Calculate magnetic induction at point O if the wire carrying
a current I has the shape shown in Fig.

(The radius of the curved part of the wire is equal to R and linear parts of the wire are very long.)

Answer

`-(mu_0I)/(4R) (sqrt2-1)hati=(mu_0I)/(8piR) hatk`

Explanation

Circuit segment shown in fig. can be considered in three
parts.
(a) A circular loop in y-z plane. Since this loop is made of
uniform wire, therefore magnetic induction at O due to it is
`B_1=0`.

(b) A straight part, parallel to x-axis. Magnetic induction due to
it is
`B_2=(mu_0I)/(4piR)(-hatk)=-(mu_0I)/(4piR)hatk`
(c) A straight part in y-z plane. Perpendicular distance of O
form axist of this straight part is `r=Rcos 45^@` as shown in figure.
Angles subtended by lines joining O and ends of this straight part with perpendicular drawn form O are `alpha=45^@` and
`beta=90^@`. Magnetic induction at O due to this part is
`B_3=(mu_0I)/(4piR)(sin alpha+sinbeta)`
`vecB_3=(mu_0I)/(4piR)(sqrt2-1)hati`
`vecB=vecB_1+vecB_2+vecB_3=-(mu_0I)/(4R)(sqrt2-1)hati-(mu_0I)/(8piR)hatk`