We are looking for the B-field due to a single current loop

along the axis of the loop (lets take this as the +z axis).

Lets place our loop of radius R in the x-y plane and have its

center coincide with the origin of our coordinate system with

the current running counterclockwise when viewed from above.

Refer to the following figure.

Figure 4: Diagram for determining B-field due to loop current on

loop current axis.

The Biot-Savart law says that:

=04

2

This gives us the B-field due to a small segment of our current.

The direction has been drawn in the figure. In order to get the

net B-field at this point (at point (0,0,z)) we must integrate

over all segments of our current loop. Convince yourself that

doing so will result in a sort of CONE of B-field vectors at

point (0,0,z) all pointed radially out and up with the same

magnitude. Therefore, because each current segment along our

current loop will gives us a radially out and up pointed B-field

vector at point (0,0,z), the net horizontal B-field must be

zero. Thus, we are only left with a net B-field that points in

the positive z-direction. Let us continue with the calculation

keeping in mind that the net B-field must be in the z direction.

The dl is the arc length of a segment of our current, so this is

given by:

=

The direction of dl x is the direction of the B-field for our segment of current, but we know that all horizontal components

must be zero, so we set this direction equal to . The r in the Biot-Savart formula is the DISTANCE from our current segment to

the field point, so this is given by:

= 2 + 2

Finally, we want to pick out the component of the B-field that

is along the z direction, because this is the only non-zero

component, so:

= = 04

( )

2 + 2

But:

=

2 + 2

Therefore:

= = 04

2

2 + 2 32

Now, in order to get the net field we must integrate over all

our current segments (over all from 0 to 2pi). Therefore:

= 04

2

2 + 2 32

2

0

= 04

2

2 + 2 32

2

0

= 04

2

2 + 2 32

2

Finally:

=

+

Keep in mind that this expression holds only for points on the

axis of the current loop.