---
title: "Uniform line charge, find E at P"
subject: "Electromagnetism"
chapter: 2
key: "semester_3/electromagnetism:line-charge-20nc-find-e-p"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/line-charge-20nc-find-e-p
site: BCT Engineering
---

# Uniform line charge, find E at P

Electromagnetism · Chapter 2: Electric field

- Asked in: 2072 Chaitra, 4 marks
- Video: [A uniform line charge of 2 µC/m is located on the z axis. Find E in cartesian coordinates at P](https://www.youtube.com/watch?v=xE5sYFqdN9w) — Guy_Teaches_STEM
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/line-charge-20nc-find-e-p
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-2

## Question

A uniform line charge density of 20 nC/m is located at y = 3 and z = 5. Find E at P(5, 6, 1).

## Short answer

The perpendicular distance from the line to P is 5 m, and the unit radial vector is 0.6a_y−0.8a_z. Field magnitude is (20×10⁻⁹)/(2πε₀·5)=71.9 V/m, so E = 71.9(0.6a_y−0.8a_z) = 43.1a_y − 57.5a_z V/m.

## Answer

The line is infinite and parallel to the x-axis (y = 3, z = 5). Its field is radial from the line: E = ρ_L/(2πε₀ρ) a_ρ, where ρ is the perpendicular distance from the line to the field point. Distance along x does not matter.

```text
ρ vector from line to P(5,6,1): R_⊥ = (6−3) a_y + (1−5) a_z = 3 a_y − 4 a_z
ρ = √(9+16) = 5 m
a_ρ = (3 a_y − 4 a_z)/5 = 0.6 a_y − 0.8 a_z

|E| = ρ_L/(2πε₀ρ)
    = (20×10⁻⁹)(17.975×10⁹)/5
    = 71.9 V/m

E = 71.9 (0.6 a_y − 0.8 a_z)
  = 43.1 a_y − 57.5 a_z  V/m
```

**Answer: E = 43.1 a_y − 57.5 a_z V/m** (magnitude 71.9 V/m).

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. The line y = 3, z = 5 is parallel to which axis?
   A. The y-axis
   B. The x-axis
   C. The z-axis
   D. None
   Answer: B. Only y and z are fixed, so x varies along the line. (Key point: Line orientation)

2. The perpendicular vector from the line to P(5, 6, 1) is:
   A. 3 a_y − 4 a_z
   B. 5 a_x + 3 a_y
   C. 3 a_y + 4 a_z
   D. 6 a_y + a_z
   Answer: A. (6 − 3) a_y + (1 − 5) a_z. (Key point: Perpendicular vector)

3. |E| equals:
   A. 43.1 V/m
   B. 359 V/m
   C. 14.4 V/m
   D. 71.9 V/m
   Answer: D. ρ_L/(2πε₀ρ) = 20×10⁻⁹ × 17.975×10⁹ / 5. (Key point: Magnitude)

4. E at P is:
   A. 57.5 a_y − 43.1 a_z V/m
   B. 43.1 a_y + 57.5 a_z V/m
   C. 43.1 a_y − 57.5 a_z V/m
   D. 71.9 a_x V/m
   Answer: C. Multiply the magnitude by a_ρ = 0.6 a_y − 0.8 a_z. (Key point: Result)

## Related in Electromagnetism

- [Point charge and four line charges, D](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Apoint-and-four-line-charges-d) — Electromagnetism, chapter 2
- [Line charge and point charge, find E](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aline-and-point-charge-find-e) — Electromagnetism, chapter 2, asked 2×
- [Potential due to sheet and line charge](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Asheet-and-line-charge-potential) — Electromagnetism, chapter 2
- [Potential from point and line charge](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Apotential-point-and-line-charge) — Electromagnetism, chapter 2
- [Line charge at y=3, z=5, E at three points](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aline-charge-three-points-find-e) — Electromagnetism, chapter 2
- [Sheet and line charge, E at origin](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Asheet-line-charge-e-origin) — Electromagnetism, chapter 2

## Related in other subjects

- [Line-to-pin capacitances for uniform string voltage](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Agrading-ring-five-insulators) — High Voltage Engineering, chapter 5
- [Lumped line energisation with trapped charge](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Alumped-line-trapped-charge) — High Voltage Engineering, chapter 3
- [Charge matrix of 400 kV line](https://www.bctengineer.com/question.md?key=semester_8%2Fhigh_voltage_engineering%3Acharge-matrix-400kv-midspan) — High Voltage Engineering, chapter 5
- [400 kV quadruple bundle line, 20 m spacing](https://www.bctengineer.com/question.md?key=semester_4%2Fpower_system_analysis_i%3Aquad-bundle-400kv-20m) — Power System Analysis I, chapter 4