AcademyElectromagnetic Induction
Academy
Displacement Current
Level 1 - Physics topic page in Electromagnetic Induction.
Principle
Displacement current lets changing electric flux produce magnetic fields even where no charge crosses a gap.
Notation
\(I_d\)
displacement current
\(\mathrm{A}\)
\(\Phi_E\)
electric flux
\(\mathrm{V\,m}\)
\(\epsilon_0\)
permittivity of free space
\(\mathrm{F\,m^{-1}}\)
\(\vec B\)
magnetic field
\(\mathrm{T}\)
\(I_{\mathrm c}\)
conduction current
\(\mathrm{A}\)
\(\mu_0\)
permeability of free space
\(\mathrm{N\,A^{-2}}\)
Method
Derivation 1: Charging capacitor gap
Between capacitor plates, no conduction current crosses the gap, but the electric field and electric flux change.
Electric flux
\[\Phi_E=\int\vec E\cdot d\vec A\]
Displacement current
\[I_d=\epsilon_0\frac{d\Phi_E}{dt}\]
Derivation 2: Consistency with charging current
For parallel plates with field \(E=Q/(\\epsilon_0A)\), the displacement current equals the wire current during charging.
Flux between plates
\[\Phi_E=EA=\frac{Q}{\epsilon_0}\]
Displacement current
\[I_d=\epsilon_0\frac{d}{dt}\left(\frac{Q}{\epsilon_0}\right)\]
Matches conduction
\[I_d=\frac{dQ}{dt}=I_c\]
Derivation 3: Ampere-Maxwell law
Maxwell's correction adds displacement current to Ampere's law.
Ampere-Maxwell law
\[\oint\vec B\cdot d\vec\ell=\mu_0(I_c+I_d)\]
Flux form
\[\oint\vec B\cdot d\vec\ell=\mu_0I_c+\mu_0\epsilon_0\frac{d\Phi_E}{dt}\]
Rules
Use displacement current when electric flux changes.
Electric flux
\[\Phi_E=\int\vec E\cdot d\vec A\]
Displacement current
\[I_d=\epsilon_0\frac{d\Phi_E}{dt}\]
Ampere-Maxwell
\[\oint\vec B\cdot d\vec\ell=\mu_0I_c+\mu_0\epsilon_0\frac{d\Phi_E}{dt}\]
Examples
Question
A capacitor has electric flux increasing at
\[5.0\times10^{10}\,\mathrm{V\,m\,s^{-1}}\]
Find \(I_d\).Answer
\[I_d=\epsilon_0\frac{d\Phi_E}{dt}=(8.85\times10^{-12})(5.0\times10^{10})=0.443\,\mathrm{A}\]
Checks
- Displacement current is not charge crossing vacuum.
- It is caused by changing electric flux.
- It produces magnetic fields in Ampere-Maxwell law.
- For an ideal charging parallel-plate capacitor, \(I_d=I_c\).