Read field direction from a potential map
You will be able to: Relate potential differences, distances and field components with correct signs.
Why are field arrows perpendicular to equipotential lines?
Along a straight horizontal strip, potential falls from 100 V to 0 V over 0.20 m. A positive charge is pushed toward lower potential. Equal-voltage lines run perpendicular to that horizontal electric field.
A useful starting point: Volts: energy per unit charge →
Words and symbols before equations
- Equipotential
- A line or surface on which V is constant.
- Displacement Δx
- Signed change in position along a chosen axis.
- Uniform-field component
- E_x=−ΔV/Δx when potential varies linearly along x.
- Field estimate
- For nonuniform fields, −ΔV/Δx estimates the average component over a short interval, not the full vector magnitude along any path.
What this picture assumes
Uniform horizontal field over 0.20 m. Left V=100 V; right V=100 V+ΔV. Graph uses actual signed potential slope. E_x=−ΔV/0.20; the field direction is perpendicular to vertical equipotential lines when E is nonzero.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- ΔV=-80 V across +0.20 m. E_x=400 V/m rightward. Equipotential lines are vertical for a nonzero uniform horizontal field; the plotted line is V versus x, not an equipotential.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
Field points toward decreasing potential. With +x to the right, a negative voltage slope gives positive E_x. In a uniform field, the magnitude along the field direction is |ΔV|/d.
Moving along an equipotential gives ΔV=0 and electric work −qΔV=0. Where E is nonzero it must be perpendicular to the equipotential; a tangential component would imply a voltage change along it.
For equal voltage increments, closer equipotentials indicate a larger perpendicular field. To map a field experimentally, measure V relative to one fixed reference at many known positions, draw equal-V contours, then estimate perpendicular voltage gradients. Probe spacing and measurement noise limit precision.
A worked example, step by step
V falls from 100 V at x=0 to 20 V at x=0.20 m, linearly. Find E_x and work on +2 μC moving between those points.
- ΔV=20−100=−80 V and Δx=+0.20 m.
- E_x=−(−80)/0.20=+400 V/m, rightward.
- ΔU=qΔV=(2×10⁻⁶)(−80)=−1.6×10⁻⁴ J.
- Electric work=+1.6×10⁻⁴ J.
Voltage slope gives a signed field component. |ΔV|/distance along an arbitrary path is not generally the full field magnitude.
Can electric work be zero along a path in a region where E is nonzero?
Compare with an explanation
Yes. Motion along an equipotential is perpendicular to E and gives no electric work.
Predict. Change one thing. Explain.
Change the voltage difference over a fixed 0.20 m gap. Compare the potential graph’s slope and field readout. Reverse the slope and predict the arrow direction.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
ΔV=-80 V across +0.20 m. E_x=400 V/m rightward. Equipotential lines are vertical for a nonzero uniform horizontal field; the plotted line is V versus x, not an equipotential.
Uniform horizontal field over 0.20 m. Left V=100 V; right V=100 V+ΔV. Graph uses actual signed potential slope. E_x=−ΔV/0.20; the field direction is perpendicular to vertical equipotential lines when E is nonzero.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant charge, field or energy 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 questionA map has 0, 20 and 40 V contours spaced 0.05 m apart, increasing toward the right. (a) Estimate E_x. (b) State direction. (c) Find electric work for +1 μC moving along the 20 V contour. (d) Name an experimental uncertainty.
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Compare with the answer and four-point rubric
- 1 point: E_x=−20/0.05=−400 V/m for this uniform spacing.
- 1 point: Leftward, toward lower potential.
- 1 point: Zero because ΔV=0.
- 1 point: Voltage noise, probe position uncertainty or insufficiently close spatial sampling.
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 1Direction of E relative to V?
Toward decreasing potential.
RECALL 2Field versus equipotential?
Perpendicular wherever the field is nonzero.
RECALL 3What does −ΔV/Δx measure?
The average field component along x; it equals the local component for a uniform field.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Read field direction from a potential map
- Uniform field along x: E_x=−ΔV/Δx.
- Equipotential displacement: ΔV=0 and W_electric=0.
- Equal voltage steps closer together imply stronger perpendicular field.
Remember: Voltage slope gives a signed field component. |ΔV|/distance along an arbitrary path is not generally the full field magnitude.
Conditions: Uniform horizontal field over 0.20 m. Left V=100 V; right V=100 V+ΔV. Graph uses actual signed potential slope. E_x=−ΔV/0.20; the field direction is perpendicular to vertical equipotential lines when E is nonzero.
Refresh Kid · AP Physics 2 Unit 2 (official Unit 10) · Objectives 10.5.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 10.5, objectives 10.5.B. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. Refresh Kid calls this the second AP Physics 2 unit; College Board numbers it Unit 10; the first unit in this course is official Unit 9. 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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