---
title: "Dielectric boundary at y=0, find fields"
subject: "Electromagnetism"
chapter: 2
key: "semester_3/electromagnetism:dielectric-boundary-y-d1-find-region2"
source: https://www.bctengineer.com/q/semester_3/electromagnetism/dielectric-boundary-y-d1-find-region2
site: BCT Engineering
---

# Dielectric boundary at y=0, find fields

Electromagnetism · Chapter 2: Electric field

- Asked in: 2070 Chaitra, 3+3 marks
- Video: [3.16 With reference to the figure 3.19, find E1 if E2=4ax-3ay+3az (V/m), where ε1=2 ε0 and ε2=8 ε0.](https://www.youtube.com/watch?v=DZP1vCKQ3dE) — EE-Vibes (Electrical Engineering Lessons) 
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/electromagnetism/dielectric-boundary-y-d1-find-region2
- Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-2

## Question

Consider the region y < 0 composed of a uniform dielectric material for which the relative permittivity (ε_r) is 3.2, while the region y > 0 is characterized by ε_r = 2. Let the flux density in region 1 be D1 = −30a_x + 50a_y + 70a_z nC/m². Find: (a) the magnitude of flux density and electric field intensity at region 2; (b) polarization (P) in region 1 and region 2.

## Short answer

The normal D_y is continuous; tangential D is scaled by ε_r2/ε_r1=0.625, giving D₂=−18.8a_x+50a_y+43.8a_z nC/m², |D₂|=69.0 nC/m². E₂=D₂/(2ε₀), |E₂|=3.90 kV/m. P₁=0.6875D₁, P₂=0.5D₂ with components as given.

## Answer

Boundary: the plane y = 0, with normal along a_y. Region 1 (y < 0): ε_r1 = 3.2. Region 2 (y > 0): ε_r2 = 2. D₁ = −30 a_x + 50 a_y + 70 a_z nC/m².

Conditions: the normal component of D is continuous; the tangential component of E is continuous, so D_t1/ε_r1 = D_t2/ε_r2.

#### (a) D₂ and E₂
```text
D_N1 = 50 a_y  →  D_N2 = 50 a_y  nC/m²
D_t1 = −30 a_x + 70 a_z
D_t2 = (ε_r2/ε_r1) D_t1 = (2/3.2)(−30 a_x + 70 a_z)
     = −18.75 a_x + 43.75 a_z

D₂ = −18.75 a_x + 50 a_y + 43.75 a_z  nC/m²
|D₂| = √(18.75² + 50² + 43.75²) = √4765.6 = 69.03 nC/m²
```
Electric field in region 2: E₂ = D₂/(ε₀ε_r2) = D₂/(2 × 8.854×10⁻¹²):

```text
E₂ = (−18.75 a_x + 50 a_y + 43.75 a_z)×10⁻⁹ / (17.708×10⁻¹²)
   = −1058.8 a_x + 2823.6 a_y + 2470.6 a_z  V/m
|E₂| = 3898 V/m ≈ 3.90 kV/m
```

#### (b) Polarization, P = D − ε₀E = D(1 − 1/ε_r)
```text
Region 1: P₁ = D₁ (1 − 1/3.2) = 0.6875 D₁
        = −20.63 a_x + 34.38 a_y + 48.13 a_z  nC/m²

Region 2: P₂ = D₂ (1 − 1/2) = 0.5 D₂
        = −9.38 a_x + 25 a_y + 21.88 a_z  nC/m²
```

**Answers:** |D₂| = 69.0 nC/m², |E₂| = 3.90 kV/m; P₁ = −20.6 a_x + 34.4 a_y + 48.1 a_z nC/m², P₂ = −9.38 a_x + 25 a_y + 21.9 a_z nC/m².

## Check yourself

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

1. At y = 0, the normal component of D1 is:
   A. −30 a_x
   B. 70 a_z
   C. all of D1
   D. 50 a_y
   Answer: D. The normal to the plane y = 0 is a_y. (Key point: Boundary normal)

2. How does D_t2 relate to D_t1?
   A. D_t2 = (ε_r1/ε_r2) D_t1
   B. D_t2 = D_t1
   C. D_t2 = (ε_r2/ε_r1) D_t1
   D. D_t2 = 0
   Answer: C. E_t continuous means D_t/ε_r is the same. (Key point: Tangential D scaling)

3. D₂ equals:
   A. −30 a_x + 50 a_y + 70 a_z
   B. −18.75 a_x + 50 a_y + 43.75 a_z nC/m²
   C. −48 a_x + 50 a_y + 112 a_z
   D. −18.75 a_x + 31.25 a_y + 43.75 a_z
   Answer: B. Multiply the tangential parts by 2/3.2 = 0.625. (Key point: D₂ result)

4. P₂ equals:
   A. 0.5 D₂
   B. 0.6875 D₂
   C. 2 D₂
   D. D₂/ε₀
   Answer: A. P = D(1 − 1/ε_r) = D(1 − 1/2) = 0.5D₂. (Key point: Polarization)

## Related in Electromagnetism

- [Dielectric boundary at z=0, region 1 data](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adielectric-boundary-z-3-2) — Electromagnetism, chapter 2
- [Dielectric boundary, E1 given, find E2 D2 P2](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adielectric-boundary-find-e2-d2) — Electromagnetism, chapter 2
- [Polarization and E in both regions](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adielectric-boundary-x-polarization-e) — Electromagnetism, chapter 2

## Related in other subjects

- [Numerical: received power, E-field, flux density, voltage](https://www.bctengineer.com/question.md?key=semester_7%2Fwireless_communication%3Afree-space-received-power-e-field) — Wireless Communication, chapter 3, asked 2×