Displacement Current
A changing electric field makes a magnetic field.
Lesson at a glanceShow the key points
- Ampère's law fails when E changes.
- i_d = ε₀ dΦ_E/dt; ∮B⃗·dl⃗ = μ₀(i_c + i_d) (40.3).
- Between charging plates i_d = i_c.
- Conduction plus displacement current is continuous.
A changing magnetic field makes an electric field. Maxwell realised the reverse must also hold, and adding that one term to Ampère’s law turned electricity and magnetism into a theory of light.
Where Ampère’s law fails
Ampère’s law, ∮B⃗·dl⃗ = μ₀i, holds only while the electric field through the loop is steady. Charge a capacitor with circular plates and put a compass between the plates: it deflects, so there is a magnetic field there. Take a loop γ lying between the plates, parallel to them. Its circulation of B is not zero, yet no charge crosses the flat surface it bounds: i = 0. And the field due to the currents in the wires alone, worked out from Biot–Savart, would give zero circulation. Something else between the plates makes magnetic field: the changing electric field.
The displacement current
Just as Faraday’s law says ∮E⃗·dl⃗ = −dΦB/dt, a changing electric flux makes a magnetic circulation:
With both a current and a changing electric field present, Maxwell’s generalised Ampère’s law is:
Show that while a parallel-plate capacitor is charging, the displacement current through the whole region between the plates equals the current in the wires.
A capacitor of plate area A is charged by a steady current i. What displacement current passes through an area A/3 between the plates, parallel to them?
Circular plates of radius 5.0 cm are charged at 0.50 A. Find B between them 3.0 cm from the axis, and outside the plates 10 cm from the axis.
Current is always continuous
If charge piles up inside a closed surface, conduction current i₁ in exceeds i₂ out: d(qinside)/dt = i₁ − i₂. By Gauss’s law ΦE = qinside/ε₀, so ε₀ dΦE/dt = d(qinside)/dt, i.e. id = i₁ − i₂.
Counting displacement current, as much current flows out of any closed surface as flows in.
- 1Ampère's law ∮B⃗·dl⃗ = μ₀i fails when E changes (charging capacitor).
- 2A changing electric flux makes B: displacement current i_d = ε₀ dΦ_E/dt.
- 3∮B⃗·dl⃗ = μ₀(i_c + i_d) (40.3).
- 4Total current (conduction + displacement) is continuous.
Check yourself
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