Steady State of a Circuit

Before a circuit is connected: No current flows through the wire conductors. In equilibrium state, all charges from battery become surface charges on the wire conductors to prevent current from flowing before connecting. The system is in equilitbrium and both net charges, current and electric field are equal to zero inside the system. 𝐼=|𝑞|𝑛𝐴𝑢|𝐸|=0⇒𝐸=0⇒𝑣=𝑢|𝐸|=0 When two wire conductor ends are placed closely, electric field due to gap faces are induced. However, electric field due to gap faces inside wire conductors are also cancelled by the redistribution of charges to maintain the system in equilibrium.

When two wire conductor ends are just in contact, the surface charge of the connected faces become zero so does the electric field between gap. As the electric field due to gap faces becomes zero, the instantaneous local net electric fields in the wire conductors become not equal to zero and causes the charges of wire conductors to move instantaneously.

In other words, the instantanteous net electric field at which two wire conductors are connected is as following

Similar to equilibrium state, the electric field disturbance due to the closing of gap will propagate throughout wire conductors at the speed of light, 𝑐≈3×10⁸𝑚/𝑠, while the moving of charge due the net local electric field is at the drift speed of the electrons, 𝑣≈5×10^-5𝑚/𝑠 in order to achieve an euilibrium state. That is, electrons always move locally with respected to the local net electric field. However, because of the charges supplied by the battery are infinite before the battery is exhausted, the system can only achieve a steady state with the net electric field inside the wire conductors is a constant.

When the system is in steady state, the electric field inside the system is constant and the current inside the system is not equal to zero and can be determine as following. 𝐸=constant⇒𝑣=𝑢|𝐸|⇒𝐼=|𝑞|𝑛𝐴𝑢|𝐸|≠0 resistor energy conservation voltage loop law battery