A particle of mass ‘m’ and charge ‘q’ is projected with a velocity $$v0$$ in a viscous medium, where a uniform and constant magnetic field of induction B exists everywhere in a direction perpendicular to the direction of projection of the particle. The force of viscous drag on the particle is given by the law f→$$=$$−bv→, where ‘b’ is a positive constant and v→ is velocity of the particle. The distance travelled by the particle during a time interval from the instant of projection until velocity vector turns by $$2$$π radians is $$mv0b(1$$−en$$π$$$$bqB)$$. Find ‘n’?

Question

A particle of mass ‘m’ and charge ‘q’ is projected with a velocity  $$v0$$ in a viscous medium, where a uniform and constant magnetic field of induction B  exists everywhere in a direction perpendicular to the direction of projection of the particle. The force of viscous drag on the particle is given by the law  f→$$=$$−bv→, where ‘b’ is a positive constant and  v→ is velocity of the particle. The distance travelled by the particle during a time interval from the instant of projection until velocity vector turns by $$2$$π radians is  $$mv0b(1$$−en$$π$$$$bqB)$$. Find ‘n’?

Infinity Learn · Accepted Answer

Along the tangent net force is ​$$F=$$−bv​⇒$$mdvdt=$$−bvDue to magnetic force, the particle move in circular path of radius R.​∴$$v=dldt=Rd$$θdt ⇒$$mdv=$$−bRdθ⇒$$dv=$$−bmmvqBdθ⇒ $$V0Vdvv=$$−bqB $$02$$$$π$$dθ⇒$$v=v0e$$−$$b2$$$$π$$qB​Now, again​$$F=$$−bv​⇒$$mvdvds=$$−bv⇒m∫$$v0vdv=$$−b∫$$0Sds$$​⇒mv−$$v0=$$−bSSo, distance covered $$S=mbv0$$−$$v=mv0b1$$−$$e2$$$$π$$bqB

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