We can use the Biot-Savart law to find the magnetic field of the current element. The law states that the magnetic field dB produced by a small segment of current I dl at a distance r from the segment is given by:

dB = μ0/4π (I dl x r) / r^2

where μ0 is the permeability of free space, x is the cross product, and r is the distance from the segment to the point where we want to find the field.

For a finite current element, we can break it up into small segments of length dl and add up the contribution of each segment to find the total magnetic field at a point. We can assume that the current element is a straight wire of length L and uniform current I.

Let's consider a point P at a distance r from the current element, as shown in the figure below:


We can choose a coordinate system such that the current element lies along the z-axis and point P lies in the x-y plane at a distance z from the origin. We can also choose dl to be a small segment of length dx along the z-axis, as shown in the figure.

The contribution of this segment to the magnetic field at point P is given by:

dB = μ0/4π (I dl x r) / r^2

where r is the distance from the segment to point P. We can use the Pythagorean theorem to find r:

r^2 = z^2 + x^2

The vector product dl x r gives a vector that is perpendicular to both dl and r, and has magnitude equal to the area of the parallelogram formed by dl and r. In this case, dl is along the z-axis and r lies in the x-y plane, so the area of the parallelogram is simply dx times the distance between dl and the x-axis, which is z.

Therefore, dl x r has magnitude equal to z dx, and direction along the y-axis. We can write it as:

dl x r = z dx j

where j is the unit vector along the y-axis.

Putting everything together, we get:

dB = μ0/4π (I z dx j) / (z^2 + x^2)

We can integrate this expression over the length of the current element to find the total magnetic field at point P. Since the current element is symmetric about the origin, we can assume that the contributions from opposite sides cancel out along the x-axis, and we only need to integrate over the positive z-axis. The limits of integration are 0 to L/2, since the current element extends from -L/2 to L/2.

The total magnetic field B at point P is given by:

B = ∫dB = μ0/4π I z j ∫dx / (z^2 + x^2) from 0 to L/2

This integral can be evaluated using a substitution u = z/x, which gives:

B = μ0/4π I z j ∫du / (1 + u^2) from z/L to ∞

The limits of integration have been changed to account for the substitution. The integral can be evaluated using the arctangent function:

B = μ0/4π I z j [ arctan(u) ] from z