AP Physics C: E&M FRQ Practice

💡AP Physics C: E&M

Practice FRQ 1 of 201/20

1. An insulating sphere of radius

R = 0.20\ \text{m}

is centered at the origin. The sphere has a spherical cavity of radius

a = 0.050\ \text{m}

that is concentric with the sphere, as shown in Figure 1. The insulating material (the region

a < r < R

) contains a uniform volume charge density

\rho = +6.0\times10^{-6}\ \text{C/m}^3

. The cavity (the region

r < a

) is empty. Ignore edge effects and assume electrostatic equilibrium.

Figure 1. Insulating sphere with concentric spherical cavity and uniform volume charge density in the insulating material.

A clean black-and-white cross-sectional diagram (a 2D slice through the center) of two concentric circles representing a sphere with a concentric spherical cavity.

Layout and geometry (must be perfectly concentric):
- Draw a large circle centered in the middle of the figure. This is the outer surface of the insulating sphere.
- Draw a smaller circle with the exact same center as the large circle. This is the cavity boundary.
- The common center point is explicitly marked with a small solid dot and labeled "origin" directly next to it.

Numeric labels that must appear exactly as text:
- Near the boundary of the large circle, place the label "R = 0.20 m" with a thin leader line pointing to the outer circle.
- Near the boundary of the small circle, place the label "a = 0.050 m" with a thin leader line pointing to the inner circle.

Region identification and charge density (must be unambiguous):
- The annular region between the two circles (the material for which a < r < R) is filled with uniform light-gray shading (or evenly spaced hatch marks) to indicate the charged insulating material.
- Inside this shaded annulus, include the text label "uniform volume charge density" on one line and directly below it "ρ = +6.0×10^−6 C/m^3".
- The central cavity region (inside the smaller circle, r < a) is left completely white (no shading) and labeled "empty cavity (r < a)".

Radial coordinate indication:
- Draw a single straight radius line from the origin to the right, ending exactly at the outer circle. Label this line "r" above the line.
- On this same radius line, mark the point where it crosses the cavity boundary with a small tick mark and place the text "r = a" next to that tick.
- Mark the point where it reaches the outer surface with a small tick mark and place the text "r = R" next to that tick.

Styling constraints:
- Circle outlines are solid black, with the outer circle slightly thicker than the inner circle.
- No extraneous objects, no 3D perspective, no grid, no background texture.
- Only the labels specified above appear as text.

Figure 2. Axes for graphing the electric field magnitude E as a function of radial distance r.

A blank set of Cartesian axes prepared for sketching E versus r.

Axes and formatting (all of these must be present):
- Horizontal axis label: "r (m)" centered below the axis.
- Vertical axis label: "E (N/C)" centered along the vertical axis.
- The axes intersect at the left end of the horizontal axis, and that intersection is labeled with the number "0" (the origin label must be visible).
- Arrowheads are drawn at the positive end of the r-axis (pointing right) and at the positive end of the E-axis (pointing up).

Exact x-axis range and ticks (must be numeric):
- The r-axis starts at "0" at the origin and extends to "0.30" at the far right end.
- Place visible tick marks and numeric labels at increments of 0.05 m: "0.05", "0.10", "0.15", "0.20", "0.25", "0.30".
- The tick at "0.05" is additionally labeled "a = 0.050" directly beneath that tick (so the axis shows both the numeric tick and the symbolic boundary).
- The tick at "0.20" is additionally labeled "R = 0.20" directly beneath that tick.

Exact y-axis range and ticks (must be numeric, even though the curve is student-sketched):
- The E-axis runs from "0" at the origin up to "8.0×10^4" at the top.
- Place visible tick marks and numeric labels every "2.0×10^4": "2.0×10^4", "4.0×10^4", "6.0×10^4", "8.0×10^4".

Guide markers for region changes (must be explicit visual guides):
- Draw a faint vertical dashed guide line through the tick at r = a (0.05 m), extending from the x-axis upward to near the top of the plotting area.
- Draw a faint vertical dashed guide line through the tick at r = R (0.20 m), extending from the x-axis upward to near the top of the plotting area.
- Under the x-axis, centered between 0 and a, add the small text "Region: r < a".
- Under the x-axis, centered between a and R, add the small text "Region: a < r < R".
- Under the x-axis, centered between R and 0.30 m, add the small text "Region: r > R".

No plotted curve:
- Do not draw any E(r) curve on these axes; this figure is axes only.
- No gridlines (other than the two dashed vertical guides at r = a and r = R). No title.

i. 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 each of the three regions

0 ≤ r < a

a < r < R

, and

r > R

. Express your answers in terms of

\rho

a

R

r

, and physical constants, as appropriate.

ii. Derive an expression for the potential difference

\Delta V = V(a) - V(R)

. Begin your derivation by writing a fundamental physics principle or an equation from the reference information, and use your result from part A(i). Express your answer in terms of

\rho

a

R

, and physical constants, as appropriate.

iii. On the axes shown in Figure 2, sketch a graph of the magnitude of the electric field

E

as a function of

r

from

r = 0

to a position that is outside the sphere (past

r = R

). Clearly indicate any changes in functional form at

r = a

and

r = R

A linear dielectric material with dielectric constant

\kappa = 4.0

is inserted so that it completely fills the cavity region

0 ≤ r < a

, as shown in Figure 3. The charge density

\rho

in the insulating material remains uniform and unchanged.

B. Derive an expression for the electric flux

\Phi_E

through a spherical Gaussian surface of radius

r_g = 0.030\ \text{m}

(which lies entirely within the dielectric-filled cavity). Express your answer in terms of

\rho

a

r_g

\kappa

, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Figure 3. Same geometry as Figure 1, with the cavity filled by a linear dielectric of dielectric constant κ.

A clean black-and-white cross-sectional diagram of the same two concentric circles as Figure 1, with identical geometry and the same center at the origin.

Layout and geometry (must match Figure 1 exactly):
- Draw a large circle centered in the middle of the figure, representing the outer radius of the sphere.
- Draw a smaller concentric circle sharing the exact same center, representing the cavity boundary.
- Mark the shared center with a small solid dot and label it "origin".

Numeric labels that must appear exactly as text:
- Outer boundary label with leader line: "R = 0.20 m" pointing to the outer circle.
- Inner boundary label with leader line: "a = 0.050 m" pointing to the inner circle.

Material regions and labels:
- The annular region between the two circles (a < r < R) is filled with uniform light-gray shading (or evenly spaced hatch marks) and labeled inside the annulus with: "insulating material" and on the next line "ρ = +6.0×10^−6 C/m^3".
- The central region inside the smaller circle (r < a) is filled with a distinct pattern from the annulus (for example, a different hatch direction or dotted fill) to indicate a different material.
- Inside the cavity region, place the text label "linear dielectric" and directly below it "dielectric constant κ".

Radial coordinate indication (same as Figure 1):
- Draw a radius line from the origin to the right, ending at the outer circle and labeled "r".
- Place a tick mark where the radius crosses the inner circle labeled "r = a".
- Place a tick mark at the outer circle labeled "r = R".

Styling constraints:
- Outer circle outline slightly thicker than inner circle outline.
- No 3D perspective, no extra objects, no field lines, no grid.
- Only the labels specified above appear as text.

💡AP Physics C: E&M

Practice FRQ 1 of 201/20

1. An insulating sphere of radius

R = 0.20\ \text{m}

is centered at the origin. The sphere has a spherical cavity of radius

a = 0.050\ \text{m}

that is concentric with the sphere, as shown in Figure 1. The insulating material (the region

a < r < R

) contains a uniform volume charge density

\rho = +6.0\times10^{-6}\ \text{C/m}^3

. The cavity (the region

r < a

) is empty. Ignore edge effects and assume electrostatic equilibrium.

Figure 1. Insulating sphere with concentric spherical cavity and uniform volume charge density in the insulating material.

Figure 2. Axes for graphing the electric field magnitude E as a function of radial distance r.

i. 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 each of the three regions

0 ≤ r < a

a < r < R

, and

r > R

. Express your answers in terms of

\rho

a

R

r

, and physical constants, as appropriate.

ii. Derive an expression for the potential difference

\Delta V = V(a) - V(R)

. Begin your derivation by writing a fundamental physics principle or an equation from the reference information, and use your result from part A(i). Express your answer in terms of

\rho

a

R

, and physical constants, as appropriate.

iii. On the axes shown in Figure 2, sketch a graph of the magnitude of the electric field

E

as a function of

r

from

r = 0

to a position that is outside the sphere (past

r = R

). Clearly indicate any changes in functional form at

r = a

and

r = R

A linear dielectric material with dielectric constant

\kappa = 4.0

is inserted so that it completely fills the cavity region

0 ≤ r < a

, as shown in Figure 3. The charge density

\rho

in the insulating material remains uniform and unchanged.

B. Derive an expression for the electric flux

\Phi_E

through a spherical Gaussian surface of radius

r_g = 0.030\ \text{m}

(which lies entirely within the dielectric-filled cavity). Express your answer in terms of

\rho

a

r_g

\kappa

, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Figure 3. Same geometry as Figure 1, with the cavity filled by a linear dielectric of dielectric constant κ.