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💡AP Physics C: Electricity and Magnetism
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💡AP Physics C: Electricity and Magnetism

FRQ 1 – Mathematical Routines
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Unit 8: Electric Charges, Fields, and Gauss's Law
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Practice FRQ 1 of 11/1

1. A solid insulating sphere of radius R=0.12 mR = 0.12\ \text{m}R=0.12 m is centered at the origin and has a uniform volume charge density ρ=+5.0×10−6 C/m3\rho = +5.0× 10^{-6}\ \text{C/m}^3ρ=+5.0×10−6 C/m3. The sphere is embedded in a homogeneous, linear dielectric material with permittivity ε=2.5 ε0\varepsilon = 2.5\,\varepsilon_0ε=2.5ε0​. A point particle of mass m=0.020 kgm = 0.020\ \text{kg}m=0.020 kg and charge q=−3.0×10−6 Cq = -3.0× 10^{-6}\ \text{C}q=−3.0×10−6 C is located on the positive xxx-axis at x=0.30 mx = 0.30\ \text{m}x=0.30 m, as shown in Figure 1. The sphere and the particle are stationary and the dielectric fills all space.

Figure 1. Uniformly charged insulating sphere at the origin and a point particle on the +x-axis; concentric spherical Gaussian surface of radius r.

Figure 1

Figure 2. Axes for a student sketch of electric-field magnitude E versus radial distance r from the center of the sphere.

Figure 2
A.
i.

Using Gauss’s law, derive an expression for the magnitude E(r)E(r)E(r) of the electric field produced by the uniformly charged insulating sphere for the region 0<r<R0 < r < R0<r<R. Express your answer in terms of ρ\rhoρ, ε\varepsilonε, rrr, and physical constants, as appropriate.

ii.

Derive an expression for the magnitude E(r)E(r)E(r) of the electric field produced by the sphere for the region r>Rr > Rr>R. Express your answer in terms of ρ\rhoρ, RRR, ε\varepsilonε, rrr, 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 the magnitude of the electric field produced by the sphere, EEE, as a function of rrr from r=0r = 0r=0 to r=0.36 mr = 0.36\ \text{m}r=0.36 m.

Figure 3. Gaussian surface of radius r = 0.20 m centered at the origin encloses the insulating sphere but not the point particle at x = 0.30 m.

Figure 3
B.

Derive an expression for the electric flux ΦE\Phi_EΦE​ through this Gaussian surface. Express your answer in terms of ρ\rhoρ, RRR, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. Consider the spherical Gaussian surface of radius rG=0.20 mr_G = 0.20\ \text{m}rG​=0.20 m centered at the origin, as shown in Figure 3. The point particle at x=0.30 mx = 0.30\ \text{m}x=0.30 m lies outside this Gaussian surface.

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Free Response Question Practice

This practice environment simulates the AP AP Physics C: Electricity and Magnetism 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."
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Tip: Answer all parts of each question. Partial credit is often available, so even if you are unsure, provide what you know.