Electric Circuit

Surface charges make the electric field that drives the current in a circuit. Transient effects precede the steady state and a battery maintains a charge separation and a potential difference. The methods used to analyze circuits are current-node rule where current into a nod equals current out of the node and voltage-loop rule where the total potential difference around a loop is zero.
In general, the metal conductor is in equilibrium when there is no current flow or the average drift velocity of electrons inside the metal conductor is zero. When there is current flow and the average drift velocity of electrons inside the metal conductor is constant. The current flow of the system is in steady state.

Kirchhoff's Current Law

Current Node Rule: all current in = all current out

Electric Field in Circuit Wire

Constant current in the wire implies constant electric field in wire. Conventional current is a function of drift velocity, while drift velocity is controlled by the magnitude of electric field and the mobility 𝑢 set by the material. 𝐼=|𝑞|𝑛𝐴𝑣; 𝑣=𝑢|𝐸|⇒𝐼=|𝑞|𝑛𝐴𝑢|𝐸| Excess charge goes to the surface of a conductor. There are bands of constant charge on the wire from +ve pole to −ve pole to provide constant 𝐸. The amount of surface charge is proprtional to the voltage.

Source and Reference

https://www.youtube.com/watch?v=XNytX9ZYkJ0&list=PLZ6kagz8q0bvxaUKCe2RRvU_h7wtNNxxi&index=14