Question

State and explain Ampere’s circuital law.

Answer 1

Hint: Ampere's circuital law states that the closed line integral of magnetic field around a current carrying conductor is equal to absolute permeability times the total current threading the conductor. We can formulate it mathematically and understand the concept well.

Complete step-by-step answer:
Amperes circulate law mathematically state that the line integral of magnetic field of induction B around any closed path in free space is equal to absolute permeability of free space \[{{\mu }_{\circ }}\]​ times the total current flowing through area bounded by the path. The circuit is shown in the given picture

Mathematically, we can that the law as, \[\phi \overset{\to }{\mathop{B}}\,.\overset{\to }{\mathop{dl}}\,={{\mu }_{\circ }}I\] where,
B is magnetic induction
I is the total current flowing through the wire
Dl is length element of path
\[{{\mu }_{\circ }}\] is permeability of the space Ampere's law is a generalisation of Biot-Savart's law and can be derived from it and is used to determine magnetic field at any point due to distribution of current. Consider a long straight current carrying conductor XY, which is placed in a vacuum. A steady current I is flowing through it from the end Y to X. Now, imagine a closed curve (amperian loop) around the conductor having radius r as shown in the picture. The loop is assumed to be made of a very large number of small elements each of which is of length dl. Its direction is along the direction of the traced loop as shown in the picture. Now, let B be the strength of the magnetic field around the conductor. Hence, we get the final expression as
\[\phi =B.dl=B.dl.\cos \theta \]
where, \[\theta \]=angle between B and dl.

Note: Ampere's law gives us another method to calculate the magnetic field due to a given current distribution in a given fashion. Ampere's law can be derived from the Biot-Savart law and Biot-Savart law can also be derived from Ampere's law. Ampere's law is more useful under certain conditions where current distribution is symmetrical. We can calculate the magnetic field in a solenoid using this law.