A current `I` flows in a long straight wire with cross-section haviing the form of a thin half-ring of radius `R` (Fig). Find the induction of the magnitude field at the point `O`.

Explanation

First of all let us find out direction of vector `vec(B)` at points `O`. For this purpose, we divide the entire conductor into elementary fragments wtih currnet `di`. It is obvius that the sun of any two symmetric frnagments gives a resultant along `vec(B)` shown in the figure and consequently, vector `vec(B)` will also be directed as shwon
So, `|vec(B)| = int dB sin varphi` ....(1)
`= int (mu_(0))/(2pi R) di sin varphi`
`= int_(0)^(n) (mu_(0))/(2pi^(2) R) sin varphi d varphi, ("as " di = (i)/(pi) d varphi)`
Hence `B = mu_(0) i//pi^(2) R`