--- title: "Displacement current density at 60 Hz" subject: "Electromagnetism" chapter: 3 key: "semester_3/electromagnetism:displacement-current-metallic-conductor-60hz" source: https://www.bctengineer.com/q/semester_3/electromagnetism/displacement-current-metallic-conductor-60hz site: BCT Engineering --- # Displacement current density at 60 Hz Electromagnetism · Chapter 3: Magnetic field - Asked in: 2073 Shrawan, 5 marks - Video: [KTU Electromagnetics EE302, Module-4 Conduction current and displacement current densities](https://www.youtube.com/watch?v=pR0xqJGyUVs) — 100 Million Thoughts - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/electromagnetism/displacement-current-metallic-conductor-60hz - Whole chapter: https://www.bctengineer.com/md/semester_3/electromagnetism/chapter-3 ## Question Find the amplitude of the displacement current density in a metallic conductor at 60 Hz, if ε = ε0, μ = μ0, σ = 5.8×10⁷ S/m and J = sin(377t − 117.1z) a_x MA/m². ## Short answer Displacement current density J_d = (ε/σ) dJ_c/dt. For J_c = sin(377t - 117.1z) MA/m^2, dJ_c/dt = 377 cos(...) MA/m^2. Amplitude = (ε0 ω/σ) * 1e6 = (8.854e-12 * 377 / 5.8e7) * 1e6 = 5.76e-11 A/m^2. ## Answer The conduction current density is J_c = σE and the displacement current density is J_d = ∂D/∂t = ε ∂E/∂t. Given J = J_c = sin(377t − 117.1z) a_x MA/m², so ```text E = J_c / σ J_d = ε ∂E/∂t = (ε/σ) ∂J_c/∂t ``` **Step 1: differentiate.** ω = 377 rad/s (60 Hz). ```text ∂J_c/∂t = 377 cos(377t − 117.1z) a_x MA/m² ``` **Step 2: substitute** ε = ε₀ = 8.854×10⁻¹² F/m, σ = 5.8×10⁷ S/m. ```text ε₀ ω / σ = (8.854×10⁻¹² × 377) / (5.8×10⁷) = 3.338×10⁻⁹ / 5.8×10⁷ = 5.755×10⁻¹⁷ ``` ```text J_d = 5.755×10⁻¹⁷ cos(377t − 117.1z) a_x MA/m² = 5.755×10⁻¹¹ cos(377t − 117.1z) a_x A/m² ``` **Step 3: ratio check.** J_d/J_c = ωε/σ = 5.755×10⁻¹⁷ ≈ 10⁻¹⁷, so displacement current is completely negligible in a metal at power frequency. **Answer: amplitude of J_d = 5.755×10⁻¹⁷ MA/m² = 5.76×10⁻¹¹ A/m²** (leading J_c by 90°). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Given J_c, how is the displacement current density J_d obtained? A. J_d = sigma dJ_c/dt B. J_d = J_c / (epsilon w) C. J_d = (epsilon/sigma) dJ_c/dt D. J_d = J_c integrated over time Answer: C. E = J_c/sigma, and J_d = epsilon dE/dt = (epsilon/sigma) dJ_c/dt. (Key point: Relation between J_d and J_c) 2. How does J_d relate in phase to J_c = sin(377t - 117.1z)? A. In phase with J_c B. 180 degrees opposite C. It is a cosine, leading J_c by 90 degrees D. It lags J_c by 90 degrees Answer: C. Differentiating sin gives cos, so J_d leads J_c by 90 degrees. (Key point: Phase of J_d) 3. What is the amplitude of J_d? A. 5.755 x 10^-17 MA/m^2 (5.76 x 10^-11 A/m^2) B. 5.755 x 10^-11 MA/m^2 C. 1 MA/m^2 D. 3.34 x 10^-9 A/m^2 Answer: A. Amplitude = epsilon0 w / sigma x 1 MA/m^2 = 5.755 x 10^-17 MA/m^2, i.e. 5.76 x 10^-11 A/m^2. (Key point: Amplitude of J_d) 4. What does J_d/J_c = w epsilon/sigma ~ 10^-17 show for a metal at 60 Hz? A. Displacement current dominates B. Displacement current is negligible C. The metal behaves as a dielectric D. Both currents are about equal Answer: B. The ratio is tiny, so conduction current completely dominates in a metal at power frequency. 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