Chapter 3 Problems
Original chapter problems, with explicitly stated assumptions and authored solutions. These retain the manuscript’s problem identities; calculus extensions and unfinished material make no AP coverage claim.
Original problem 1
An isolated spherical metal conductor of radius R carries total charge Q at electrostatic equilibrium. Find E and V inside and outside using V(∞)=0.
Authored clarification: No external fields are present. The sphere’s symmetry gives uniform surface charge.
Worked solution with reasons for each step · Original reading location
Archived original prompt
A charge Q is placed on a spherical metal conductor of radius R. Determine the electric field and potential outside and inside of the conductor.
No original chapter-problem answer was supplied. This worked solution is authored for the explicitly stated reviewed prompt.
Original problem 2
Optional original-source extension. Prerequisites: infinite-sheet superposition and conductor electrostatic equilibrium. This item makes no AP coverage claim.
A thin infinite insulating sheet at z=0 has uniform surface density σ. Two neutral, isolated conducting slabs are placed symmetrically, one on either side. Determine the field in the vacuum gaps adjacent to the insulating sheet.
Authored clarification: The source does not specify finite versus infinite geometry, grounding, net slab charge or sheet density. This symmetric infinite-sheet model fixes those conditions; a different electrical connection can change the answer.
Worked solution with reasons for each step · Original reading location
Archived original prompt
No original chapter-problem answer was supplied. This worked solution is authored for the explicitly stated reviewed prompt.
Original problem 3
Four identical 2 mF parallel-plate capacitor units are electrically combined in parallel. Find their total capacitance.
Authored clarification: Merely placing capacitors next to one another does not specify a circuit. Here their corresponding terminals are connected in parallel; different connections give different results.
Worked solution with reasons for each step · Original reading location
Archived original prompt
No original chapter-problem answer was supplied. This worked solution is authored for the explicitly stated reviewed prompt.
Original problem 4
Optional original-source extension. Prerequisites: the ideal parallel-plate model and the meaning of an unconstrained maximum. This item makes no AP coverage claim.
A capacitor must fit inside a cube of side L. Does this alone determine which geometry—parallel plate, coaxial cylinder or concentric spheres—has the greatest capacitance?
Authored clarification: No minimum electrode spacing, dielectric, conductor thickness or breakdown limit was supplied. The task is to diagnose whether a maximum is defined.
Worked solution with reasons for each step · Original reading location
Archived original prompt
No original chapter-problem answer was supplied. This worked solution is authored for the explicitly stated reviewed prompt.
Original problem 5
Optional original-source extension. Prerequisites: series and parallel capacitance for two distinct dielectric layouts. This item makes no AP coverage claim.
An air-filled parallel-plate capacitor has C₀=1 F. A dielectric K=4 fills (a) one quarter of plate area through the full gap, or (b) one quarter of gap thickness over the entire area. Find C for both, neglecting fringing.
Authored clarification: The original “fourth” is ambiguous. Both physically distinct layouts are solved explicitly.
Worked solution with reasons for each step · Original reading location
Archived original prompt
No original chapter-problem answer was supplied. This worked solution is authored for the explicitly stated reviewed prompt.
Original problem 6
A vacuum parallel-plate capacitor has plate area A, separation d and charges ±Q. Derive its electric energy density in terms of Q, A and ε₀, neglecting fringing.
Authored clarification: Energy density refers to the nearly uniform gap field, and A is one plate’s facing area.
Worked solution with reasons for each step · Original reading location
Archived original prompt
No original chapter-problem answer was supplied. This worked solution is authored for the explicitly stated reviewed prompt.
Original problem 7
Optional original-source extension. Prerequisites: long-wire magnetic fields and force on a current-carrying wire. This item makes no AP coverage claim.
Two long parallel wires separated by distance s carry equal maintained currents I in opposite directions. They remain electrically insulated. Describe their interaction and give the magnetic force per unit length.
Authored clarification: The original does not say whether wires are electrically joined or merely placed adjacent. This problem supplies two distinct insulated long wires; joining them would require a circuit specification.
Worked solution with reasons for each step · Original reading location
Archived original prompt
No original chapter-problem answer was supplied. This worked solution is authored for the explicitly stated reviewed prompt.
Original problem 8
At constant 20 V, a circuit’s total resistance increases from 1 Ω to 10 Ω. Find its initial and final Joule power and the final-to-initial ratio.
Authored clarification: The ideal source holds the voltage at 20 V while the total resistance changes. The reviewed wording asks for both powers and their ratio.
Worked solution with reasons for each step · Original reading location
Archived original prompt
No original chapter-problem answer was supplied. This worked solution is authored for the explicitly stated reviewed prompt.