Massless $$non-conducting$$ rod AB of length $$2l$$ is placed in uniform time varying magnetic field confined in a cylindrical region of radius (R$$ $$l) as shown in the figure. The center of the rod coincides with the centre of the cylindrical region. The rod can freely rotate in the plane of the figure about an axis coinciding with the axis of the cylinder. Two particles, each of mass m and charge q are attached to the ends A and B of the rod. The time varying magnetic field in this cylindrical region is given by $$B=B01-t2$$ where $$B0$$ is a constant. The field is switched on at time t $$=$$ $$0$$. Consider: $$B0$$ $$=100T$$, I $$=$$ $$4$$ cm, $$qm=4$$$$π$$$$100C/kg$$. Calculate the time (in$$ $$sec$$$$) in which the rod will reach position CD shown in the figure for the first time.

Question

Massless $$non-conducting$$ rod AB of length $$2l$$ is placed in uniform time varying magnetic field confined in a cylindrical region of radius (R$$ > $$l) as shown in the figure. The center of the rod coincides with the centre of the cylindrical region. The rod can freely rotate in the plane of the figure about an axis coinciding with the axis of the cylinder. Two particles, each of mass m and charge q are attached to the ends A and B of the rod. The time varying magnetic field in this cylindrical region is given by $$B=B01-t2$$ where $$B0$$ is a constant. The field is switched on at time t $$=$$ $$0$$. Consider: $$B0$$ $$=100T$$, I $$=$$ $$4$$ cm, $$qm=4$$$$π$$$$100C/kg$$. Calculate the time (in$$ $$sec$$$$) in which the rod will reach position CD shown in the figure for the first time.

Infinity Learn · Accepted Answer

The magnetic field is decreasing with time. Using Faraday's $$law(or$$ Lenz's $$law)$$ one can see that the emf induced in a closed path will be in clockwise sense. It means that the induced electric field is clockwise. Therefore, end A will reach point C. Magnitude of induced electric field $$isE=l2dBdt=l2B02=B0l4=B0$$×$$0.044=1V/m$$τ$$=2Eq$$×l∴α$$=2EqlI=2Eql2$$×$$ml2=Eqml$$∴θ$$=12$$α$$t2t2=2$$θα$$=2$$×π$$2$$α$$=$$π$$mlEq=$$πlE×$$1q/m=$$π×$$0.041$$×$$14$$$$π$$$$100$$∴  $$t2=1$$      ∴$$t=1s$$

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