A long solid aluminum cylinder of radius `a = 5.0 cm` rotates about its axis in unidrom magnetic field with induction `B = 10 mT`. The angluar velocity of rotation equlas `omega = 45 rad//s` with `omega uarr uarr B` Neglecting the magnetic field of appearing chagres, find their spcae and surfaface densities.

Explanation

Choose `vec(omega) uarr uarr vec(B)` along the z-axis and choose `vec(r)`, as the cylindrical polar radius of a reference point (perpendicular distance from the axis). This point has the velocity.
`vec(v) = vec(omega) xx vec(r)`,
and experiences a `(vec(v) xx vec(B))` force, which must be counterbalanced by an electric field,
`vec(E) = -(vec(omega) xx vec(r)) xx vec(B) = -(vec(omega). vec(B)) vec(r)`.
There must appear a space charge density,
`rho = epsilon_(0) div vec(E) = -3 epsilon_(0) vec(omega) vec(B) = -8 pC//m^(3)`
Since the cylinder, as a whole is electrically neutral the surface of the cylinder must acquire a positive charge of surface density,
`sigma = + (2 epsilon_(0) (vec(omega). vec(B)) pi a^(2))/(2pi a) = epsilon_(0) a vec(omega).vec(B) = +- 2 pC//m^(2)`