Question

Using Ampere’s circuital law, obtain an expression for magnetic flux density ‘B’ at a point near an infinitely long and straight conductor, carrying a current I.

Answer 1

Consider a point P at a distance from a straight infinity long wire (conductor) carrying a current I in free space.

Because of the axial symmetry about the straight wire, the magnetic induction has the same magnitude B at all points on a circle in a transverse plane and centred on the wire. We therefore choose an Amperian loop, a circle of radius r centred on the wire with its plane perpendicular to the wire.

`vec"B"` is everywhere tangential to the circular Amperian loop.

Angle θ = 0° (between `vec"B" and vec"dl"`) at all points on the loop.

`oint vec"B" * vec"dl" = oint "Bdl" = "B" oint "dl" = "B"(2pi"r")`

`therefore oint vec"B" * vec"dl" = mu_0"I"`

Where `mu_0` is the permeability of free space.

B = `(mu_0 "I")/(2pi"r")`