According to the LAW OF BIOT &
SAVART,
the current element shown in the Figure contributes

to the magnetic field at the "test point" shown, where

is the distance from the current element to the test point,

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

but this would leave us with integrals that cannot be solved by inspection.

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

and since

giving

we can write Eq. (1) as

or

where .

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):

(note that

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 ():

and very close to the rod ():

The last result can be used as the field due to an

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.

Jess H. Brewer - Last modified: Mon Nov 16 17:56:05 PST 2015