---
title: "Derive Poisson's and Laplace's equations"
subject: "Electromagnetism"
chapter: 2
key: "semester_3/electromagnetism:poisson-laplace-equations-derive"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/poisson-laplace-equations-derive
site: BCT Engineering
---

# Derive Poisson's and Laplace's equations

Electromagnetism · Chapter 2: Electric field

- Asked in: 2075 Chaitra, 2 marks; 2074 Asoj, 1 marks (asked 2 times)
- Video: [Poisson's and Laplace Equations: Derivations and Explanations](https://www.youtube.com/watch?v=lVRIw36CAWs) — Engineering Funda
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/poisson-laplace-equations-derive
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-2

## Question

Derive Poisson's and Laplace's equations.

## Short answer

Poisson's equation: ∇²V = −ρ_v/ε, derived from Gauss's law ∇·D = ρ_v, D = εE, and E = −∇V for homogeneous medium. Laplace's equation: ∇²V = 0, obtained when ρ_v = 0.

## Answer

#### Derivation of Poisson's equation

Start with three basic relations of electrostatics:

1. Gauss's law, point form: ∇ · D = ρ_v
2. Constitutive relation: D = εE
3. Relation between field and potential: E = −∇V

Substituting (2) and (3) into (1):

```text
∇ · (ε E) = ρ_v
∇ · (−ε ∇V) = ρ_v
```

For a homogeneous medium (ε constant):

```text
∇ · ∇V = ∇²V = −ρ_v / ε            ← Poisson's equation
```

#### Laplace's equation

In a region with no free charge, ρ_v = 0, and Poisson's equation reduces to

```text
∇²V = 0                              ← Laplace's equation
```

#### Forms of ∇²V

```text
Cartesian:     ∇²V = ∂²V/∂x² + ∂²V/∂y² + ∂²V/∂z²

Cylindrical:   ∇²V = (1/ρ) ∂/∂ρ (ρ ∂V/∂ρ) + (1/ρ²) ∂²V/∂φ² + ∂²V/∂z²

Spherical:     ∇²V = (1/r²) ∂/∂r (r² ∂V/∂r) + (1/(r² sinθ)) ∂/∂θ (sinθ ∂V/∂θ)
                     + (1/(r² sin²θ)) ∂²V/∂φ²
```

Poisson's and Laplace's equations are used to find the potential in a region when boundary conditions (or charge distribution) are known, for example in capacitors.

## Check yourself

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

1. Which three relations are combined to derive Poisson's equation?
   A. Ampere's law, Faraday's law and Ohm's law
   B. ∇·D = ρ_v, D = εE and E = -∇V
   C. ∇×E = 0, B = μH and J = σE
   D. Continuity equation, Gauss's law for magnetism and D = εE
   Answer: B. Substituting D = εE and E = -∇V into ∇·D = ρ_v gives Poisson's equation. (Key point: Starting relations)

2. For a homogeneous medium, Poisson's equation is:
   A. ∇²V = -ρ_v/ε
   B. ∇²V = ρ_v/ε
   C. ∇²V = 0
   D. ∇V = -ρ_v/ε
   Answer: A. ∇·(-ε∇V) = ρ_v with constant ε gives ∇²V = -ρ_v/ε. (Key point: Poisson's equation)

3. When does Poisson's equation reduce to Laplace's equation ∇²V = 0?
   A. When the potential is constant in time
   B. When the medium is non-homogeneous
   C. When the field E is uniform everywhere
   D. When the region has no free charge, ρ_v = 0
   Answer: D. Setting ρ_v = 0 in Poisson's equation gives Laplace's equation. (Key point: Laplace's equation)

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