---
title: "Line charge at y=3, z=5, E at three points"
subject: "Electromagnetism"
chapter: 2
key: "semester_3/electromagnetism:line-charge-three-points-find-e"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/line-charge-three-points-find-e
site: BCT Engineering
---

# Line charge at y=3, z=5, E at three points

Electromagnetism · Chapter 2: Electric field

- Asked in: 2067 Shrawan, 6 marks
- Video: [2 .1 find electric field intensity due to line charge at y=3 z=5 and linear charge density is 30nC/m](https://www.youtube.com/watch?v=AUHA2QC4T7M) — Engineering Knowledge
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/line-charge-three-points-find-e
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-2

## Question

An infinitely long uniform line charge is located at y = 3, z = 5. If ρ_L = 30 nC/m, find E at (i) P_A(0, 0, 0) (ii) P_B(0, 6, 1) (iii) P_C(5, 6, 1).

## Short answer

E = ρ_L/(2πε₀ρ) a_ρ with ρ_L/(2πε₀)=539.3 V. P_A: ρ=5.83 m, E=−47.6 a_y−79.3 a_z V/m. P_B: ρ=5 m, E=64.7 a_y−86.3 a_z V/m. P_C same as P_B.

## Answer

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

ρ_L/(2πε₀) = (30×10⁻⁹)(17.975×10⁹) = 539.3 V

#### (i) P_A(0, 0, 0)
```text
ρ vector = (0−3) a_y + (0−5) a_z = −3 a_y − 5 a_z
ρ = √34 = 5.831 m
E = 539.3/5.831 × (−3 a_y − 5 a_z)/5.831
  = 92.48 × (−0.5145 a_y − 0.8575 a_z)
  = −47.6 a_y − 79.3 a_z  V/m
```

#### (ii) P_B(0, 6, 1)
```text
ρ vector = (6−3) a_y + (1−5) a_z = 3 a_y − 4 a_z ,  ρ = 5 m
E = (539.3/5) × (0.6 a_y − 0.8 a_z)
  = 107.9 × (0.6 a_y − 0.8 a_z)
  = 64.7 a_y − 86.3 a_z  V/m
```

#### (iii) P_C(5, 6, 1)
Its y and z coordinates equal those of P_B, so the perpendicular distance is the same and x makes no difference for an infinite line:

```text
E = 64.7 a_y − 86.3 a_z  V/m
```

| Point | ρ (m) | E (V/m) |
|---|---|---|
| P_A | 5.83 | −47.6 a_y − 79.3 a_z |
| P_B | 5 | 64.7 a_y − 86.3 a_z |
| P_C | 5 | 64.7 a_y − 86.3 a_z |

## Check yourself

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

1. What does the field of an infinite line charge depend on?
   A. Distance along the line
   B. Perpendicular distance ρ from the line only
   C. Distance from the origin
   D. Total length
   Answer: B. E = ρ_L/(2πε0ρ) a_ρ depends only on the perpendicular distance, so position along the line does not matter. (Key point: Field of infinite line)

2. For a line at y = 3, z = 5 parallel to x, the distance to the origin is:
   A. 8 m
   B. √34 m
   C. 5 m
   D. √(9+25+x²) m
   Answer: B. Only y and z count: (−3,−5) gives ρ = √34 = 5.83 m. (Key point: Distance at P_A)

3. Why is E at P_C(5,6,1) equal to that at P_B(0,6,1)?
   A. They have the same y and z, so same perpendicular distance and direction
   B. The line is finite
   C. Coincidence of the numbers
   D. E is zero at both
   Answer: A. For an infinite line along x, changing x changes nothing. (Key point: P_C versus P_B)

4. At P_B(0,6,1) the unit vector from the line is:
   A. 0.6 a_y − 0.8 a_z
   B. 0.8 a_y − 0.6 a_z
   C. −0.6 a_y + 0.8 a_z
   D. 0.6 a_y + 0.8 a_z
   Answer: A. The vector (6−3, 1−5) = (3,−4) over magnitude 5 gives 0.6 a_y − 0.8 a_z, pointing away from the line. (Key point: Direction of E)

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