The magentic induction in vacumm at a plaen surface of a magentic is equal to `B` and the vector `B` forms an angle `theta` with the normal `n` of the surface (Fig). The permeability of the magnetic is equal to `mu`. Find:
(a) the flux of the vector `H` through the spherical surface `S` of radius `R`, whose centre lies on the surface of the magentic,
(b) the circulation of the vector `B` around the square path `I'` widh side `l` located as shwon in the fig.

Explanation

(a) `oint vec(H) .d. vec(S) = oint ((vec(B))/(mu_(0)) - vec(J)).d vec(S) = -oint vec(J). d. vec(S)`, since `oint vec(B).d.vec(S) = 0`
Now `vec(J)` is nonvanishing only in the bottom half of the sphere.

Here, `B'_(n) = B cos theta, H'_(t) = (1)/(mu_(0)) B sin theta, B'_(t) = mu B sin theta, H'_(n) = (B)/(mu mu_(0)) cos theta`
`j_(n) = (B cos theta)/(mu_(0)) (1 - (1)/(mu))` and `J_(t) = (mu - 1)/(mu_(0)) B sin theta`.
Only `J_(n)` contributes the surface intergal and
`oint vec(J) . d. vec(S) = -oint_(lower) vec(J).d.vec(S) = oint_(lower) vec(J_(n)) dS = (pi R^(2) B cos theta)/(mu_(0)) (1 - (1)/(mu))`
(b) `oint_(r) vec(B).d.vec(r) = (B_(t) - B'_(t)) l = (1 - mu) B l sin theta`