---
title: "Given V = 2x²y − 5z, find fields"
subject: "Electromagnetism"
chapter: 2
key: "semester_3/electromagnetism:given-potential-2x2y-5z"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/given-potential-2x2y-5z
site: BCT Engineering
---

# Given V = 2x²y − 5z, find fields

Electromagnetism · Chapter 2: Electric field

- Asked in: 2067 Shrawan, 5 marks
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/given-potential-2x2y-5z
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-2

## Question

Given the potential field V = 2x²y − 5z and a point P(−4, 3, 6), find at P: (i) V (ii) E (iii) a_E (iv) D (v) ρ_v.

## Short answer

At P, V = 66 V. E = 48 a_x − 32 a_y + 5 a_z V/m; a_E = 0.829 a_x − 0.553 a_y + 0.086 a_z. D = 425 a_x − 283 a_y + 44.3 a_z pC/m². ρ_v = −106 pC/m³.

## Answer

Given V = 2x²y − 5z (volts) and the point P(−4, 3, 6). Use ε₀ = 8.854×10⁻¹² F/m.

#### (i) V at P
```text
V = 2(−4)²(3) − 5(6) = 2(16)(3) − 30 = 96 − 30 = 66 V
```

#### (ii) E = −∇V
```text
∂V/∂x = 4xy  = 4(−4)(3) = −48
∂V/∂y = 2x²  = 2(16)    = 32
∂V/∂z = −5

E = −(−48 a_x + 32 a_y − 5 a_z) = 48 a_x − 32 a_y + 5 a_z  V/m
```

#### (iii) Unit vector a_E
```text
|E| = √(48² + 32² + 5²) = √(2304 + 1024 + 25) = √3353 = 57.9 V/m
a_E = E/|E| = 0.829 a_x − 0.553 a_y + 0.086 a_z
```

#### (iv) D = ε₀E
```text
D = 8.854×10⁻¹² (48 a_x − 32 a_y + 5 a_z)
  = 425 a_x − 283 a_y + 44.3 a_z  pC/m²
```

#### (v) Volume charge density
ρ_v = ∇·D = −ε₀∇²V.

```text
∇²V = ∂²V/∂x² + ∂²V/∂y² + ∂²V/∂z² = 4y + 0 + 0 = 4(3) = 12
ρ_v = −ε₀(12) = −8.854×10⁻¹² × 12 = −1.06×10⁻¹⁰ C/m³ = −106 pC/m³
```

**Answers:** V = 66 V; E = 48 a_x − 32 a_y + 5 a_z V/m; a_E = 0.829 a_x − 0.553 a_y + 0.086 a_z; D = 425 a_x − 283 a_y + 44.3 a_z pC/m²; ρ_v = −106 pC/m³.

## Check yourself

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

1. V at P(−4,3,6) for V = 2x²y − 5z is:
   A. 66 V
   B. 126 V
   C. −66 V
   D. 96 V
   Answer: A. 2(16)(3) − 30 = 96 − 30 = 66 V. (Key point: Evaluating V)

2. E at P is:
   A. −48 a_x + 32 a_y − 5 a_z
   B. 48 a_x − 32 a_y + 5 a_z
   C. 48 a_x + 32 a_y + 5 a_z
   D. 4 a_x + 2 a_y − 5 a_z
   Answer: B. E = −∇V = −(4xy, 2x², −5) = −(−48, 32, −5) = (48, −32, 5) V/m. (Key point: E from V)

3. D at P is obtained as:
   A. D = E/ε0
   B. D = ε0 E
   C. D = μ0 E
   D. D = ∇V
   Answer: B. D = ε0E in free space, giving about 425 a_x − 283 a_y + 44.3 a_z pC/m². (Key point: Relation of D and E)

4. ρ_v is found using:
   A. ρ_v = ε0∇²V
   B. ρ_v = −ε0∇²V = −106 pC/m³
   C. ρ_v = ∇V
   D. ρ_v = 0 since V is given
   Answer: B. ρ_v = ∇·D = −ε0∇²V, and ∇²V = 4y = 12, so ρ_v = −8.854×10^−12 × 12 ≈ −106 pC/m³. (Key point: Volume charge density)

## Related in Electromagnetism

- [Electric dipole potential and field](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adipole-potential-and-field-derive) — Electromagnetism, chapter 2, asked 4×
- [Energy stored for given potential field](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aenergy-stored-potential-field) — Electromagnetism, chapter 2
- [Potential gradient gives electric field](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Apotential-gradient-gives-electric-field) — 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
- [Vector field to cylindrical at point](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Avector-field-transform-cylindrical) — Electromagnetism, chapter 1
- [H of infinite filament via vector potential](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Ah-infinite-filament-vector-potential) — Electromagnetism, chapter 3

## Related in other subjects

- [Irrotational field and scalar potential F=(2x+z²)i+zj+(y+2xz)k](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Airrotational-field-and-scalar-potential-f-three) — Mathematics III, chapter 2, asked 3×
- [pn junction: potential, widths, field, Fermi levels](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_engineering_material%3Apn-1e16-1e17-fermi-levels) — Electrical Engineering Material, chapter 6, asked 2×
- [Field and electron acceleration from potential](https://www.bctengineer.com/question.md?key=semester_1%2Fengineering_physics%3Apotential-field-point-electron-acceleration) — Engineering Physics, chapter 7
- [Work done by a force field F=(2x−y+2z)i+(x+y−z)j…](https://www.bctengineer.com/question.md?key=semester_3%2Fmathematics_iii%3Awork-done-by-a-force-field-two) — Mathematics III, chapter 2