Chapter 2 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
Four point charges each equal to −Q, Q>0, are at (±a,±a), with a=1 m. Find the potential at the origin using V(∞)=0.
Authored clarification: The original says “negative charges of charge q”; Q here is the positive magnitude, so each source is −Q.
Worked solution with reasons for each step · Original reading location
Archived original prompt
Four negative charges of charge q are placed at the positions (-1,-1), (-1, 1), (1,-1), and (1,1). Find the electric potential at the point (0,0).
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: ideal infinite-sheet fields, signed potential differences and energy conservation. This item makes no AP coverage claim.
Infinite sheets at x=0 and x=d>0 have surface densities +σ and +2σ. Find V(d)−V(0) and the energy change for a test charge q moved from 0 to d. Which charge sign could accelerate from rest in that direction?
Authored clarification: σ>0 denotes surface density, not a finite total charge. The motion starts at the +σ sheet and ends at the +2σ sheet.
Worked solution with reasons for each step · Original reading location
Archived original prompt
Two infinite plates are placed near one another. One plate has a charge of 2σ the other has a charge of σ. What is the energy associated with moving a particle with charge q from the plate with charge +σ to the plate with charge +2σ? What type of charge would the particle need to move between the plate without outside help?
No original chapter-problem answer was supplied. This worked solution is authored for the explicitly stated reviewed prompt.
Original problem 3
Optional original-source extension. Prerequisites: scalar potential superposition for ideal thin rings and a reference at infinity. This item makes no AP coverage claim.
Two thin uniformly charged rings of radius R share a center and lie in perpendicular planes. Their total charges are Q₁ and Q₂. Find the potential at the common center with V(∞)=0.
Authored clarification: The original does not give the ring charges; Q₁ and Q₂ are explicit independent parameters.
Worked solution with reasons for each step · Original reading location
Archived original prompt
Two charged ring of radius z are placed so that they are perpendicular and their centers coincide. What is the electric potential at the center of this setup?
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: continuous surface charge, a definite integral and electric potential energy. This item makes no AP coverage claim.
A thin disk of radius R and uniform total charge Q lies in the xy plane. Find V(h) on its axis for h≥0 with V(∞)=0, and ΔU for a test charge qₜ moved quasistatically from the center just above the disk to height h.
Authored clarification: The disk charge Q and moving test charge qₜ are distinct. The initial location and potential reference are now specified.
Worked solution with reasons for each step · Original reading location
Archived original prompt
A disk of radius z with a uniformly distributed charge q sits on the ground. What is the electric potential and electric potential energy associated with moving a charge a height h above its center? What do you expect to see when h>>z?
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: an ideal infinite-line field, a radial potential integral and logarithms. This item makes no AP coverage claim.
An infinitely long wire has uniform line charge density λ. Determine V(d) with the finite reference V(r₀)=0.
Authored clarification: The reference radius r₀>0 and line density λ are essential; an ideal infinite line does not admit V(∞)=0.
Worked solution with reasons for each step · Original reading location
Archived original prompt
Find the potential a distance d away from a charged infinitely long wire.
No original chapter-problem answer was supplied. This worked solution is authored for the explicitly stated reviewed prompt.
Original problem 6
Define an equipotential and describe the equipotential surfaces of an isolated point charge Q using V(∞)=0.
Authored clarification: Treat Q as a nonzero isolated point charge in three dimensions. Describe the full surfaces and a cross section in a plane through the charge.
Worked solution with reasons for each step · Original reading location
Archived original prompt
Define what an equipotential is and then the draw equipotentials for a point charge.
No original chapter-problem answer was supplied. This worked solution is authored for the explicitly stated reviewed prompt.
Original problem 7
Explain the relation between equipotentials and electric field lines. For a negative point charge −Q, Q>0, does potential increase or decrease as radius increases?
Authored clarification: The source sentence ends unfinished. This explicitly completes the comparison using increasing radius and V(∞)=0.
Worked solution with reasons for each step · Original reading location
Archived original prompt
Explain how the equipotential relates to electric field lines. Do the equipotentials of a negative charge increase of
No original chapter-problem answer was supplied. This worked solution is authored for the explicitly stated reviewed prompt.
Original problem 8
Optional original-source extension. Prerequisites: partial derivatives and the potential-gradient relation. This item makes no AP coverage claim.
A potential is . Find the electric field.
Authored clarification: This transcribes the original formula image with coefficients a and b to make its units explicit; the source image corresponds to a=b=1 in its chosen units.
Worked solution with reasons for each step · Original reading location
Archived original prompt
The potential of some electric field is defined by the equation
Find the electric field associated with this potential.
No original chapter-problem answer was supplied. This worked solution is authored for the explicitly stated reviewed prompt.