A long line produces a one-over-distance field
You will be able to: Derive the infinite-line field by integration and state its approximation limits.
Why does a line charge’s field fall as 1/r rather than 1/r²?
Near the middle of a wire much longer than your distance from it, more distant sections also contribute. The combined field falls more slowly with distance than the field of one point charge.
A useful starting point: Integrate directions around a charged arc →
Words and symbols before equations
- Infinite line
- An ideal uniformly charged line extending without end; a local approximation for a sufficiently long real wire.
- Radial distance r
- Shortest distance from the line, not a distance measured along it.
- Source coordinate x
- Position along the wire measured from the nearest point.
- Linear density λ
- Charge per unit wire length in C/m.
What this picture assumes
Ideal infinitely long uniform thin line in vacuum; r > 0. Positive radial field points away from the line. A finite real wire only approximates this near its middle and away from its ends.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- λ = 3 nC/m; at r = 0.5 m, E_r = 108 N/C. Positive is away from the wire.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
At distance r from a line along x, paired elements at ±x cancel their along-wire components. The outward component is dE_r = kλr dx/(x² + r²)³ᐟ².
Integrate x from −∞ to +∞. The antiderivative is x/[r²√(x² + r²)], so E_r = 2kλ/r = λ/(2πε₀r). Positive λ gives an outward radial field and negative λ an inward field.
A truly infinite line has no finite total charge, so do not substitute a total Q into kQ/r². For a finite line, the 1/r approximation works near its middle when distance is small compared with length; far away a finite line returns to point-charge behavior. Gauss’s law will give the same ideal result later.
A worked example, step by step
An ideal line has λ = +3 nC/m. Find E at r = 0.50 m and at r = 1.0 m.
- Use the infinite-line formula because the stated model is uniform and unbounded.
- At 0.50 m, E_r = 2(9)(3)/0.50 = 108 N/C.
- At 1.0 m, E_r = 54 N/C.
- Doubling r halves the field. This differs from the quarter-strength result for a point charge.
An infinite line has no finite total Q. Distinguish its 1/r field from a finite rod’s distant 1/r² field.
Would the exact 1/r law describe a short wire from very far away?
Compare with an explanation
No. Its finite total charge then behaves approximately as a point source.
Predict. Change one thing. Explain.
Double r and predict the field ratio. Compare with the finite-rod lesson to identify the role of source geometry.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
λ = 3 nC/m; at r = 0.5 m, E_r = 108 N/C. Positive is away from the wire.
Ideal infinitely long uniform thin line in vacuum; r > 0. Positive radial field points away from the line. A finite real wire only approximates this near its middle and away from its ends.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant charge, vector superposition, electric field, flux or symmetry 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 line gives 72 N/C outward at r = 0.25 m. Find λ, then predict E at r = 0.75 m.
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Compare with the answer and four-point rubric
- 1 point: Use λ = E_r r/(2k).
- 1 point: λ = 72(0.25)/(18 × 10⁹) = +1 nC/m.
- 1 point: The new distance is three times larger.
- 1 point: E = 24 N/C outward at 0.75 m.
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 1Which components cancel in the integration?
Those parallel to the line.
RECALL 2Why is the field radial?
Symmetry removes any preferred direction along or around the line.
RECALL 3What happens at the ideal line itself?
The thin-line model is singular; it does not describe a finite-radius interior.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
A long line produces a one-over-distance field
- E_r = ∫₋∞∞ kλr dx/(x² + r²)³ᐟ².
- E_r = 2kλ/r = λ/(2πε₀r).
- Valid for r > 0; an ideal infinitely thin line is singular on its axis.
Remember: An infinite line has no finite total Q. Distinguish its 1/r field from a finite rod’s distant 1/r² field.
Conditions: Ideal infinitely long uniform thin line in vacuum; r > 0. Positive radial field points away from the line. A finite real wire only approximates this near its middle and away from its ends.
Refresh Kid · AP Physics C: Electricity and Magnetism Unit 1 (official Unit 8) · Objectives 8.4.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 8.4, objectives 8.4.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026. This is E&M Unit 1: Electric Charges, Fields, and Gauss’s Law, numbered Unit 8 in the official combined Physics C sequence. Topics 8.1–8.6 retain their official identifiers. Quantitative force examples use at most four point charges. Field integrals use the specified rods, ring, arc and infinite wire; Gauss-law field calculations use spherical, cylindrical or planar symmetry. Optional projected 3D views clarify area normals and geometry; camera rotation never changes the physics. 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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