8.3. Electromagnetic Induction in Circuit Approximation

It is possible to apply the circuit approximation even in the presence of regions where the term \(\partial_t \mathbf{b}\) cannot be neglected, such as in electromagnetic circuits involving transformers, motors, or electric generators.

In these situations, if it is possible to identify a connected region \(V_0\) in space where \(\partial_t \mathbf{b} = \mathbf{0}\), and therefore \(\nabla \times \mathbf{e} = \mathbf{0}\), it is possible to define the electric field in terms of a potential \(\varphi\) in \(V_0\):

\[\mathbf{e} = - \nabla \varphi \qquad , \qquad \mathbf{r} \in V_0 \ .\]

It is possible to calculate the potential differences at the terminals of a system where \(\partial_t \mathbf{b} \ne 0\), enclosed in the volume \(V_k\), using Faraday’s law:

\[\oint_{\ell_k} \mathbf{e} \cdot \hat{\mathbf{t}} = - \frac{d}{dt} \int_{S_k} \mathbf{b} \cdot \hat{\mathbf{n}} \ ,\]

where the closed path \(\ell_k = \ell_k^{cond} \cup \ell_k^{mors}\) describes the conductor in \(V_k\) closed by the geometric line between the terminals. If the resistivity of the conductor in \(V_k\) can be neglected, \(\int_{\ell_k^{cond}} \mathbf{e} \cdot \hat{\mathbf{t}} = 0\), the voltage difference at the terminals is:

\[\Delta v_k = \int_{\ell^{mors}_k} \mathbf{e} \cdot \hat{\mathbf{t}} = - \frac{d}{dt} \int_{S_k} \mathbf{b} \cdot \hat{\mathbf{n}}\]