# Consider a straight piece of length x of a wire carrying a current i. Let P be a point on the perpendicular bisector of the piece, situated at a distance d from its middle point. Show that for d >> x, the magnetic field at P varies at 1/d2 whereas for d <<x, it varies as 1/d.

Given:
Length of the piece: x
Current in the piece : i
Distance between midpoint of piece and P : d

P is on the perpendicular bisector of AB meeting at O.
Formula used:
By Biot-Savart Law:
Here, dB is the magnitude of magnetic field element, μ0 is the permeability of free space and μ0= 4π × 10-7 T mA-1,dl is the length element, r is the distance between the current carrying wire and the required point.
We know that, magnetic field at a point on the perpendicular bisector is
Here, a is the length of the wire. In this case a=x
d is the distance between point P and the midpoint of AB.
For d
x
x2 can be neglected.

B α 1/d2
Hence for d
x magnetic field is inversely proportional to the square of the distance d.
For d
x
d2 can be neglected.

B α 1/d
Hence, when d
x the magnetic field is inversely proportional to the distance d.

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