#### Maxwell's Correction to Ampère's Law; the Displacement Current
emd.Me.dc

The term which should be zero (but isn't) in divcurlB can be rewritten using the continuity equation as \[ {\boldsymbol \nabla} \cdot {\bf J} = -\frac{\partial \rho}{\partial t} = - \frac{\partial}{\partial t} (\varepsilon_0 {\boldsymbol \nabla} \cdot {\bf E}) = -{\boldsymbol \nabla} \cdot \left( \varepsilon_0 \frac{\partial {\bf E}}{\partial t} \right). \] The extra term would thus be eliminated if we were to put

- Gr (7.36)

\[ {\boldsymbol \nabla} \times {\bf B} = \mu_0 {\bf J} + \mu_0 \varepsilon_0 \frac{\partial {\bf E}}{\partial t} \tag{AmpMax}\label{AmpMax} \]

Note: this changes nothing in magnetostatics. Aesthetic appeal: \[ \boxed{ \mbox{A changing electric field induces a magnetic field.} } \] Real confirmation of Maxwell's theory: 1888, Hertz's experiments on propagation of electromagnetic waves.

Maxwell baptized this term the

Displacement current

- Gr (7.37)

\[ {\bf J}_d \equiv \varepsilon_0 \frac{\partial {\bf E}}{\partial t}. \tag{Jd}\label{Jd} \]

Resolves the charging capacitor plate problem: if plates close together, field is \[ E = \frac{\sigma}{\varepsilon_0} = \frac{1}{\varepsilon_0} \frac{Q}{A} \] where \(A\) is the area. Between the plates, \[ \frac{\partial E}{\partial t} = \frac{1}{\varepsilon_0 A} \frac{dQ}{dt} = \frac{1}{\varepsilon_0 A} I. \] Checking AmpMax, \[ \oint {\bf B} \cdot d{\bf l} = \mu_0 I_{\mbox{enc}} + \mu_0 \varepsilon_0 \int d{\bf a} \cdot \frac{\partial {\bf E}}{\partial t} \label{Gr(7.38)} \] Flat surface: OK, \(E = 0\) and \(I_{\mbox{enc}} = I\). Balloon surface: \(I = 0\) but \(\int d{\bf a} \cdot (\partial {\bf E}/\partial t) = I/\varepsilon_0\). Answers are consistent.

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Author: Jean-Sébastien Caux

Created: 2022-03-24 Thu 08:42