Question

A direct current of density `j` flows along a round unifrom staright wire with cross-section radius `R`. Find the magnectic induction vector of this current at the point whose position relative to the axis of the wire is defined by a radius vector `r`. The magentic permeabilility is assumed to be equla to unity thoughhout all the space.

Answer 1

By circulation theorem inside the conductor
`B_(varphi) 2pi r = mu_(0) j_(x) pi r^(2)`, or `B_(varphi) = mu_(0) j_(x) r//2`
i.e., `vec(B) = (1)/(2) mu_(0) vec(j) xx vec(r)`
Similarly outside the conductor,
`B_(varphi) 2pi r = mu_(0) j_(x) pi R^(2)` or, `B_(varphi) = (1)/(2) mu_(0)j_(x) (R^(2))/(r)`
So, `vec(B) = (1)/(2) mu_(0) (vec(j) xx vec(r)) (R^(2))/(r^(2))`