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NEET PHYSICSMOVING CHARGES AND MAGNETISMMedium

Question

A metallic rod of mass per unit length of 0.5 kg m⁻¹ is lying horizontally on a smooth inclined plane which makes an angle of 30° with the horizontal. The rod is not allowed to slide down by flowing a current through it when a magnetic field of induction of 0.25 T is acting on it in the vertical direction. What is the current flowing through the rod to keep it stationary?

A

7.14 A

B

5.98 A

C

14.76 A

D

11.32 A

Step-by-Step Solution

  1. Force Analysis: The rod is in equilibrium on the inclined plane under the influence of three forces:
  • Gravity (mgmg) acting vertically downwards.
  • Normal reaction (NN) perpendicular to the plane.
  • Magnetic force (FmF_m) acting horizontally. Since the magnetic field B⃗\vec{B} is vertical and the current l⃗\vec{l} is horizontal, the force F⃗=I(l⃗×B⃗)\vec{F} = I(\vec{l} \times \vec{B}) is horizontal . The magnitude is Fm=IlBsin⁡(90∘)=IlBF_m = IlB\sin(90^\circ) = IlB.
  1. Equilibrium Condition: For the rod to remain stationary, the component of forces down the incline must balance the component up the incline.
  • Component of weight down the incline: mgsin⁡θmg \sin \theta .
  • Component of magnetic force up the incline: Since FmF_m is horizontal, its component along the incline is Fmcos⁡θF_m \cos \theta.
  1. Equation: mgsin⁡θ=Fmcos⁡θmg \sin \theta = F_m \cos \theta mgsin⁡θ=(IlB)cos⁡θmg \sin \theta = (IlB) \cos \theta
  2. Calculation: Rearranging for current II: I=mglBsin⁡θcos⁡θ=ml⋅gBtan⁡θI = \frac{mg}{lB} \frac{\sin \theta}{\cos \theta} = \frac{m}{l} \cdot \frac{g}{B} \tan \theta Given: ml=0.5 kg m−1\frac{m}{l} = 0.5 \text{ kg m}^{-1}, g=9.8 m s−2g = 9.8 \text{ m s}^{-2}, B=0.25 TB = 0.25 \text{ T}, θ=30∘\theta = 30^\circ. I=0.5×9.80.25×tan⁡30∘I = \frac{0.5 \times 9.8}{0.25} \times \tan 30^\circ I=4.90.25×0.5774I = \frac{4.9}{0.25} \times 0.5774 I=19.6×0.5774≈11.32 AI = 19.6 \times 0.5774 \approx 11.32 \text{ A}

Exam Context & Concepts Covered

This question aligns with the NEET PHYSICS syllabus, specifically targeting concepts from MOVING CHARGES AND MAGNETISM. Mastering this topic is crucial for scoring well in the upcoming medical entrance examinations. Solving conceptually related problems will help you understand the nuances of these concepts and improve your problem-solving speed.

PHYSICSMOVING CHARGES AND MAGNETISMmetalliclengthhorizontallysmoothinclined

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