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A long solid aluminum cylinder of radiusa = 5.0 cm rotates aboutits axis in unidrommagneticfield with induction B = 10 mT. Theangluarvelocityof rotationequlas omega = 45 rad//s withomega uarr uarr B Neglecting the magneticfield of appearingchagres, find theirspcaeand surfaface densities. |
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Answer» Solution :CHOOSE `vec(omega) uarr uarr vec(B)` alongthe z-axisand choose`vec(r)`, as the cylindrical polarradius of a reference point(perpendiculardistance fromthe AXIS). This pointhas the velocity. `vec(v) = vec(omega) XX vec(r)`, and experiences a `(vec(v) xx vec(B))` FORCE, which must becounterbalancedby an electric FIELD, `vec(E) = -(vec(omega) xx vec(r)) xx vec(B) = -(vec(omega). vec(B)) vec(r)`. There must appear a spacecharge density, `rho = epsilon_(0) div vec(E) = -3 epsilon_(0) vec(omega) vec(B) = -8 pC//m^(3)` Since the cylinder, as a wholeis electrically neutralthe surfaceof the cylinder must acquirea 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)` |
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