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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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Practice FRQ 1 of 201/20

1. A solid insulating sphere of radius R=0.060 mR=0.060\ \text{m}R=0.060 m is centered at the origin and has a uniform volume charge density ρ=+4.0×10−6 C/m3\rho=+4.0× 10^{-6}\ \text{C/m}^3ρ=+4.0×10−6 C/m3. Concentric with the sphere is a thin conducting spherical shell with inner radius a=0.090 ma=0.090\ \text{m}a=0.090 m and outer radius b=0.110 mb=0.110\ \text{m}b=0.110 m. The conductor has a net charge Qshell=−5.0×10−9 CQ_{\text{shell}}=-5.0× 10^{-9}\ \text{C}Qshell​=−5.0×10−9 C. The region outside the shell is air. The gravitational field near Earth is g=9.8 m/s2g=9.8\ \text{m/s}^2g=9.8 m/s2. Figure 1 shows the configuration of this system.

Figure 1. Concentric charged insulating sphere and conducting spherical shell (cross-sectional view through the center).

Figure 1

Figure 2. Axes for graphing electric-field magnitude E versus 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 R<r<aR<r<aR<r<a. Express your answer in terms of ρ\rhoρ, RRR, rrr, and physical constants, as appropriate.

ii.

Derive expressions for the charge QinnerQ_{\text{inner}}Qinner​ on the inner surface of the conducting shell (at r=ar=ar=a) and the charge QouterQ_{\text{outer}}Qouter​ on the outer surface (at r=br=br=b). Express your answers in terms of ρ\rhoρ, RRR, QshellQ_{\text{shell}}Qshell​, 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 EEE as a function of radial distance rrr for 0≤r0≤ r0≤r to a position outside the shell. Clearly indicate and label the regions 0<r<R0<r<R0<r<R, R<r<aR<r<aR<r<a, a<r<ba<r<ba<r<b, and r>br>br>b.

Figure 3. Same geometry as Figure 1, with a dielectric filling the region between the insulating sphere and the conductor.

Figure 3
B.

Derive an expression for the magnitude Ediel(r)E_{\text{diel}}(r)Ediel​(r) of the electric field in the dielectric-filled region R<r<aR<r<aR<r<a. Express your answer in terms of ρ\rhoρ, RRR, rrr, κ\kappaκ, and physical constants, as appropriate. Begin your derivation by writing a fundamental physics principle or an equation from the reference information. A dielectric material of relative permittivity κ=3.5\kappa=3.5κ=3.5 completely fills the spherical region R<r<aR<r<aR<r<a, as shown in Figure 3. The insulating sphere (radius RRR and charge density ρ\rhoρ) and the conducting shell (net charge QshellQ_{\text{shell}}Qshell​) are unchanged.

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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:

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