---
title: "Poynting vector, S = E×B/μ0"
subject: "Engineering Physics"
chapter: 9
key: "semester_1/engineering_physics:poynting-vector-proof"
source: https://www.bctengineer.com/q/semester_1/engineering_physics/poynting-vector-proof
site: BCT Engineering
---

# Poynting vector, S = E×B/μ0

Engineering Physics · Chapter 9: Electromagnetic Waves

- Asked in: 2073 Shrawan, 5 marks; 2072 Chaitra, 5 marks; 2071 Chaitra, 5 marks; 2067 Asar, 5 marks (asked 4 times)
- Video: [Poynting vector](https://www.youtube.com/watch?v=g6IjaH7bjnI) — Lynne Lawson
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_1/engineering_physics/poynting-vector-proof
- Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-9

## Question

Define Poynting vector. Prove that S = (E × B)/μ0, where the symbols have their usual meanings.

## Short answer

The Poynting vector S = (E×B)/μ₀ is the electromagnetic energy flux (W/m²). Using Maxwell's equations, one obtains ∇·S = −∂u/∂t − E·J, which is the energy conservation law.

## Answer

#### Definition
The **Poynting vector S** is the rate of flow of electromagnetic energy per unit area, in the direction of propagation of the wave. Its unit is W/m². It is given by **S = E × H = (E × B)/μ₀**.

#### Proof
We use energy conservation with Maxwell's equations (in vacuum, in a region with current density J):

∇×E = −∂B/∂t and ∇×B = μ₀J + μ₀ε₀ ∂E/∂t.

1. Use the vector identity ∇·(E×B) = B·(∇×E) − E·(∇×B).
2. Substitute: ∇·(E×B) = −B·∂B/∂t − E·[μ₀J + μ₀ε₀ ∂E/∂t].
3. Divide by μ₀:

   ∇·(E×B)/μ₀ = −(1/μ₀) B·∂B/∂t − ε₀ E·∂E/∂t − E·J
4. Note B·∂B/∂t = ½ ∂(B²)/∂t and E·∂E/∂t = ½ ∂(E²)/∂t. So

   ∇·[(E×B)/μ₀] = −∂/∂t [ε₀E²/2 + B²/(2μ₀)] − E·J
5. The bracket is the energy density u = u_E + u_B. E·J is the power given to the charges per unit volume. Define S = (E × B)/μ₀:

   **∇·S = −∂u/∂t − E·J**
6. Integrate over a volume and use the divergence theorem: ∮S·dA = −d/dt ∫u dV − ∫E·J dV.

The left side is the energy flowing out through the surface per second. The right side is the decrease in stored field energy minus the work done on the charges per second. So S·dA is the power crossing dA, and S = (E×B)/μ₀ is the energy flux, pointing along E × B (the direction of travel).

## Check yourself

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

1. The Poynting vector S represents:
   A. the total energy stored in the field
   B. the force per unit area on a charge
   C. the magnetic flux density
   D. the rate of energy flow per unit area (W/m²), along the direction of propagation
   Answer: D. S = (E × B)/μ₀ is the energy flux. (Key point: Definition)

2. The proof starts with the identity:
   A. ∇·(E×B) = B·(∇×E) − E·(∇×B)
   B. ∇·(E×B) = E·(∇×B) + B·(∇×E)
   C. ∇·(E×B) = E·B
   D. ∇×(E×B) = 0
   Answer: A. Then Faraday's and Ampere-Maxwell equations are substituted. (Key point: Vector identity)

3. The bracket ε₀E²/2 + B²/(2μ₀) in the result is:
   A. the Poynting vector
   B. the electromagnetic energy density u
   C. the current density
   D. the power supplied to charges
   Answer: B. u = u_E + u_B. (Key point: Energy density)

4. The result ∇·S = −∂u/∂t − E·J means:
   A. field energy always grows
   B. S is zero in free space
   C. energy flowing out equals the fall in field energy plus work done on charges
   D. energy flows along B only
   Answer: C. Hence S·dA is the power crossing dA. (Key point: Final result)

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