Ampere's circuital law

 Ampere's circuital law states that the line integral of a magnetic field around any closed loop is equal to the permeability of free space multiplied by the net electric current enclosed by that loop.

While the Biot-Savart Law gives the magnetic field at a point due to a current element using vector calculus, Ampere’s law provides a simpler alternative when symmetry is present. For instance, in the case of a long straight wire, a toroid, or a solenoid, the law allows for a direct and often less mathematically intensive calculation of the magnetic field. The law states that the line integral of the magnetic field around any closed loop is proportional to the net current enclosed by that loop. However, it is important to note that Ampere’s law is strictly valid for steady currents, meaning the current must remain constant with time. The complications that arise in time-varying situations lead to the introduction of the concept of displacement current, which extends the law to dynamic fields, a topic explored in later chapters.

We draw a circular loop with radius ' r' passing through point  'P', with the current-carrying wire lying on its axis. This loop is called an Amperian loop.

The Amperian loop can be of any shape, but for practical calculations, it is often chosen with symmetry to simplify the mathematics.

It is important to choose the direction of the current and the direction in which you traverse the loop properly; this is determined by the right-hand rule. If you curl the fingers of your right hand in the direction you traverse the loop, then your right thumb points in the direction in which the current is considered positive.