A thin conducting rod of rectangular cross section and of length 'ℓ'$$=25$$ meter and mass ‘m’ $$=$$ $$4$$ kg lies on the horizontal table. It can translate along $$x-axis$$. Coefficient of friction between the rod and the table is 'μ'$$=12$$. If the current in the conductor is $$2$$ ampere, then the minimum magnitude of uniform constant magnetic field strength (in$$ $$Tesla) that should exist in space such that conducting rod just starts to translate along $$x-axis$$ is P .Value of $$P=$$ $$-------$$ (take$$ g $$=$$ $$10$$ $$m/sec2)

Question

A  thin conducting  rod  of  rectangular cross section and of length  'ℓ'$$=25$$  meter  and  mass ‘m’  $$=$$  $$4$$  kg  lies  on  the  horizontal  table.  It can translate along $$x-axis$$. Coefficient of friction between the  rod  and  the  table  is  'μ'$$=12$$. If the current in the conductor is $$2$$ ampere, then  the minimum magnitude of uniform constant magnetic field strength (in$$ $$Tesla) that should exist in space such that conducting rod just starts to translate along $$x-axis$$ is P .Value of $$P=$$  $$-------$$ (take$$ g  $$=$$ $$10$$  $$m/sec2)

Infinity Learn · Accepted Answer

Let say magnetic field acts such that magnetic force F makes an angle θ with horizontal.Vertical :​ $$N=mg-Fsin$$θHorizontal :​ $$fL=Fcos$$θ⇒μ$$N=Fcos$$θ⇒μ(mg-Fsin$$θ$$)=Fcos$$θ⇒$$F=$$μmgμ$$sin$$θ$$+$$$$cos$$θFor F to be minimum, $$($$μ$$sin$$θ$$+$$$$cos$$θ$$) should be maximum∴ddθ$$($$μ$$sin$$θ$$+$$$$cos$$θ$$)=0$$⇒μ$$cos$$θ−$$sin$$θ$$=0$$⇒μ$$=$$$$sin$$θ$$cos$$θ⇒$$sin$$θ$$=$$μ$$1+$$μ$$2$$ and $$cos$$θ$$=11+$$μ$$2$$ ∴$$Fmin=$$μ$$mg1+$$μ$$2$$⇒iℓ$$Bmin=$$μ$$mg1+$$μ$$2$$⇒$$Bmin=$$μmgiℓ$$1+$$μ$$2=(0.5)(40)(45)(52)=2T$$

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