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Q.

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

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answer is 6.

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Detailed Solution

Induced Electric field  ∫E→.dr→=dϕdt ⇒E2πx=πx2dBdt ⇒E=x2dBdt     ⇒E=3kxt22For small elemental part  dτ=3kxt22×q2πxdxπr2×x⇒τ=3kt2qr2∫0rx3dx=3kqt2r24 Torque due to friction force  τ=∫0rdτ=∫0rμm2πxdxπr2gx=23μmgrFor rotational equilibrium, both torques balances each other.3kqt2r24=23μmgr ⇒t=12sec
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