Chapter 2

Maxwell's equations

Density of current and intensity in steady-state


In a steady-state (independent of time) :

so .

Thus, it can be deduced that :

  • The total intensity exiting a given closed surface is equal to zero in steady-state :

  • In steady-state, the intensity has the same value through any section of a given field tube :

Kirchhoff's circuit laws

Let us consider a closed surface formed by one field tube (called current tube) and two surfaces and .

These two surfaces are based on two loops of and have the same orientation.

Let and be the intensities, or respectively the flux of the density of current through and  :

(We can notive that the two vectors and are perpendicular).

So :

In steady-state, the intensity of the electric current has the same value through any section of a circuit branch.

We can easily deduce Kirchhoff's circuit laws (flux or current density vector conservation).

Local charge conservation law
Maxwell's equations