2025 AP Physics C: Electricity and Magnetism Free-Response Questions

Question 1: Version J 1. An isolated, air-filled, charged capacitor consists of two conducting, coaxial, cylindrical shells that each have length L. The inner shell has radius R₁ and the outer shell has radius R₂, as shown in Figure 1, where R₁ ≪ R₂ ≪ L. The surface charge densities (amounts of charge per unit area) of the inner and outer shells are +σ₁ and −σ₂, respectively. The absolute values of the total charges on the shells are equal. Figure 1: Side View and Cross-Sectional View of the cylindrical shells, labeled +σ₁, −σ₂, L, R₁, and R₂. Note: Figures not drawn to scale. A. i. Using Gauss’s law, derive an expression for the magnitude E of the electric field as a function of the radial distance r from the center of the capacitor for the region R₁ < r < R₂. Express your answer in terms of R₁, σ₁, r, and physical constants, as appropriate. ii. Derive an expression for the absolute value |ΔV| of the potential difference between the outer and inner shells in terms of R₁, R₂, σ₁, 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 shell. Figure 2: Axes showing E as a function of r, with marked positions R₁ and R₂. B. A material of dielectric constant κ is inserted into the isolated, charged capacitor such that the material fills the region R₁ < r < R₂, as shown in Figure 3. Figure 3: Cross-Sectional View showing the dielectric material labeled κ filling the region R₁ < r < R₂, with radii R₁ and R₂. Derive an expression for the capacitance C of the capacitor with the material inserted in terms of L, R₁, R₂, κ, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.