Pre-Quantum Electrodynamics

• PM 9.1, 9.2
• Gr 7.3.2

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

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

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.