A thin conducting rod of rectangular cross section and of length $' ℓ' = 2 \sqrt{5}$ 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 $' μ' = (\frac{1}{2})$ . 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 $m / \sec^{2}$ )

Question

A thin conducting rod of rectangular cross section and of length $' ℓ' = 2 \sqrt{5}$ 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 $' μ' = (\frac{1}{2})$ . 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 $m / \sec^{2}$ )

Infinity Learn · Accepted Answer

Let say magnetic field acts such that magnetic force F makes an angle θ with horizontal.Vertical :​ N=mg-FsinθHorizontal :​ fL=Fcosθ⇒μN=Fcosθ⇒μ(mg-Fsinθ)=Fcosθ⇒F=μmgμsinθ+cosθFor F to be minimum, (μsinθ+cosθ) should be maximum∴ddθ(μsinθ+cosθ)=0⇒μcosθ−sinθ=0⇒μ=sinθcosθ⇒sinθ=μ1+μ2 and cosθ=11+μ2 ∴Fmin=μmg1+μ2⇒iℓBmin=μmg1+μ2⇒Bmin=μmgiℓ1+μ2=(0.5)(40)(45)(52)=2T

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