A uniform disc of radius r and mass m is charged uniformly with charge q This disc is placed flat on a rough horizontal surface having coefficient of friction on μ. A uniform magnetic field is present in a circular region (a$$$$r) but varying at $$kt3$$ as shown in fig. Then the time (in$$ $$second) after which the disc begin to rotate is $$2x$$. Find $$x=$$? Given $$r=1m$$,$$m=18kg$$,$$q=(1/3)c$$,μ$$=0.1$$,$$k=4(S$$.I.$$units)$$,$$g=10m/s2$$

Question

A uniform disc of radius r and mass m is charged uniformly with charge q  This disc is placed flat on a rough horizontal surface having coefficient of friction on μ.  A uniform magnetic field is present in a circular region  (a$$>$$r) but varying at  $$kt3$$ as shown in fig. Then the time (in$$ $$second) after which the disc begin to rotate is  $$2x$$. Find  $$x=$$? Given  $$r=1m$$,$$m=18kg$$,$$q=(1/3)c$$,μ$$=0.1$$,$$k=4(S$$.I.$$units)$$,$$g=10m/s2$$

Infinity Learn · Accepted Answer

Induced Electric field  ∫E→.dr→$$=d$$ϕdt ⇒$$E2$$$$π$$$$x=$$π$$x2dBdt$$ ⇒$$E=x2dBdt$$     ⇒$$E=3kxt22For$$ small elemental part  dτ$$=3kxt22$$×$$q2$$$$π$$xdx$$π$$$$r2$$×x⇒τ$$=3kt2qr2$$∫$$0rx3dx=3kqt2r24$$ Torque due to friction force  τ$$=$$∫$$0rd$$τ$$=$$∫$$0r$$μ$$m2$$$$π$$xdx$$π$$$$r2gx=23$$μmgrFor rotational equilibrium, both torques balances each $$other.3kqt2r24=23$$μmgr ⇒$$t=12sec$$

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