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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 201/20

1. An insulating solid sphere of radius R=0.120 mR = 0.120\ \text{m}R=0.120 m is centered at the origin. The material has a uniform volume charge density ρ=+6.00×10−6 C/m3\rho = +6.00× 10^{-6}\ \text{C/m}^3ρ=+6.00×10−6 C/m3 everywhere within the sphere except for a spherical cavity of radius a=0.0400 ma = 0.0400\ \text{m}a=0.0400 m centered at the origin (the cavity is empty space). The region outside the sphere is filled with a linear dielectric medium with relative permittivity κ=2.50\kappa = 2.50κ=2.50, as shown in Figure 1. The electric permittivity of free space is ε0\varepsilon_0ε0​ and the gravitational constant is GGG. Ignore polarization effects inside the charged insulator (treat the given ρ\rhoρ as the free charge density in the material).

Figure 1. Cross-section of a charged insulating sphere with a central spherical cavity, surrounded by a dielectric medium.

Figure 1

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

Figure 2
A.
i.

Using Gauss's law, derive an expression for the magnitude E(r)E(r)E(r) of the electric field for the region a<r<Ra < r < Ra<r<R. Express your answer in terms of ρ\rhoρ, aaa, rrr, and physical constants, as appropriate.

ii.

Derive an expression for the magnitude E(r)E(r)E(r) of the electric field for the region r>Rr > Rr>R. Your expression must explicitly include the effect of the dielectric medium with relative permittivity κ\kappaκ. Express your answer in terms of ρ\rhoρ, aaa, RRR, rrr, κ\kappaκ, and physical constants, as appropriate.

iii.

On the axes shown in Figure 2, sketch a graph of EEE as a function of rrr from r=0r = 0r=0 to a position that is outside the sphere. Clearly indicate and label the behavior in each region 0≤r<a0 ≤ r < a0≤r<a, a<r<Ra < r < Ra<r<R, and r>Rr > Rr>R.

Figure 3. A charged particle on the +x axis at r = 0.200 m from the center of the charged sphere in a dielectric medium.

Figure 3
B.

Derive an expression for the ratio FeFg\dfrac{F_e}{F_g}Fg​Fe​​ of the magnitude of the electric force on the particle to the magnitude of the gravitational force on the particle due to the charged sphere's mass. The sphere's (charged material) mass is M=2.00 kgM = 2.00\ \text{kg}M=2.00 kg. Express your answer in terms of the given quantities and physical constants. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. A particle with charge q=+2.00×10−9 Cq = +2.00× 10^{-9}\ \text{C}q=+2.00×10−9 C and mass m=3.00×10−6 kgm = 3.00× 10^{-6}\ \text{kg}m=3.00×10−6 kg is placed at rest on the +x axis at a distance r=0.200 mr = 0.200\ \text{m}r=0.200 m from the center of the sphere, as shown in Figure 3.

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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 carefullybefore 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.