Question

{"given":"A magnetic needle is initially parallel to a magnetic field. Work done W is required to turn it through 60°. We need to find the torque required to maintain the needle at this 60° position.","key_observation":"The work done in rotating a magnetic dipole in a magnetic field is given by W = -MB(cos θ₂ - cos θ₁). The torque on a magnetic dipole is τ = MB sin θ. We can use the work done to find MB, then calculate the torque at the final position.","option_analysis":[{"label":"(A)","text":"$\\sqrt{3}$ W","verdict":"correct","explanation":"Step 1: Calculate MB from work done:\n$$W = -MB(\\cos \\theta_2 - \\cos \\theta_1) = -MB(\\cos 60° - \\cos 0°)$$\n$$W = -MB(\\frac{1}{2} - 1) = MB \\cdot \\frac{1}{2}$$\n$$MB = 2W$$\nStep 2: Calculate torque at 60°:\n$$\\tau = MB \\sin 60° = 2W \\times \\frac{\\sqrt{3}}{2} = \\sqrt{3}W$$"},{"label":"(B)","text":"W","verdict":"incorrect","explanation":"This would be the torque if sin 60° = 1/2, but sin 60° = √3/2. The calculation gives τ = MB sin 60° = 2W × (√3/2) = √3 W, not W."},{"label":"(C)","text":"$\\frac{\\sqrt{3}}{2}$ W","verdict":"incorrect","explanation":"This represents the value of sin 60° times W, but doesn't account for the factor of 2 that comes from MB = 2W. The correct torque is τ = 2W × (√3/2) = √3 W."},{"label":"(D)","text":"2W","verdict":"incorrect","explanation":"This would be the torque if sin 60° = 1, but sin 60° = √3/2. The torque is τ = MB sin θ = 2W × (√3/2) = √3 W, not 2W."}],"answer":"(A)","formula_steps":[]}