AP Physics C: E&M FRQ Practice

💡AP Physics C: E&M

Guided Practice

Practice FRQ 1 of 201/20

1. A solid insulating sphere of radius

R = 0.040\ \text{m}

is centered at the origin and has a uniform volume charge density

\rho = +6.0× 10^{-6}\ \text{C/m}^3

. Concentric with the sphere is a thin conducting spherical shell with inner radius

a = 0.060\ \text{m}

and outer radius

b = 0.080\ \text{m}

. The conducting shell has a net charge

Q_{\text{shell}} = -1.0× 10^{-9}\ \text{C}

. The region everywhere is filled with a linear dielectric material of permittivity

\varepsilon = 2.5\,\varepsilon_0

, as shown in Figure 1.

Figure 1. Charged insulating sphere and concentric conducting spherical shell embedded in a uniform dielectric.

A clean black-and-white cross-sectional schematic (a 2D cut through the center) of concentric spherical objects centered on the origin.

Overall layout and center reference:
- The diagram is centered on a single point marked with a small solid dot at the exact center of the page.
- Next to the dot, place the label "origin" and the symbol "O".
- All circles share this same center.

Insulating sphere (inner solid sphere):
- Draw a circle representing the outer surface of a solid insulating sphere.
- Label this region (inside the circle) "solid insulating sphere".
- Inside the sphere, place the text "uniform volume charge density" on one line and directly beneath it "ρ = +6.0×10^−6 C/m^3".
- The sphere boundary is a single solid circular outline.

Conducting spherical shell (concentric annulus in cross-section):
- Draw two additional concentric circles outside the insulating sphere to represent the conducting shell’s inner and outer surfaces.
- The smaller of these two circles is the inner surface; the larger is the outer surface.
- Shade or lightly hatch ONLY the annular region between these two circles to indicate conducting material.
- Place the label "conducting spherical shell" centered within the hatched annulus.
- On the conducting shell region, place the text "net charge on shell" and directly beneath it "Q_shell = −1.0×10^−9 C".

Radii and exact numerical labels:
- Draw three straight radius markers as thin lines from the origin to the right, each terminating exactly on one of the three key boundaries.
- The shortest radius marker ends on the insulating sphere boundary and is labeled "R = 0.040 m".
- The next radius marker ends on the inner surface of the conducting shell and is labeled "a = 0.060 m".
- The longest radius marker ends on the outer surface of the conducting shell and is labeled "b = 0.080 m".
- Each label is placed just beyond the end of its radius marker, with the variable and value on the same line.

Dielectric medium (everywhere):
- Indicate that the entire space in the figure (inside the insulating sphere, the gap between sphere and shell, the shell’s exterior region, and all visible background) is filled with dielectric.
- Place a single background label in the open space outside the outer radius circle: "linear dielectric everywhere" on one line, and beneath it "ε = 2.5 ε0".

Spatial relationships that must be visually unambiguous:
- The insulating sphere boundary is the smallest circle.
- There is a clear empty-looking gap (no hatching) between the insulating sphere boundary at R and the shell’s inner boundary at a.
- The conducting material occupies only the region from a to b (the hatched annulus).
- The region outside b is unhatched background labeled as dielectric.

Styling constraints:
- Use solid black outlines for boundaries.
- Use light, evenly spaced hatching for the conducting shell material only.
- No electric field lines and no charges drawn as discrete symbols on surfaces in this figure.
- No title other than the caption; no grid.

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

A blank set of Cartesian axes for a student sketch of electric field magnitude as a function of radial distance.

Axes formatting (all required text must be visible):
- Horizontal axis labeled "r (m)".
- Vertical axis labeled "E (N/C)".
- The axes intersect at the lower-left corner of the plotting area, and the intersection is labeled with the number "0".
- Arrowheads appear on the positive end of the r-axis (pointing right) and on the positive end of the E-axis (pointing up).

X-axis (r) numeric range and ticks:
- The r-axis starts at 0 and ends at 0.10.
- Tick marks are drawn every 0.01.
- The tick labels shown are: 0, 0.01, 0.02, 0.03, 0.04, 0.05, 0.06, 0.07, 0.08, 0.09, 0.10.
- Three special tick marks are emphasized (slightly longer than the others) and labeled directly below the axis with both symbol and value:
- At r = 0.040, label "R = 0.040".
- At r = 0.060, label "a = 0.060".
- At r = 0.080, label "b = 0.080".

Y-axis (E) numeric range and ticks:
- The E-axis starts at 0 and ends at 1.6×10^4.
- Tick marks are drawn every 2.0×10^3.
- The tick labels shown are: 0, 2.0×10^3, 4.0×10^3, 6.0×10^3, 8.0×10^3, 1.0×10^4, 1.2×10^4, 1.4×10^4, 1.6×10^4.

Plot area styling:
- No curve is drawn (completely blank plotting region).
- No grid lines.
- Medium-weight axis lines; lighter tick marks.
- No additional annotations besides the axis labels, numeric tick labels, and the three emphasized x-axis markers at R, a, and b.

i. Using Gauss's law, derive an expression for the magnitude

E(r)

of the electric field for the region

R < r < a

. Express your answer in terms of

\rho

R

r

\varepsilon

, and physical constants, as appropriate.

ii. Derive an expression for the electric flux

\Phi_E

through a spherical surface of radius

r = 0.070\ \text{m}

centered at the origin. Express your answer in terms of the given quantities 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 shell

\left(r > b\right)

Figure 3. Conducting shell surfaces at r = a and r = b emphasized for charge distribution.

A black-and-white cross-sectional schematic of the same concentric geometry as Figure 1, centered on the same origin, with explicit emphasis on the inner and outer surfaces of the conducting shell.

Overall geometry (must match Figure 1 exactly):
- A central dot marks the origin at the exact center of the page, labeled "origin O".
- Three concentric circles are drawn, in increasing radius order: the insulating sphere boundary at R, the shell inner surface at a, and the shell outer surface at b.

Insulating sphere:
- The smallest circle is the insulating sphere boundary.
- Inside it, label "solid insulating sphere" and include the text "ρ = +6.0×10^−6 C/m^3".

Conducting shell with highlighted surfaces:
- The annular region between the circles at a and b is the conducting material; lightly hatch the annulus.
- Make the inner boundary circle at r = a thicker than all other outlines and label it "inner surface (r = a = 0.060 m)".
- Make the outer boundary circle at r = b thicker than all other outlines and label it "outer surface (r = b = 0.080 m)".
- Place two leader arrows (thin arrows) pointing to the thickened circles:
- One arrow points to the inner surface circle and is labeled "Q_inner on inner surface".
- One arrow points to the outer surface circle and is labeled "Q_outer on outer surface".

Net charge label on the conductor:
- Inside the hatched annulus (conducting material), include the text "net charge on shell" and directly beneath it "Q_shell = −1.0×10^−9 C".

Radii labels:
- Draw radius markers from the origin to the right for R, a, and b, each ending exactly on its corresponding circle.
- Label them exactly: "R = 0.040 m", "a = 0.060 m", "b = 0.080 m".

Dielectric label:
- In the open space outside the outer circle, include: "linear dielectric" and beneath it "ε = 2.5 ε0".

Styling constraints:
- No electric field lines.
- No free-charge plus/minus symbols placed on the surfaces (charges are indicated only by Q_inner, Q_outer, and Q_shell labels).
- No grid and no extra decorative elements.

B. Derive an expression for the total charge

Q_{\text{inner}}

on the inner surface of the conducting shell (at

r=a

) and the total charge

Q_{\text{outer}}

on the outer surface (at

r=b

). Express your answers in terms of

\rho

R

Q_{\text{shell}}

, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. The configuration remains as shown in Figure 3. The conducting shell is in electrostatic equilibrium and the dielectric permittivity is uniform at

\varepsilon = 2.5\,\varepsilon_0

everywhere.