Physics · Magnetism And Matter · NEET
The minus sign is a choice of reference, not an error. We fix U = 0 at θ = 90° (dipole perpendicular to B). When the dipole turns toward the field (θ < 90°), the field does positive work on it, so its stored energy drops below zero, giving negative U. Negative energy just means the dipole is in a bound, lower state than the θ = 90° reference. At θ = 0°, U = -mB is the most negative and most stable.
By convention (NCERT), the reference position is where the dipole is perpendicular to the field, because there the torque is maximum but the dipole is neither strongly pulled to align nor to anti-align. Setting U = 0 there makes the formula clean: U = -mB cosθ. If you shifted the zero to θ = 0°, every energy value would just change by a constant mB, but energy differences (which is all that matters physically) stay the same.
Torque τ = mB sinθ tells you how strongly the field twists the dipole at a given angle; it is a vector effect that is maximum at 90° and zero at 0° and 180°. Potential energy U = -mB cosθ tells you how much energy is stored at that angle; it is a scalar, minimum at 0° and maximum at 180°. Torque is the negative rate of change of U with angle: τ = -dU/dθ.
Not exactly — they are linked but different. U is the energy at one position. The work you must do to turn the dipole from angle θ1 to θ2 equals the change in energy: W = U2 - U1 = mB(cosθ1 - cosθ2). So work is a difference of two U values. That is why the next concept, work done to rotate a bar magnet, builds directly on this formula.
At θ = 0° (aligned): U = -mB cos0° = -mB, the minimum, stable equilibrium. At θ = 90° (perpendicular): U = -mB cos90° = 0, the reference. At θ = 180° (anti-aligned): U = -mB cos180° = +mB, the maximum, unstable equilibrium. Remember: cos runs +1 → 0 → -1, so U runs -mB → 0 → +mB.
A bar magnet is hung by a thin cotton thread in a uniform horizontal magnetic field and is in equilibrium. The energy required to rotate it by 60° is W. The torque required to keep the magnet in this new (60°) position is:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
U = -mB cosθ, which in vector form is U = -m·B, where m is the magnetic moment, B is the field, and θ is the angle between them.
At θ = 0°, when the dipole is aligned with the field. Then U = -mB, the lowest possible value, and this is the position of stable equilibrium.
The work done against the field torque τ = mB sinθ to rotate from θ = 90° (reference) to θ is U = ∫ mB sinθ' dθ' from 90° to θ = -mB cosθ. Integrating the torque gives the energy.
Joule (J). Since m is in A·m² and B is in tesla (T), the product mB has units A·m²·T = J.
It links three high-frequency ideas — torque, work to rotate, and stable/unstable equilibrium — so one clean formula U = -mB cosθ unlocks a whole cluster of Magnetism and Matter questions.
