Equal-potential contours reveal field direction and strength
You will be able to: Read equipotential spacing, estimate a field from measured voltage differences, and interpret an optional potential surface.
How can a voltage map reveal a field without drawing source charges?
A map with 2 V between neighboring contours can show weak or strong fields. If those contours are 1 m apart, the normal field is about 2 N/C. If they are only 0.25 m apart, it is about 8 N/C.
A useful starting point: Field is negative slope, not potential height →
Words and symbols before equations
- Equipotential contour
- A curve connecting locations with equal V.
- Normal direction
- Perpendicular to a contour; the field points toward decreasing V.
- Finite-difference estimate
- Approximation E_n ≈ −ΔV/Δn for a small normal spacing.
- Potential surface
- A graph whose horizontal coordinates are positions and whose height represents volts, not physical elevation.
What this picture assumes
Local example V = C + g(x²+y²); E = (−2gx,−2gy). Contour increments are 2 V. The optional 3D graph uses height for V−C, with a labeled vertical voltage scale; it is not physical terrain or a trajectory. Camera rotation changes only the projection.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Probe (1,0) m: V=2 V, E=(-4,0) N/C, magnitude 4 N/C. The circle center has E=0.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
Along an equipotential contour, a small tangent displacement gives dV = 0, so the field has no tangential component. Where E is nonzero it is perpendicular to the contour and points toward smaller V.
For equal voltage increments, closer contour spacing indicates a larger field. Estimate the normal component with −ΔV/Δn. Unequal voltage increments must be accounted for before comparing spacing.
The model uses V(x,y) = C + g(x² + y²), a local example with circular contours. E = (−2gx, −2gy). Its optional 3D graph plots V as height. It is not a physical bowl, a particle path, or a full infinite-space source model. A voltage-mapping experiment can sample positions, draw equal-value contours and repeat readings to assess uncertainty.
A worked example, step by step
Measured potentials at x = 0.90 m and 1.10 m along y = 0 are 1.62 V and 2.42 V. Estimate Eₓ at x = 1.00 m.
- Use a centered difference with Δx = 1.10 − 0.90 = 0.20 m.
- ΔV = 2.42 − 1.62 = +0.80 V.
- Eₓ ≈ −0.80/0.20 = −4.0 N/C.
- The field points toward lower V, leftward. Real data would require position and voltage uncertainty; these example values are synthetic.
Compare contour spacing only for equal voltage steps. Graph height represents potential, not geometric altitude.
Is electric work zero for a move along one equipotential?
Compare with an explanation
Yes for a fixed charge: W_e = −qΔV = 0, though the force may be nonzero and normal to the path.
Predict. Change one thing. Explain.
Change g and track both voltage labels and the field at the probe. Rotate the optional 3D graph without changing the probe or physical field. Interpret contour spacing using the labeled voltage increments.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Probe (1,0) m: V=2 V, E=(-4,0) N/C, magnitude 4 N/C. The circle center has E=0.
Optional 3D view: potential as graph height
Height represents V − C. The graph is not physical terrain. Rotate the camera without changing E.
Local example V = C + g(x²+y²); E = (−2gx,−2gy). Contour increments are 2 V. The optional 3D graph uses height for V−C, with a labeled vertical voltage scale; it is not physical terrain or a trajectory. Camera rotation changes only the projection.
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 questionDesign a voltage-mapping analysis using measured positions and voltages. Explain how to estimate field direction and strength, and name one uncertainty to evaluate.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Record voltage relative to a fixed reference at a grid of known positions.
- 1 point: Draw or interpolate equal-potential contours and label their values.
- 1 point: Estimate normal field as −ΔV/Δn; point toward decreasing V.
- 1 point: Repeat voltage readings and assess probe-position/spacing uncertainty; label interpolated results as estimates.
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 1What does closer spacing mean for equal voltage steps?
A stronger normal electric field.
RECALL 2Why label every contour?
Unequal voltage steps can make spacing comparisons misleading.
RECALL 3Does camera rotation change E?
No. It only changes the projected potential graph.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Equal-potential contours reveal field direction and strength
- E is normal to equipotentials where E ≠ 0.
- E_n ≈ −ΔV/Δn for small normal separation.
- For V = C + g(x²+y²), E = (−2gx, −2gy).
Remember: Compare contour spacing only for equal voltage steps. Graph height represents potential, not geometric altitude.
Conditions: Local example V = C + g(x²+y²); E = (−2gx,−2gy). Contour increments are 2 V. The optional 3D graph uses height for V−C, with a labeled vertical voltage scale; it is not physical terrain or a trajectory. Camera rotation changes only the projection.
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.
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