What is ampere’s law?
According to this law “The line integral of magnetic field B along a closed path due to current is equal to the product of the permeability of free space and the current enclosed by the closed path”.
Mathematically it is expressed as:
Where
μ_{0}=permeability of free space
i=current flowing through the conductor.
Consider a straight conductor in which current i is flowing.The current produces the magnetic field B around the conductor.The magnetic field lines are in the form of concentric circles.
Ampere showed that the flux density B at any point near the conductor is directly proportional to the current i and inversely proportional to the distance ‘r’ from the conductor ,so:
Where is the length of the path called circumference of the circle.
Divide the circle representing the magnetic field line into large number of small elements each of length dl.The quantity B.dl is calculated for each element as:
B.dl=Bdlcos= Bdlcos0=Bdl
For complete circle:
Integral form of ampere’s law:
Differential form of ampere law:
Since the integral form of ampere’s law is:
The above relation is known as differential form of ampere’s circuital law.
Applications of ampere’s circuital law:

Field due to a solenoid:
Consider a solenoid having n turns per unit length.When current passes,through the solenoid,magnetic field is produced inside the solenoid which is directed along the axis of solenoid.The magnetic field in space outside is so weak that it is considered as zero.
In order to calculate the value of magnetic field B inside the solenoid using ampere’s law,we consider a closed path abcda in the form of rectangle.This closed path is known as Amperian path,as seen in the figure.
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