The continuity equation for conductors in simple DC circuits is seldom found and derived in textbooks though derivations in case of semiconductors are found in several textbooks. The charge density and current density functions are related by the continuity equation (see Electricity and Magnetism by Edson Ruther Peck, McGraw Hill, 1953) which maybe derived by applying the principle of conservation of charge. Since most textbooks on circuit theory do not discuss this important aspect of the conduction process in the dc steady state in particular, I have discussed this in textbook 4 (see last frame of video 1 to be discussed below). In its most general form the equation of continuity is ∂J_x/∂x + ∂J_y/∂y + ∂J_z/∂z + ∂ρ/∂t = 0. (Eq. 1) as derived from the conservation of electric charge law. The current density J in an isotropic medium is given by the relation J = σE (Eq. 2) where E is the electric field intensity and where σ is the conductivity of the medium. It is also written from Eq. 1 as ∂J_x/∂x + ∂J_y/∂y + ∂J_z/∂z = 0. (Eq. 3) when there is no excess charge in the conductor or that there is no unpaired charge density (lattice ion and conduction electron). In the absence of emf in a region in the circuit (say, away from the source or battery and within a small section of the conductor or a resistor), the total electric field E, may be expressed in terms of a scalar potential function U; E_x = - ∂U/∂x E_y = - ∂U/∂x, and E_z = - ∂U/∂z (Eq. 4) Eqs. (2), (3) and (4) characterize the current flow within a region of a homogeneous, linear, isotropic conductor where there is no emf. If a dc circuit of a battery and a wire is laid in a straight line along the x-axis then evidently, the presence of surface charges will guarantee that the total field E will be a constant E_x along the axis in the region. Therefore, the solution of Eq. (3) gives J_x = a constant, so using Eq. 2, we get J_x = σE_x = I/A (Eq. 5) where σ is the conductivity of the wire, I is the current in the circuit and A the cross-sectional area of the wire. Eq. 5 is the equation of continuity applicable to the steady-state in a simple DC circuit. Electrostatics and circuits belong to one science not two and it is instructive to understand Current, the conduction process and Voltage at the fundamental level as in the following two videos: i. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-REsWdd76qxc.html and ii. ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-8BQM_xw2Rfo.html The last frame References in video #1 lists textbook 4 in which a supplementary article “Charge Densities and Continuity and Prop of em signals in wires.pdf” in the pdf files folder in the CD discusses these topics in more detail with several diagrams using a unified approach and includes a description of the application of the general continuity equation in special situations like conductors in isolation and in semiconductors.
