Physics · Magnetism And Matter · NEET
Stable is at theta = 0 (m parallel to B). At 0 degrees the potential energy U = -mB cos(0) = -mB is the LOWEST possible value. Nature always prefers lowest energy, so the magnet settles here and returns if disturbed. Unstable is at theta = 180 degrees (m anti-parallel to B), where U = +mB is highest.
Torque tau = mB sin theta is zero at theta = 0 AND theta = 180, so torque alone cannot decide. Use ENERGY. Compute U = -mB cos theta. If U is MINIMUM it is stable; if U is MAXIMUM it is unstable. A quick test: give a tiny nudge. At 0 degrees the torque that appears pushes m back toward B (restoring) = stable. At 180 degrees the torque pushes m further away = unstable.
U = -mB cos theta. The cos theta term is largest (+1) at theta = 0, and because of the minus sign, that makes U most negative (-mB), i.e. minimum. A system is stable at its lowest energy point, just like a ball rests at the bottom of a valley. To move the magnet away from theta = 0 you must DO work, so it resists = stable.
No. At theta = 90 degrees torque tau = mB sin(90) = mB is MAXIMUM, not zero. Equilibrium needs zero net torque, which happens only at theta = 0 and theta = 180. At 90 degrees the dipole feels the strongest turning effect and immediately rotates toward theta = 0.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
theta = 0 (m parallel to B) which is stable, and theta = 180 degrees (m anti-parallel to B) which is unstable. Both have zero torque because tau = mB sin theta = 0 at these angles.
Stable (theta = 0): U = -mB (minimum). Unstable (theta = 180 degrees): U = +mB (maximum). The energy difference between them is 2mB, which is the work needed to flip the magnet completely.
W = U_final - U_initial = (+mB) - (-mB) = 2mB. This is the maximum work possible in rotating the dipole, because you go from the lowest energy to the highest energy state.
A compass needle is a small magnetic dipole. It settles at theta = 0, the stable equilibrium with minimum potential energy, so its north pole points along Earth's magnetic field. Push it and it swings back and forth about this position.
This standard energy analysis (U = -mB cos theta) assumes a UNIFORM field, where the dipole feels torque but no net force. In a non-uniform field there is also a net translational force, which is covered in force vs torque on a magnet in uniform vs non-uniform field.
