According to the LAW OF BIOT &
the current element shown in the Figure contributes
We choose to look at the rod and the point in their common plane. Thus the direction of can be seen by inspection to be into the page. In fact circulates around the rod in circular loops according to the Right Hand Rule.
To integrate equation (1) we need to convert all variables to match the differential (over which we integrate).
We could use Eq. (2) to express R in terms of
z (where r and h are constants) and use
If we want to solve this problem without reference to external aids (like tables of integrals), it is better to convert into angles and trigonometric functions as follows:
Equation (3) can be rewritten
Integrating this differential is trivial; we are left with just the difference between u at the limits of integration (the top and bottom of the rod):
This equation expresses a completely general solution to this problem.
Let's check to see what this gives for the field directly out from the
midpoint of the rod - i.e. for h = L/2:
Let's also check to see what we get (at the midpoint)
very far from the rod ():
Note that the general formula (10)
(and the right-hand rule to determine directions)
can be used to find the net from a circuit
composed of any arrangement of straight wire segments
carrying a current I, by the principle of superposition.
Note also, however, that the result is a vector sum.