A current carrying conductor will have magnetic field around it. It generates induction of certain magnitude and here we are interested in finding that at a certain point due to different structure.

We can use the combination of the Biot-servert’s law and
ampere law to find the magnetic field at different points around a current
carrying conductor.

For example, let us assume a finite straight conductor and we
are interested in finding the magnetic field induction at a location that is in
perpendicular to the current carrying conductor. Let us assume that the each
end of the straight conductor is making some known angle at the point where we
need to measure the field.

We can write the equation as shown below. We can extend this
derivation to an infinite wire carrying current and it coincides with the
derivation we have made in the previous case.

We can extend this discussion to a square shaped coil
carrying current. The equation is same as the previous case. Any way the square
is having four identical shapes and to find the total value of the magnetic
field, we shall multiple the previous case value with four.

We can also do the same with a equilateral triangle shaped
current carrying conductor. We need to find the magnetic field at the center of
the triangle due to one part using the previous case and to get the total value
of the magnetic induction, we shall multiply with three. All this is shown in
the diagram below.

We can also find the magnetic field at a point due to current
carrying conductor in the shape of a sector. Let us assume that we know the
angle at the center and we can find the magnetic field at the center due to
current carrying conductor as shown in the diagram below.

In the above diagram one more problem is solved. There we
need to measure the magnetic field due to a current carrying conductor. It has
a straight line part and also a circular part as shown in the diagram above.

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