Demonstrate that the law if refraction of direct current lines at the boundary between two conducting media has the form `tan alpha_(1) sigma_(2)//sigma_(1)`, where `sigma_(1)` and `sigma_(2)` are the conductivies of the media, `alpha_(2)` and `alpha_(1)` are the angles between the current lines and the normal of the boundary surface.

Explanation

By charge conservation, current `j`, leaving the medium (1) must enter the medium (2) Thus
`j_(1) cos alpha_(1) = j_(2) cos alpha_(2)`
Another relation follows from
`E_(1t) - E_(2t)`,
which is a consequence of `oint vec(E) d vec(r) = 0`
Thus `(1)/(sigma_(1)) j_(i) sin alpha_(1) = (1)/(sigma_(2)) j_(2) sin alpha_(2)`
or, `(tan alpha_(1))/(sigma_(1)) = (tan alpha_(2))/(sigma_(2))`
or, `(tan alpha_(1))/(tan alpha_(2)) = (sigma_(1))/(sigma_(2))`