizontal tick marks: every 2 cm, labeled at 0, 2, 4, 6, 8, 10, 12, 14, 16, 18.
- Vertical axis label: "E (N/C)".
- Vertical axis range: from 0 to 1500 N/C.
- Vertical tick marks: every 300 N/C, labeled at 0, 300, 600, 900, 1200, 1500.
- The origin at the intersection of the axes is labeled "0".
- Add arrowheads at the positive ends of both axes.

Required radial reference markers:
- Draw three thin vertical guide lines (light gray or dashed) rising upward from the r-axis at the labeled tick positions corresponding exactly to:
- R1 at 4 cm (this is the tick labeled 4).
- R2 at 10 cm (tick labeled 10).
- R3 at 12 cm (tick labeled 12).
- At the top of each guide line, place a small label directly above it reading exactly "R1", "R2", and "R3" respectively.

No plotted curve:
- Do not draw any E(r) curve in this figure; it must be a blank set of axes for students to sketch on.

Styling:
- No grid.
- Axes are solid black, medium thickness.
- Guide lines are lighter and thinner than the axes so they are clearly reference markers, not data.
- No title beyond the caption; no legend.

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 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 electric flux $\Phi_E$ through a spherical surface of radius $r = 15.0\ \text{cm}$ centered on the spheres. Express your answer 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 conducting shell.

Figure 3. Same concentric-sphere geometry as Figure 1, with a dielectric (dielectric constant κ) filling only the gap between R1 and R2.

Draw the same 2D cross-sectional concentric-sphere diagram as in Figure 1 (same center dot, same three concentric circles, same radius dimension arrows and the same printed numerical values).

Geometry and labels (must match Figure 1 exactly):
- Keep the three circles and the horizontal radius line with the three dimension arrows labeled exactly:
- "R1 = 4.0 cm"
- "R2 = 10.0 cm"
- "R3 = 12.0 cm"
- Keep the insulating sphere label inside r<R1: "ρ = +2.0×10^-6 C/m^3".
- Keep the conductor region between R2 and R3 hatched and labeled "Conductor".
- Keep the conductor net charge label: "Net charge on shell: Q_shell = −6.0×10^-10 C" with a leader line pointing to the conducting shell.

Dielectric insertion (the only change from Figure 1):
- The annular region between the R1 and R2 circles (and only this region) is filled with a distinct shading pattern (for example, light stippling or light crosshatching) that is clearly different from the conductor hatching and different from the insulator fill.
- Place the text centered within this annular region: "Dielectric (κ)".
- Also include the inequality "R1 < r < R2" directly next to the dielectric label to make the dielectric’s extent unambiguous.

Outside region:
- Outside the largest circle, label the exterior as "Outside (r > R3)".

Styling constraints:
- Ensure three region textures are visually distinct:
- Insulating sphere (r<R1): very light uniform gray fill.
- Dielectric (R1<r<R2): stippled or crosshatched lightly.
- Conductor (R2<r<R3): diagonal hatching (heavier than dielectric shading).
- No field lines.
- All labels must be readable and not overlap circle boundaries; use leader lines if needed.

Derive an expression for the magnitude $E_\kappa(r)$ of the electric field for the region $R_1 < r < R_2$ after the dielectric is inserted. Express your answer in terms of $\kappa$ , $\rho$ , $R_1$ , $r$ , and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. A dielectric material of dielectric constant $\kappa = 4.0$ is inserted such that it completely fills the region $R_1 < r < R_2$ , as shown in Figure 3. The charges on the insulating sphere and on the conducting shell remain the same as in the original situation, and the conductor remains in electrostatic equilibrium.