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💡AP Physics C: E&M
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💡AP Physics C: E&M

FRQ 1 – Mathematical Routines
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Unit 8: Electric Charges, Fields, and Gauss's Law
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This mapping reflects College Board's exam structure - each FRQ type tests specific skills that are taught in particular units.

Practice FRQ 1 of 201/20
1. A nonconducting (insulating) solid sphere of radius R=0.20 mR = 0.20\ \text{m}R=0.20 m is centered at the origin. The sphere contains a spherical cavity of radius a=0.050 ma = 0.050\ \text{m}a=0.050 m whose center is located on the +x+x+x-axis at x=d=0.080 mx = d = 0.080\ \text{m}x=d=0.080 m. The insulating material (everywhere outside the cavity) has a uniform volume charge density ρ=+6.0×10−6 C/m3\rho = +6.0× 10^{-6}\ \text{C/m}^3ρ=+6.0×10−6 C/m3. The sphere is in vacuum (permittivity ε0\varepsilon_0ε0​), as shown in Figure 1.

Figure 2. Axes for sketching the magnitude of electric field E versus radial distance r for a uniformly charged insulating sphere.

A blank set of 2D graph axes only (no curve pre-drawn). Axes formatting (must be exact): - Horizontal axis: labeled exactly "r (m)". - Horizontal axis range: from 0 to 0.40. - Horizontal tick marks: every 0.05. - Horizontal tick labels shown at each tick: 0, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35, 0.40. - Vertical axis: labeled exactly "E (N/C)". - Vertical axis range: from 0 to 2.0×10^5. - Vertical tick marks: every 0.5×10^5. - Vertical tick labels shown at each tick: 0, 0.5×10^5, 1.0×10^5, 1.5×10^5, 2.0×10^5. - The origin is labeled "0" at the intersection of the axes. - Put arrowheads on the positive ends of both axes. Required reference marking on the r-axis: - Directly above the tick labeled 0.20, add a small vertical dashed guide line (light) and label it exactly "R = 0.20 m". This is a reference marker, not a curve. No other markings: - No gridlines. - No title. - No curve drawn. (These axes are designed so the student can sketch: a straight-line increase from r=0 to r=R, then a decreasing 1/r^2-type curve for r>R; the axes themselves must remain blank except for labels/ticks and the R marker.)
A. Consider the special case in which the cavity is not present. That is, the entire sphere of radius R is filled with insulating material of uniform charge density ρ\rhoρ.
i. Using Gauss's law, derive an expression for the magnitude E(r)E(r)E(r) of the electric field as a function of the distance rrr from the center for the region 0<r<R0<r<R0<r<R. Express your answer in terms of ρ\rhoρ, rrr, and physical constants, as appropriate.
ii. Derive an expression for the electric potential difference ΔV=V(R)−V(0)\Delta V = V(R)-V(0)ΔV=V(R)−V(0) for the uniformly charged sphere described in part A. 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 the magnitude of the electric field EEE as a function of rrr from r=0r=0r=0 to a position that is outside the sphere r>Rr>Rr>R for the uniformly charged sphere described in part A.

Figure 1. Insulating sphere with an off-center spherical cavity (all dimensions and charge density shown).

Clean 3D-style cutaway diagram with a clearly indicated Cartesian coordinate system. Overall layout and coordinate axes: - Place a right-handed set of axes through the center of the large sphere. - Draw the +x-axis as a horizontal arrow pointing to the right, labeled "+x". Mark the origin at the sphere’s center with a solid dot labeled "O" and the text "origin". - Optionally include small +y and +z arrows (thin) to indicate 3D orientation, but the +x-axis must be dominant and clearly labeled. Large sphere (charged insulating material): - Draw a large sphere centered exactly on O. - Label the sphere radius with a leader line from O to the outer surface along the +x direction, labeled exactly: "R = 0.20 m". - The sphere material (the region that is not the cavity) is shaded lightly or filled with a faint color wash. - Place a text label inside the material region (not in the cavity) reading exactly: "uniform volume charge density ρ = +6.0×10^-6 C/m^3". Spherical cavity (vacuum) inside the sphere: - Draw a smaller spherical cavity completely inside the large sphere. - The cavity center lies on the +x-axis to the right of O. - Mark the cavity center with a small solid dot on the +x-axis and label that point "C". - Next to that dot, add the text label exactly: "x = d = 0.080 m" to indicate the center location measured from the origin along +x. - Draw a radius indicator for the cavity: a short line from C to the cavity surface (any direction, but keep it unobstructed), labeled exactly: "a = 0.050 m". - The cavity region is unshaded/white and explicitly labeled "cavity (vacuum)". Spatial relationships that must be visually obvious: - The cavity is offset toward the +x side (right side) of the large sphere, not centered. - The cavity boundary does not touch the outer boundary of the large sphere (clearly separated). - The cavity center C is between O and the outer surface along +x. Environment: - Outside the large sphere, add a small label near the exterior reading exactly: "surroundings: vacuum (ε0)". Line/label styling: - Sphere outlines: solid black, medium thickness. - Dimension leader lines: thin black lines with arrowheads at the measured endpoints. - All numeric values must appear exactly as written: R = 0.20 m, a = 0.050 m, d = 0.080 m, ρ = +6.0×10^-6 C/m^3.

Figure 3. Gaussian surfaces centered at the origin for r < R and r > R with the off-center cavity shown.

Two side-by-side subpanels of the same sphere-and-cavity geometry, each showing a different spherical Gaussian surface centered at the origin. Common geometry in BOTH subpanels: - Draw the same large insulating sphere centered at a marked origin point O. - Include a prominent +x-axis arrow pointing right, passing through O, labeled "+x". - Show the internal spherical cavity offset on the +x-axis. Mark the cavity center with a dot labeled "C" and the text label "d = 0.080 m" placed next to it. - Label the large sphere radius with the text "R = 0.20 m" placed near the outer boundary. - Label the cavity radius with the text "a = 0.050 m" placed near the cavity boundary. - Shade the insulating material region lightly and leave the cavity region unshaded/white. - Add the text label inside the material region: "ρ (uniform, positive)". Left subpanel (case 1): - Title text at the top of the left subpanel: "Case 1: r < R". - Draw a spherical Gaussian surface centered at O, shown as a dashed circle/sphere outline (dashed line) that lies entirely inside the large sphere. - The Gaussian surface must be smaller than the large sphere so it does not intersect the outer boundary. - Label this dashed surface with a leader line: "Gaussian surface, radius r". Right subpanel (case 2): - Title text at the top of the right subpanel: "Case 2: r > R". - Draw a spherical Gaussian surface centered at O, shown as a dashed circle/sphere outline (dashed line) that fully encloses the entire large sphere. - The dashed Gaussian surface must be larger than the large sphere so the sphere is completely inside it. - Label this dashed surface with a leader line: "Gaussian surface, radius r". Visual clarity requirements: - In both subpanels, the Gaussian surface is centered exactly at O (not at the cavity center). - The cavity remains in the same offset position toward +x in both subpanels. - Use consistent scaling between the two subpanels for the sphere and cavity; only the Gaussian surface size changes. Line styles: - Outer sphere boundary: solid black. - Cavity boundary: solid black. - Gaussian surface: dashed black line. - Axes: thin black arrows.
B. Using your results from part A and the principle of superposition, derive an expression for the electric field vector E⃗\vec{E}E at the center of the cavity (the point at x=dx=dx=d) as shown in Figure 3. Express your answer in terms of ρ\rhoρ, ddd, and physical constants, as appropriate. Your final answer must include direction. Now return to the original object, which includes the spherical cavity shown in Figure 1. A student models the charge distribution using superposition: a uniformly charged sphere of radius RRR with charge density +ρ+\rho+ρ centered at the origin, plus a uniformly charged sphere of radius aaa with charge density −ρ-\rho−ρ centered at x=dx=dx=d (representing the missing charge in the cavity).
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Free Response Question Practice

This practice environment simulates the AP AP Physics C: E&M Free Response Questions section. Here are some guidelines:

  • Read each question carefully before responding. Pay attention to command verbs like "identify," "explain," "analyze," or "evaluate."
  • Use the timer to practice time management. You can pause, restart, or hide the timer as needed.
  • Mark for Review if you want to come back to a question later.
  • Your responses are saved automatically as you type. You can also use the drawing tool for questions that require diagrams or graphs.
  • Use the toolbar for formatting options like bold, italic, subscript, and superscript.
  • Navigate between questions using the Previous and Next buttons at the bottom of the screen.

Tip: Answer all parts of each question. Partial credit is often available, so even if you are unsure, provide what you know.