Build a rod’s field by adding tiny charges
You will be able to: Set up and evaluate an axial line-charge integral with explicit limits.
How do we calculate a field when charge is spread along a rod?
Imagine a straight charged strip as many small pieces. At a point beyond its end, the nearer pieces produce stronger fields than the farther pieces. Treating all charge as if it were at one location is not exact nearby.
A useful starting point: Charge mobility changes the field inside matter →
Words and symbols before equations
- Linear density λ
- Charge per length, in C/m; uniform λ gives total Q = λL.
- dq
- A small charge element; for a rod dq = λ dx.
- Integration variable x
- A coordinate that moves along the source; distinct from the fixed observation point.
- Definite integral
- The limit of adding small contributions over stated bounds.
What this picture assumes
Uniform positive rod on x = 0 to L, observation point x = −a. Midpoint elements approximate a continuous line. Exact field is leftward. All graph distances are in meters; a is strictly positive.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Q = 1 nC; exact Eₓ = -4.5 N/C; 10-piece midpoint sum -4.493 N/C; difference 0.006537 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
Place a positive uniform rod on x = 0 to L and the observation point at x = −a, where a > 0. A piece at x is a distance a + x away, and its field at the observation point points left.
Its signed contribution is dEₓ = −kλ dx/(a + x)². Add from x = 0 to L. Since ∫(a + x)⁻² dx = −1/(a + x), the result is Eₓ = −kλ[1/a − 1/(a + L)].
Far away, a much larger than L, this approaches −kQ/a². For a nonuniform rod, retain λ(x) inside the integral; Q = ∫λ(x) dx. The model’s finite midpoint sum illustrates convergence to the continuous answer.
A worked example, step by step
A 1 m rod carries uniform λ = +1 nC/m. Find the field 1 m to the left of its near end.
- Use x ∈ [0, 1] m and a = 1 m; every contribution points −x.
- Eₓ = −kλ ∫₀¹ dx/(1 + x)².
- Eₓ = −9[1/1 − 1/2] = −4.5 N/C.
- The field points left. Putting all Q at the near end would wrongly give 9 N/C.
The distance to each element varies. Keep source coordinate x separate from the fixed gap a.
Why not use E = kQ/r² at the rod’s center for every distance?
Compare with an explanation
The element distances are different; a point-charge approximation is accurate only sufficiently far away.
Predict. Change one thing. Explain.
Change the number of small pieces while keeping the physical rod fixed. Compare the midpoint sum with the exact result; changing numerical resolution should converge, not change the physical answer.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Q = 1 nC; exact Eₓ = -4.5 N/C; 10-piece midpoint sum -4.493 N/C; difference 0.006537 N/C.
Uniform positive rod on x = 0 to L, observation point x = −a. Midpoint elements approximate a continuous line. Exact field is leftward. All graph distances are in meters; a is strictly positive.
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 questionFind the field at x = −1 m for a rod from 0 to 2 m with λ = +2 nC/m. State your limits and direction.
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Compare with the answer and four-point rubric
- 1 point: dq = λ dx and the distance is 1 + x.
- 1 point: Eₓ = −kλ ∫₀² dx/(1 + x)².
- 1 point: Eₓ = −18(1 − 1/3) = −12 N/C.
- 1 point: The field points left, away from every positive source element.
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 dq represent?
A small piece of charge, not a new independent source parameter.
RECALL 2What are the bounds here?
The source extends from x = 0 to L.
RECALL 3What should a refined numerical sum do?
Approach the integral for the same physical charge distribution.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Build a rod’s field by adding tiny charges
- dq = λ(x) dx; Q = ∫λ(x) dx.
- Uniform axial rod: Eₓ = −kλ[1/a − 1/(a + L)].
- a > 0; point is outside the rod on its extension.
Remember: The distance to each element varies. Keep source coordinate x separate from the fixed gap a.
Conditions: Uniform positive rod on x = 0 to L, observation point x = −a. Midpoint elements approximate a continuous line. Exact field is leftward. All graph distances are in meters; a is strictly positive.
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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