Find the magnetic induction due to a long straight conductor using Ampere's circuital law.

Explanation

Magnetic field due to the current carrying wire of infinite length using ampere.s law: Consider a straight conductor of infinite length carrying current I and the direction of magnetic field lines. Since the wire is geometrically cylindrical in shape and symmetrical about its `a xx ` is, we construct an Amperian loop in the form of a circular shape at a distance r from the centre of the conductor . from the ampere.s law, we get
`oint_(C ) vec(B). d vec(l) = mu_(0)` I
Where d`vec(l)` is the element along the amperican loop (tangent to the circular loop ) . Hence, the angle between magnetic field vector and line element is zero. Therefore,
`oint_(C) B.dl = mu_(0) `l
Where l is the current enclosed by the amperian loop. Due to the symmetry. The magnitude of the magnetic field is uniform over the Amperian loop, we can take B out the integration.
`oint_(C) B.dl = mu_(0) `I
For a circular loop , the circumference is 2`pi`r, which implies,
`B oint_(n)^(2 pi r) dl = mu_(0)` I
`vec(B). 2 pi r = mu_(0) I " " rArr " " vec(B) = (mu_(0)I)/(2 pi r)`
In vector form, the magnetic field is
`vec(B) = (mu_(0)I)/(2 pi r)hat(n)`
Where `hat(n)` is the vector along the tangent to the Amperian loop. This perfectly agrees with the result obtained from Biot-Savart.s law as given in equation
`vec(B) = (mu_(0)I)/(2 pi a) hat(n)`