Ampere's law: explanation, definition, formula

Definition:-

"Ampere's Law states that magnetic field around a conductor in which electric current is flowing, is directly proportional to the amount of electric current flowing through conductor."

Explanation:

Ampere's law provide us method or formula to calculate magnetic field, produced by electric current.
Just like electric field is directly proportional to the amount of charge, the magnetic field is directly proportional to the amount of electric current from which it is producing.

Formula:-

The most difficult in ampere's law is to derive its equation or make the formula for calculating the magnetic field around a long conducting wire.
So let's start this,
Consider we have long straight conducting wire, when electric current flows through it, the magnetic fields produces around the wire in concentric circle. As shown in below pic.

Every circle is for some specific part of wire and as we move forward in wire, the distance between these circles increases, that mean magnetic field is decreasing. Direction of magnetic field can be found by Fleming's right hand rule.(with direction of current giving)
Now to make the formula, consider there all the parts of wire( which have there magnetic field circuit) having length L and there magnetic field around these parts or elements is B for (every part).

So the equation will become,

And after removing proportionality,

where 𝝁 is the premiability of material or conductor.
B is the magnetic field and delta L is the sum of all length parts and I is the current flowing through the conductor,
sigma shows that the sum of all the length elements or parts with their magnetic field.
Where delta L could be replace by 2πr for particular(single part) length element.
So Then magnetic filed would become,