A magnetic field B is confined to a region r ≤ a and points out of the paper (the z-axis), r = 0 being the centre of the circular region. A charged ring (charge = Q) of radius b, b> a and mass m lies in the x-y plane with its centre at the origin. The ring is free to rotate and is at rest. The magnetic field is brought to zero in time Δt. Find the angular velocity ω of the ring after the field vanishes.

<p><span data-teams="true">This is a long answer type question as classified in NCERT Exemplar</span></p><p>Since, the magnetic field is brought to zero in time? t, the magnetic flux linked with the ring also reduces from maximum to zero. This, in turn, induces an emf in ring by the</p><p>phenomenon of EMI. The induces emf causes the electric field E generation around the</p><p>ring.</p><p>The induced emf= electric field E (2 πb) (Because V =E d&nbsp; ) . (i)</p><p>By Faraday'slaw of EMI</p><p>The induced emf= rate of change of magnetic flux</p><p>=rate of change of magnetic field × area= <span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mi>B</mi> <mi>π</mi> <msup> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mrow> </mfrac> </math> </span>……. (ii)</p><p>From (i) and (ii)</p><p>2 <span contenteditable="false"> <math> <mi>π</mi> <mi>b</mi> <mi>E</mi> </math> </span>=emf=<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mi>B</mi> <mi>π</mi> <msup> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mrow> </mfrac> </math> </span></p><p>Since, the charged ring experienced a electric force= QE</p><p>This force try to rotate the coil, and the torque is given by</p><p>Torque =&nbsp; b (Force)</p><p>= Qeb= Q {<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mi>B</mi> <mi>π</mi> <msup> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> <mi>π</mi> <mi>b</mi> <mi>d</mi> <mi>t</mi> </mrow> </mrow> </mfrac> </math> </span> }b</p><p>= Q<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mi>B</mi> <mi>π</mi> <msup> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> <mi>d</mi> <mi>t</mi> </mrow> </mrow> </mfrac> </math> </span></p><p>dL= torque (dt)= Q<span contenteditable="false"> <math> <mfrac> <mrow> <mrow> <mi>B</mi> <msup> <mrow> <mrow> <mi>a</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </math> </span></p><p>as intital momentum is zero</p><p>so final momentum is = mb²w= QBa²/2</p><p>w= QBa²/2mb²</p>

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