Magnetic Field Generated by a Short Current

According to Biot-Savart law the magnetic field due to a current carrying segment is given by:
(1.1)

In our case we can write:
(1.2)
The directionality of the cross product is determined by the right hand rule and in this case the magnetic field points out of the page.
However:
(1.3)
And
(1.4)
Thus, equation (1.1) now becomes:
(1.5)

All that is left is to integrate between :
(1.6)
To solve the integral we use:

Therefore, equation (1.6) becomes:
(1.7)
The integrand is an even function, so we can use:
(1.8)

This is a simple integration, and we will use the fact that:

(1.9)
Back to equation (1.8) we obtain:

(1.10)

Note that when we let we regain the field for infinite wire:

What is left is to plug in the numbers:

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