A thin non conducting disc of mass M = 2 kg , charge Q = 2 × 10 - 2 C and radius R = 1 6 m is placed on a frictionless horizontal plane with its centre at the origin of the coordinate system. A non-un

Question

A thin non conducting disc of mass M = 2 kg , charge Q = 2 × 10 - 2 C and radius R = 1 6 m is placed on a frictionless horizontal plane with its centre at the origin of the coordinate system. A non-uniform, radial magnetic field B → = B 0 r ^ exists in space, where B 0 = 10 T and r ^ is a unit vector in the radially outward direction. The disc is set in motion with an angular velocity ω = x × 10 2 rad s - 1 , about an axis passing through its centre and perpendicular to its plane, as shown in the figure. At what value of x , the disc will lift off from the surface.

TestHub · Accepted Answer

The force an any small part of the disc is in the vertically upward direction

$d F = (\frac{Q}{π R^{2}} 2 π r d r) ω r B_{0}$

$d F = \frac{2 Q ω B_{0}}{R^{2}} r^{2} d r$

$F = \frac{2}{3} Q ω B_{0} R = M g$

$\Rightarrow ω = \frac{3 M g}{2 Q B_{0} R} = \frac{3 \times 2 \times 10}{2 \times 2 \times 10^{- 2} \times 10 \times \frac{1}{6}} = 9 \times 10^{2} r a d s^{- 1}$

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