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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.
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Solution
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")`
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