Calculate the magnetic field at point O for the current-carrying wire segment shown in Fig. The wire consists of two straight portions and a circular arc of radius a, which subtends an angle `theta`.

Explanation

The magnetic field at O due to the current in the straight
segments `A A'` and `C C'` is zero because `dvecsxxhatr=0` for these paths.
Because we can ignore segments `A A'` and `C C'`, this example is
categorized as an application of the
Biot -Savart law to the curved wire
segment AC.
Each length element `dvecs` along path
Ac is at the same distance a from O,
and the current in each contributes
a field element `dvecB`directed into the pate at O. Furthermore, at every
point on AC, `dvecs` is perpendicular
to `hatr`, hence, `|dvecsxxhatr|=ds`
From the above equation, find the magnitude of the field at O
due to the current in an element of length ds:
`dB=(mu_0)/(4pi) (Ids)/(a^2)`

Integrating this expression over the curved path AC, nothing that I
and a are constants:
`B=(mu_0I)/(4pia^2) int ds=(mu_0I)/(4pia^2)s`
From teh geometry, note that s 5 au and substitute:
`B=(mu_0I)/(4pia^2)(a theta)=(mu_0I)/(4pia) theta`
The above equation gives the magnitude of the magnetic field
at O. The direction of `vecB` is into the page at O because `dvecsxxhatr`
is into the page for every length element.