Distinguish potential cancellation from field cancellation

Establish the model and evidence

Electric potential V is energy per unit charge, measured in joules per coulomb or volts. With zero potential chosen at infinity, a point source contributes V=kQ/r. Add the signed scalar contributions from sources. Unlike field arrows, potential contributions do not require direction components. A negative potential is meaningful relative to the chosen reference, not an indication of negative distance.

Potential differences determine electric energy changes: ΔU=qΔV for a test charge whose presence does not rearrange the sources. Changing the reference adds a common constant to potentials but does not change ΔV, electric forces or observable energy changes. A positive charge tends to accelerate toward lower potential; a negative charge can accelerate toward higher potential while still losing potential energy.

The relationship between field and potential can be read without calculus. Along a short displacement parallel to a uniform field, ΔV=−EΔx. Potential falls in the field direction. For a nonuniform field, a local graph slope indicates the negative component of the field; one must not replace a curved profile by a single global uniform-field value. Equal-potential surfaces meet a nonzero field at right angles.

Equal opposite sources illustrate the crucial distinction. At their perpendicular bisector, scalar potentials cancel, yet their horizontal fields add from the positive source toward the negative source. Conversely, the midpoint between equal positive sources has zero field but a positive potential relative to infinity. Neither zero by itself proves the other. The activity reports field components beside the scalar potential profile to expose that confusion.

Worked evidence

With +2 nC at (−0.20,0) m and −2 nC at (+0.20,0) m, the point (0,0.15) m is 0.25 m from both. Potentials +71.92 and −71.92 V cancel. Horizontal field components each equal 230.144 N/C, giving Ex=460.288 N/C and Ey=0.

Predict and investigate

At the midpoint profile position for equal opposite sources, does zero potential mean a positive probe has no acceleration?

Capture right charge −2 nC at x=−0.05, 0 and +0.05 m. Use the potential trend to predict the x field direction, then compare the component ledger. Change the right charge to +2 and identify which cancellation changes.

Transfer to changed conditions

A uniform field points in +x with magnitude 300 N/C. Compare potentials at x=0 and x=0.040 m, and find the energy change of a −3 nC probe moved between them.

Compare the worked transfer

ΔV=−300×0.040=−12 V. ΔU=(−3×10⁻⁹)(−12)=+36 nJ. The electric force does −36 nJ of work during that imposed displacement; a quasistatic external agent supplies +36 nJ.

Explain the boundary

Explain why an equipotential displacement has zero electric work, even when the local field is not zero.

Translate a potential map into a field map

Predict, draw and check: three straight vertical equipotential lines lie at x=0, 0.020 and 0.040 m, labeled 12, 8 and 4 V. Draw equal horizontal field arrows toward +x, perpendicular to the lines. The field is 200 V/m under the stipulated uniform interpolation, with no y component. A +2 nC probe moved from the first to last line changes electric potential energy by −16 nJ; moving along one line changes it by zero. Reverse the voltage labels and reverse the field, not the line geometry.

A potential difference can also be maintained by chemical charge separation in a source, rather than by merely placing a point source nearby. The energy per charge of that process belongs in the circuit source account. Field arrows point toward decreasing potential, but a negative charge’s force points opposite those arrows. (Current clauses 10.5.A.3.i and 10.5.B.)

Sources and authorship

Physical relationship reference. Current AP Physics 2 framework, checked 30 October 2026. These cases, diagrams and numerical records are original. All experiments are bounded browser calculations or synthetic data, not measured laboratory evidence. Do not construct or handle charged capacitors, high-voltage sources or particle-beam equipment for these activities.

Clause-level source/action reconciliation checked 31 October 2026 against the linked current framework and Fall 2026 clarification. Added packets are original virtual comparisons; no new practical or learner-completion claim.