Magnetic Potential Energy, Equilibrium, and Small Oscillations
For a dipole in uniform B, U = -m dot B = -mB cos theta. Near theta = 0, tau is approximately -mB theta, so a body of rotational inertia J has period 2pi sqrt(J/(mB)).
Why this shows up in the exam
Stable-versus-unstable orientation questions · Current-loop magnetic SHM · Work required to rotate a loop
Learn the idea
A magnetic dipole is stable parallel to B, unstable antiparallel, and can oscillate about stable alignment. The field creates an energy valley when the dipole points with it. A small displacement from the bottom produces a restoring torque, just like a torsional oscillator.
🧠 Memory hook: Parallel is the energy valley; antiparallel is the hilltop.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- U = -m B cos(theta) — dipole potential energy
- tau approximately -m B theta — small-angle restoring torque near stable alignment
- T = 2 pi sqrt(J/(m B)) — small-oscillation period about stable equilibrium
How to approach it
- 1Write U(theta)
- 2Classify extrema by nearby energy
- 3For small oscillations identify J and use the linear torque
Common slip-ups that cost marks
- •Calling zero torque automatically stable
- •Using loop mass instead of rotational inertia
- •Applying the SHM period for large angles
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A charge of 2 microC moves perpendicular to a 3 T magnetic field at 4 x 10^5 m/s. Find the magnetic force.
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