--- title: "Ampere's law, Maxwell's modification" subject: "Engineering Physics" chapter: 9 key: "semester_1/engineering_physics:ampere-law-maxwell-modification-displacement" source: https://www.bctengineer.com/q/semester_1/engineering_physics/ampere-law-maxwell-modification-displacement site: BCT Engineering --- # Ampere's law, Maxwell's modification Engineering Physics · Chapter 9: Electromagnetic Waves - Asked in: 2070 Chaitra, 5 marks - Video: [Maxwell's Modification of Ampere's Circuital Law](https://www.youtube.com/watch?v=g4WUA2Tg5Wo) — Mokka Physics - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_1/engineering_physics/ampere-law-maxwell-modification-displacement - Whole chapter: https://www.bctengineer.com/md/semester_1/engineering_physics/chapter-9 ## Question Write and explain Ampere's law in magnetism. How did Maxwell modify it? Based on this modified equation, explain the term displacement current. Prove displacement current is equal to conduction current. ## Short answer Ampere's law ∮B·dl = μ₀ I fails for a charging capacitor; Maxwell added the displacement current term ε₀ dΦ_E/dt to give ∮B·dl = μ₀(I + ε₀ dΦ_E/dt). The displacement current I_d = ε₀ dΦ_E/dt equals the conduction current dq/dt, ensuring continuity. ## Answer #### Ampere's law The line integral of the magnetic field B round any closed path equals μ₀ times the conduction current passing through the area bounded by the path: **∮B·dl = μ₀ I** It gives B for symmetric current distributions (long wire, solenoid, toroid). #### Maxwell's modification Ampere's law fails for a charging capacitor. Take a circular loop around the wire. If the surface bounded by the loop cuts the wire, ∮B·dl = μ₀I. If the surface is a bulge passing between the plates, no conduction current crosses it and the law gives zero. The same loop cannot give two answers. ```text wire ----> || || ----> wire || || S1: flat surface cuts the wire -> I enclosed S2: bulged surface between plates -> I = 0 (?) ``` Maxwell added a term: a changing electric flux acts like a current, the **displacement current** I_d = ε₀ dΦ_E/dt. **∮B·dl = μ₀(I + I_d) = μ₀I + μ₀ε₀ dΦ_E/dt** #### Displacement current It is not a flow of charge. It is the current-like term due to a time-varying electric field. A changing E produces a magnetic field, just as a changing B produces an electric field (Faraday). This leads to electromagnetic waves. #### Proof that I_d equals the conduction current Let the capacitor plate area be A and the charge on the plates q. The field between plates is E = q/(ε₀A). 1. Electric flux through the plate area: Φ_E = EA = q/ε₀. 2. I_d = ε₀ dΦ_E/dt = ε₀ (1/ε₀) dq/dt = dq/dt. 3. The conduction current in the wire is I = dq/dt. So **I_d = I**. The current is continuous through the capacitor, and both surfaces S1 and S2 now give the same ∮B·dl = μ₀I. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Ampere's law states that ∮B·dl equals: A. ε₀ times the enclosed charge B. zero for every closed path C. μ₀ times the conduction current through the loop D. μ₀ times the magnetic flux Answer: C. Original Ampere's law only counts conduction current. (Key point: Ampere's law) 2. Why does Ampere's law fail for a charging capacitor? A. The wire has no magnetic field B. The capacitor stores no charge C. B is infinite between the plates D. A surface passing between the plates has no conduction current, so the same loop gives two different answers Answer: D. The flat surface cuts the wire (I) but the bulged one passes between plates (0). (Key point: Why Ampere's law fails) 3. What did Maxwell add to Ampere's law? A. A displacement current I_d = ε₀ dΦ_E/dt B. A magnetic monopole current C. A term proportional to dΦ_B/dt D. A factor of 2 on the right side Answer: A. The corrected law is ∮B·dl = μ₀I + μ₀ε₀ dΦ_E/dt. (Key point: Maxwell's modification) 4. In the proof, Φ_E = q/ε₀ for the plate area. What follows? A. I_d = q/ε₀ B. I_d = ε₀ d(q/ε₀)/dt = dq/dt = I C. I_d = ε₀ q D. I_d = 0 since E is constant Answer: B. The displacement current equals the rate of change of charge, which is the conduction current. 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