--- title: "Boundary condition, find E2" subject: "Electromagnetism" chapter: 2 key: "semester_3/electromagnetism:boundary-condition-find-e2-medium" source: https://www.bctengineer.com/q/semester_3/electromagnetism/boundary-condition-find-e2-medium site: BCT Engineering --- # Boundary condition, find E2 Electromagnetism · Chapter 2: Electric field - Asked in: 2067 Mangsir, 6 marks - Video: [EMFT | Lec 82 | Solved Problem on Boundary Condition of Electric Field Part 1 | R K Classes |](https://www.youtube.com/watch?v=1_sclzKlCjs) — R K Classes - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electromagnetism/boundary-condition-find-e2-medium - Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-2 ## Question Use the boundary condition to find E2 in medium 2 with the boundary located at plane y = 0. Medium 1 is a perfect dielectric characterized by ε_r1 = 3, medium 2 is a perfect dielectric characterized by ε_r2 = 5, and the electric field in medium 1 is E1 = 3a_x + 2a_y + a_z. ## Short answer Tangential E_x, E_z continuous: E_x2=3, E_z2=1. Normal D continuous: ε_r2 E_y2 = ε_r1 E_y1 → 5 E_y2 = 3×2 → E_y2=1.2. E₂ = 3 a_x + 1.2 a_y + a_z. ## Answer Boundary at y = 0, so the normal is a_y and the tangential components are those along a_x and a_z. Medium 1: ε_r1 = 3. Medium 2: ε_r2 = 5. E₁ = 3 a_x + 2 a_y + a_z. Boundary conditions for perfect dielectrics (no free surface charge): - Tangential E is continuous: E_t1 = E_t2. - Normal D is continuous: ε_r1 E_N1 = ε_r2 E_N2. ```text Tangential: E_x2 = E_x1 = 3 , E_z2 = E_z1 = 1 Normal: ε_r2 E_y2 = ε_r1 E_y1 5 E_y2 = 3 × 2 E_y2 = 6/5 = 1.2 ``` **Answer: E₂ = 3 a_x + 1.2 a_y + a_z** (same units as E₁, e.g. V/m). Check: D₁ = 3ε₀(3, 2, 1) has D_y = 6ε₀; D₂ = 5ε₀(3, 1.2, 1) has D_y = 6ε₀, so the normal flux density is continuous. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. At the boundary y = 0, which components of E carry over unchanged? A. E_y B. E_x and E_z C. All components D. None Answer: B. Tangential E is continuous; tangential directions at y = 0 are x and z, so E_x2 = 3 and E_z2 = 1. (Key point: Tangential E) 2. Which condition gives E_y2 for a perfect dielectric with no free charge? A. E_y1 = E_y2 B. ε_r1 E_y1 = ε_r2 E_y2 (normal D continuous) C. ε_r2 E_y1 = ε_r1 E_y2 D. D_x1 = D_x2 Answer: B. Normal D is continuous when there is no surface charge: 3(2) = 5 E_y2, so E_y2 = 1.2. (Key point: Normal D) 3. What is E2? A. 3 a_x + 2 a_y + a_z B. 3 a_x + 3.33 a_y + a_z C. 3 a_x + 1.2 a_y + a_z D. 1.8 a_x + 1.2 a_y + 0.6 a_z Answer: C. E_x and E_z are unchanged and E_y falls to 6/5 = 1.2 because the higher permittivity reduces the normal E. (Key point: Final E2) ## Related in Electromagnetism - [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 - [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 at y=0, find fields](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Adielectric-boundary-y-d1-find-region2) — Electromagnetism, chapter 2 - [9.4 GHz plane wave in dielectric](https://www.bctengineer.com/question.md?key=semester_3%2Felectromagnetism%3Aplane-wave-9-4ghz-dielectric) — Electromagnetism, chapter 4, asked 2× ## Related in other subjects - [3rd order Butterworth LPF, R1 = 1 Ω, R2 = 4 Ω](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Abutterworth-lp-third-order-unequal-termination) — Filter Design, chapter 5, asked 4× - [IDMT grading: R1, R2 with 1000/5 CTs](https://www.bctengineer.com/question.md?key=semester_6%2Fswitchgear_and_protection%3Aidmt-r1-r2-1000-5) — Switchgear and Protection, chapter 7, asked 2× - [Show i1 = i2 = -V/(R1+R2+R3) at t=0+](https://www.bctengineer.com/question.md?key=semester_3%2Felectrical_circuit_theory%3Aswitch-a-to-b-initial-values) — Electrical Circuit Theory, chapter 2 - [Significance of reflection coefficient; R1 = 1 Ω, R2 = 4 Ω](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Areflection-significance-unequal-termination) — Filter Design, chapter 5