State Ampere's circuital law. By using it derive an expression for magnetic field intensity at a point due to a straight current carrying conductor.

Explanation

Ampere.s Circuital Law : This law states that line integral of magnetic field over a closed circuital is equal to `mu_0` (absolute permeability of free space) times the current threading the circuit.
`oint vecB. vec(dl) = mu_0 I`
Proof: Consider a long straight conductor PQ carrying current I. Magnetic field lines are produced around the long straight current carrying conductor PQ as concentric circles.
Magnetic field due to infinitely long straight conductor at distance `a = B = (mu_0)/(4pi) (2I)/(a) " "...(1)`
Consider a circle of radius .a. around the current carrying infinitely long straight conductor. Let XY be a small elementary portion of length dl.

`because vecB` acts along the tangent to the circle.
`therefore vecB` and `vec(dl)` are in same direction i.e., angle between them is `0^@`.
`therefore oint vecB. vec(dl) = oint B dl cos phi`
`= oint B dl cos 0^@`
`= oint (mu_0)/(4pi) (2I)/(a) dl (1) " " ` [Using eqn. (1)]
`= (mu_0)/(4pi) (2I)/(a) oint dl`
`= (mu_0)/(4pi) (2I)/a (l)`
where `l = 2pia` = Length of the circular loop. =
`implies oint vecB. vec(dl) = (mu_0)/(4pi) (2I)/(a)(2 pi a)`
`implies oint vecB . vec(dl) = mu_0 I`
which is Ampere.s circuital law.