AP Physics C: E&M FRQ Practice

💡AP Physics C: E&M

Practice FRQ 1 of 111/11

3. In Experiment 1, students are asked to determine the permittivity $\varepsilon$ of an unknown, uniform dielectric sheet by measuring how the electric field between two large, parallel conducting plates changes when the dielectric is inserted. The plates are mounted horizontally and separated by a distance $d = 6.0\,\text{mm}$ using insulating spacers. The students can charge the top plate and bottom plate so that the charges on the plates have equal magnitude and opposite sign. They have access to an electrometer (to measure charge), a high-voltage power supply, a voltmeter, a meterstick, and an electronic balance. The region between the plates is initially air (take $\varepsilon_0 = 8.85× 10^{-12}\,\text{F/m}$ ).

Figure 1. Parallel-plate capacitor apparatus for measuring dielectric permittivity with fixed plate separation d = 6.0 mm.

Black-and-white physics apparatus diagram with clean, thin lines and clear labels.

Overall layout and orientation:
- The diagram is landscape-oriented.
- The parallel plates are drawn as two long, perfectly horizontal conducting rectangles spanning most of the diagram width.
- The top plate is located in the upper third of the diagram; the bottom plate is located in the lower third, directly below the top plate, with a uniform gap between them.
- A small upward arrow labeled "up" is placed near the left margin to enforce that the plates are mounted horizontally (top plate above bottom plate).

Plates and charges:
- The upper conductor is labeled "Top conducting plate" centered above it.
- The lower conductor is labeled "Bottom conducting plate" centered below it.
- On the top plate, place several small plus signs distributed along its length to indicate positive charge; next to the plate, include the text "+Q".
- On the bottom plate, place several small minus signs distributed along its length to indicate negative charge; next to the plate, include the text "−Q".
- The +Q and −Q labels must be vertically aligned to emphasize equal magnitude and opposite sign.

Plate separation and spacers:
- Show exactly two insulating spacers, one near the left end of the plates and one near the right end.
- Each spacer is drawn as a vertical rectangular block bridging the gap between the plates (touching both plates).
- Each spacer is labeled "Insulating spacer" with a leader line.
- Between the plates, near the center of the gap, draw a vertical double-headed dimension arrow from the inner surface of the top plate straight down to the inner surface of the bottom plate.
- Next to this dimension arrow, place the text "d = 6.0 mm" exactly.

Region between plates (air and dielectric insertion):
- The region between the plates is shown as an open space labeled "air" centered in the gap.
- Also show a removable dielectric sheet drawn as a thin, perfectly rectangular slab that can occupy the entire region between the plates.
- Depict the dielectric sheet in a second, adjacent position to the right of the plates: draw the same rectangular slab outside the gap (to indicate it can be inserted/removed), labeled "Dielectric sheet (uniform)".
- Add a rightward arrow from the external slab toward the gap, labeled "insert".
- Additionally, show the dielectric sheet in the inserted position: within the gap, draw a rectangle that fills the full plate overlap area between the plates (touching the inner face of the top plate and the inner face of the bottom plate), and label it "Dielectric fully inserted".
- The inserted dielectric must be shown without changing the gap distance d (the spacer blocks remain the same size and the dimension arrow remains labeled d = 6.0 mm).

Measuring instruments and connections:
- Draw a voltmeter as a circle with a bold "V" inside, positioned to the left of the plates at mid-height between them.
- Connect the voltmeter to the plates with two leads:
- One lead goes from the voltmeter’s upper terminal to the left side of the top plate.
- The other lead goes from the voltmeter’s lower terminal to the left side of the bottom plate.
- Label the voltmeter "Voltmeter" directly beneath the circle, and place the label "measures potential difference V" beside it. The symbol "V" must appear as the measured quantity.

- Draw an electrometer as a rectangular box labeled "Electrometer" positioned to the right of the plates, slightly above the midline.
- From the electrometer, draw a single lead connecting to the right side of the top plate (clearly touching the top plate).
- Near that connection, place the text "measures charge magnitude Q" so that Q is explicitly associated with the electrometer measurement.

Power supply (from prompt instrumentation list):
- Draw a high-voltage power supply as a box labeled "High-voltage power supply" positioned below the bottom plate, centered horizontally.
- From the power supply, draw two leads:
- One lead goes to the right side of the top plate.
- One lead goes to the right side of the bottom plate.
- Place a small open switch symbol in series with one of the supply leads (on the lead to the top plate) to indicate controllable charging.

Style constraints:
- All labels are horizontal, printed text.
- All connecting wires are thin straight lines with right-angle corners (no curved wires).
- No numeric values appear except the required "d = 6.0 mm".
- The only symbols on the plates are "+Q" on the top plate and "−Q" on the bottom plate, with multiple + and − signs to reinforce charge distribution.

Figure 2. Blank Cartesian grid for student graph (axes labeled; no data plotted).

A blank square plotting grid occupying most of the figure, with bold x- and y-axes and lighter interior grid lines.

Axes and placement:
- The vertical axis is a bold line located along the left edge of the grid region.
- The horizontal axis is a bold line located along the bottom edge of the grid region.
- The axes intersect at the bottom-left corner of the grid, forming the origin.
- Both axes have arrowheads indicating positive direction: rightward arrowhead on the horizontal axis and upward arrowhead on the vertical axis.

Tick marks and numeric scales (explicit, evenly spaced):
- Horizontal axis has labeled tick marks at: 0, 10, 20, 30, 40, 50.
- The label "0" is printed at the origin.
- Each tick label is printed directly below its tick mark.
- Vertical axis has labeled tick marks at: 0, 50, 100, 150, 200.
- The label "0" is printed at the origin (shared with the x-axis origin label).
- Each tick label is printed directly left of its tick mark.

Axis titles (left blank for students’ chosen quantities, as the AP task requires choice in Part C):
- The y-axis title area shows a placeholder line of text centered along the left side, rotated vertically: "Vertical axis: ____________________".
- The x-axis title area shows a placeholder line of text centered below the axis: "Horizontal axis: ____________________".
- No units are pre-printed in the axis titles (students will add units).

Grid appearance:
- Light, evenly spaced grid lines form square cells.
- There are exactly five equal major intervals between 0 and 50 on the x-axis, and exactly four equal major intervals between 0 and 200 on the y-axis, matching the labeled ticks.

No plotted elements:
- No points, no best-fit line, and no handwritten markings are present.
- No graph title is present.

Describe a procedure for collecting data that would allow the students to use a graph to determine $\varepsilon$ of the dielectric sheet, including any steps necessary to reduce experimental uncertainty.

Describe how the collected data could be graphed and how that graph would be analyzed to determine $\varepsilon$ . Include how the electric field between the plates is related to measured quantities and how Gauss’s law applies.

Q (nC)	V_air (V)	V_dielectric (V)
10.0	34	15
20.0	69	31
30.0	106	48
40.0	137	62
50.0	173	77

In Experiment 2, the students use the same plate separation $d = 6.0\,\text{mm}$ and the same pair of plates. They collect data for several values of the measured plate charge magnitude $Q$ . For each value of $Q$ , they measure the potential difference with air between the plates, $V_{\text{air}}$ , and then they fully insert the unknown dielectric sheet and measure the potential difference again, $V_{\text{dielectric}}$ , while keeping $Q$ unchanged. The data are shown in Table 1.

Indicate two quantities, either measured quantities from Table 1 or additional calculated quantities, that could be graphed to produce a straight line that could be used to determine $\varepsilon$ .

Vertical axis: Horizontal axis:

ii.

On the grid provided, create a graph of the quantities indicated in part C(i).

•

Use Table 2 to record the measured or calculated quantities that you will plot.

•

Clearly label the axes, including units as appropriate.

•

Plot the points you recorded in Table 2.

iii.

Draw a best-fit line for the data graphed in part C(ii).

Using the best-fit line that you drew in part C(iii), calculate an experimental value for the permittivity $\varepsilon$ of the dielectric sheet. Assume the plates are large enough that fringing is negligible, so the field between the plates is uniform. The plate separation is $d = 6.0\,\text{mm}$ . The plate area is $A = 0.020\,\text{m}^2$ . Use $\varepsilon_0 = 8.85× 10^{-12}\,\text{F/m}$ . Use your best-fit line from part C(iii) to determine the slope $m$ needed for the calculation.