A direct current `I = 10 A` a flows in a long straight round conductor. Find the magnetic flux through a half of wire's cross-section per one meter of its length.

Explanation

We need the flux through the shaded area.
Now by Ampere's theorem.
`B_(varphi) 2pi r = mu_(0) (I)/(pi R^(2)) pi r^(2)`
or, `B_(varphi) = (mu_(0))/(2pi) I (r)/(R^(2))`
The flux through the shaded region is,
`varphi_(1) = int_(0)^(R) 1.dr B_(varphi) (r)`
`= int_(0)^(R) dr (mu_(0))/(2pi) I (r)/(R^(2)) = (mu_(0))/(4pi) I`.