A $$non-uniform$$ magnetic field exists in the free space and given as B→$$=B0$$ℓ$$2x2k^$$, where $$B0$$ and ℓ are positive constants. A particle having positive charge ‘q’ and mass ‘m’ is projected with speed ‘ $$v0$$’ along positive $$x-axis$$ from the origin. The maximum distance of the charged particle from the $$y-$$ axis before it turns back due to the magnetic field is D. (Ignore$$ any interaction other than magnetic $$field). If $$D=($$αmℓ$$2v0qB0)1/$$β, where α, β are positive integers then find the value of α times β ?

Question

A $$non-uniform$$ magnetic  field  exists in  the free space  and  given  as  B→$$=B0$$ℓ$$2x2k^$$, where   $$B0$$  and  ℓ   are  positive constants.  A  particle  having  positive  charge  ‘q’  and  mass  ‘m’  is  projected  with  speed  ‘ $$v0$$’  along positive $$x-axis$$ from the origin. The maximum distance of the charged particle from the $$y-$$ axis  before  it  turns  back  due  to  the  magnetic  field is D.  (Ignore$$  any  interaction  other  than  magnetic $$field). If  $$D=($$αmℓ$$2v0qB0)1/$$β, where  α,  β are positive integers then find the value of  α times  β ?

Infinity Learn · Accepted Answer

Since, magnetic filed and velocity are perpendicular, speed of particle will never change. At the moment of turning back, the direction of velocity becomes parallel to y axis.F→$$=qv$$→×B→$$=q(vxi^+vyj^)(B0$$ℓ$$2x2k^)$$​⇒F→$$=(qvxB0$$ℓ$$2x2j^)+(qvyB0$$ℓ$$2x2i^)=Fyj^+Fxi^$$​​⇒$$Fy=qB0$$ℓ$$2x2vx$$ ​⇒$$mdvydt=qB0$$ℓ$$2x2dxdt$$ ​⇒∫$$0v0dvy=qB0m$$ℓ$$2$$∫$$0xmaxx2dx$$​⇒$$v0=qB0m$$ℓ$$2$$  $$xmax33$$​⇒$$xmax=(3m$$ℓ$$2v0qB0)1/3$$∴α$$=3$$, β$$=3So$$, α times β$$=9$$

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