Ampere’s circuital law:
This law states that the line integral of the magnetic field of induction along a closed path in a vacuum is equal to \(\mu_o\) times the total current threading the closed path.
\(\int\vec{B}.d\vec{I}\)= \(\mu_oI\)
Magnetic field (of induction) due to an infinitely long current carrying conductor:
Let us consider a long straight wire XY carrying a current I. Let P be a point at a perpendicular distance r from the point O on the wire.
Let us imagine a circular path of radius r, centre O so that the point P lies on the path.
Now, from Ampere’s Circuital law,
\(\int\vec{B}.d\vec{I}\)= \(\mu_oI\)
\(\implies\)\(\int\vec{B}.dl\) = \(\mu_oI\) [Since angle between \(\vec{B}\) and \(d\vec{I}\) is zero]
\(\implies\) \({B\int}dl\) = \(\mu_oI\)
\(\implies\) \(B.2πr=μ_0I\) [Since the total length of \(d\vec{I}\) is equal to circumference of the circle considered]
\(\implies\) B = \(\frac{\mu_oI}{2\pi r}\)
which is the required expression for magnetic field of induction due to a long straight current carrying wire.
