Eletric Current

Electron current is the current that electrons exit batter at -ve terminal, and enter battery at -ve terminal. Conventional current is the current that fictitous positive charges exit battery at +ve terminal, and enter battery at -ve terminal. Electron Current 𝑖=𝑛𝐴𝐿𝐿/𝑣=𝑛𝐴𝑣=electronssecond
Conventional Current 𝐼=|𝑞|𝑛𝐴𝑣=Coulombssecond=Amperes
where 𝑛=electrons𝑚3= Density of electrons in the wire
𝐴=[𝑚2]= Cross sectional area of the wire
𝑣=𝑚𝑠= Average velocity of electrons in the wire.

Biot-Savart Law of Current in Wire

The Biot-Savart Law of Current in Wire is ∆𝐵=𝜇04𝜋𝐼∆𝑙×𝑟|𝑟|2 The Biot-Savart Law of current in wire can be dervied from Biot-Savart Law of charge particle. The current in the wire can be considered as many point charges in the wire. 𝑄𝑣=𝑁𝑞𝑣=𝑛𝐴∆𝑙𝑞𝑣=𝑞|𝑣|𝑛𝐴∆𝑙=𝐼∆𝑙

Properties of Biot-Savart Law

Magnetic field depends on frame of reference. The velocities of charges must be used the same reference frame as the observer.
Biot-Savart Law is an approximation. The changes of B-field or E-field propagate at speed of light. The law assumes that 𝑣≪𝑐, and the measurement is not taken so quickly.

Source and Reference

https://www.youtube.com/watch?v=6AcPmW4xRyo&list=PLZ6kagz8q0bvxaUKCe2RRvU_h7wtNNxxi&index=12