Quasi-Stationary Currents and Current Circuits

Abstract

In engineering, the currents that play an important role are those whose strength is the same at any instant in any section of the conducting wire. In other words, the current is equal at any position of the wire but it may be time-varying. Such currents are called quasi-stationary if, additionally, the condition is fulfilled that the current filaments are lines unchanging in time. So, the current density j(r, t) should be of the type

$$ j(r,t) = j(r)g(t) $$

(14.1)

Obviously, one can talk about current tubes constant in time and about the current in the individual tubes. For any segment of the tube one has

$$ \int_{surface} {j \cdot nda = 0} {\text{ }}and{\text{ }}therefore{\text{ }}divj = 0 $$

(14.2)

Considering the Maxwell equation

$$ curl{\text{ }}H = \frac{{4\pi }}{c}j + \frac{1}{c}\frac{{\partial D}}{{\partial t}} $$

(14.3)

it becomes clear that equation (14.2) is valid always if the displacement current density 1 /(4)D/t may be neglected compared to the conduction current density j, because then

$$ curl{\text{ }}H = \frac{{4\pi }}{c}j $$

(14.4)

and therefore div J = c/(4π) div curl H = 0.

Ohm’s law for stationary currents reads I R = V _e , where V_e is the externally impressed voltage. If the induction flux through the surface enclosed by the conductor changes, then according to Faraday an additional voltage is induced in the conductor

$$ {V_i} = - \frac{1}{c}\frac{{\partial \phi }}{{\partial t}} $$

(14.5)