Signed field area gives the potential change
You will be able to: Integrate a signed field and include the initial potential.
How do we find voltage when the field varies with position?
A field can push strongly at first and weakly later. A voltage change depends on all those contributions, not just the field at either endpoint. On an E-versus-x graph, add the signed area and reverse its sign.
A useful starting point: The electric work depends on endpoints, not detours →
Words and symbols before equations
- Eₓ(x)
- The signed field component along x, measured in N/C = V/m.
- dx
- A small signed displacement along the path.
- Initial potential V₀
- The specified potential at x = 0; needed to obtain V(x), not just ΔV.
- Signed area
- Area above the E = 0 axis is positive; area below is negative.
What this picture assumes
Prescribed one-dimensional electrostatic field Eₓ = a + bx on 0 ≤ x ≤ 4 m. V(x) = V₀ − ax − bx²/2. Negative field areas retain their sign; the field graph and potential graph have different units.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- To x=3 m, signed field area = 3 V; ΔV = -3 V; V = 7 V. Local Eₓ = -2 N/C.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
For Eₓ(x) = a + bx, integrate from 0 to x: ΔV = −∫₀ˣ(a + bs)ds = −ax − ½bx². The source-coordinate letter s prevents confusion with the final x limit.
Then V(x) = V₀ − ax − ½bx². The slope of V at any point is −Eₓ there. Where E crosses zero, V has a horizontal tangent; a sign change in E can make V turn around.
If the positive and negative field areas cancel over an interval, the endpoints have equal potential even if the field between them is nonzero. Include units: field times distance is (N/C)m = J/C = V.
A worked example, step by step
Eₓ = 4 − 2x N/C with x in meters. V(0) = 10 V. Find V(3 m).
- Use V(3) − V(0) = −∫₀³(4 − 2x) dx.
- The signed field area is [4x − x²]₀³ = 12 − 9 = 3 V.
- ΔV = −3 V, so V(3) = 7 V.
- E crosses zero at x = 2 m; the negative area from 2 to 3 m partly cancels the earlier positive area.
The field graph’s height is E, its signed area is −ΔV, and an initial V is needed for an absolute potential value.
If net signed field area is zero, must E be zero everywhere?
Compare with an explanation
No. Positive and negative contributions can cancel.
Predict. Change one thing. Explain.
Move the upper integration limit past the field’s zero crossing. Predict whether V is decreasing or increasing from the local field sign before reading the value.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
To x=3 m, signed field area = 3 V; ΔV = -3 V; V = 7 V. Local Eₓ = -2 N/C.
Prescribed one-dimensional electrostatic field Eₓ = a + bx on 0 ≤ x ≤ 4 m. V(x) = V₀ − ax − bx²/2. Negative field areas retain their sign; the field graph and potential graph have different units.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant potential, energy, work, field-gradient or line-integral relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionEₓ = 6 − 2x N/C and V(0) = 5 V. Find V(2 m) and Eₓ at 2 m. Interpret the potential slope there.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: ΔV = −[6x − x²]₀² = −8 V.
- 1 point: V(2) = 5 − 8 = −3 V.
- 1 point: Eₓ(2) = 2 N/C.
- 1 point: dV/dx = −2 V/m there, so potential is locally decreasing with x.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Why keep the negative sign in the integral?
Potential decreases in the electric-field direction.
RECALL 2What supplies the integration constant?
A known potential at one reference point.
RECALL 3Can equal endpoint potentials hide a nonzero field between?
Yes. Signed field areas may cancel.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Signed field area gives the potential change
- V_B − V_A = −∫_A^B E·dℓ.
- For Eₓ = a + bx: V(x) = V₀ − ax − ½bx².
- Eₓ = −dV/dx.
Remember: The field graph’s height is E, its signed area is −ΔV, and an initial V is needed for an absolute potential value.
Conditions: Prescribed one-dimensional electrostatic field Eₓ = a + bx on 0 ≤ x ≤ 4 m. V(x) = V₀ − ax − bx²/2. Negative field areas retain their sign; the field graph and potential graph have different units.
Refresh Kid · AP Physics C: Electricity and Magnetism Unit 2 (official Unit 9) · Objectives 9.2.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 9.2, objectives 9.2.B. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. This is E&M Unit 2: Electric Potential, numbered Unit 9 in the official combined Physics C sequence. Topics 9.1–9.3 retain their official identifiers. Models assume electrostatic fields and state whether source charges are fixed or free. Potential integrals use the specified rods, ring, arc and infinite line or cylinder. Infinite-line examples use a finite reference radius, not zero potential at infinity. The optional 3D equipotential surface uses height to represent volts, not a particle trajectory. Checked with the Fall 2026 clarifications. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
Want to work through this with a tutor?
Bring your question about Signed field area gives the potential change. Your explanation and answers remain free to access.