Dipoles in Non-Uniform Magnetic Fields
An ideal fixed dipole in a non-uniform field has force F = grad(m dot B) and torque m cross B. In a uniform field the gradient force vanishes, although torque may remain.
Why this shows up in the exam
Motion of magnetic needles near magnets · Magnetic material separation · Determining whether both force and torque occur
Learn the idea
A field gradient can pull a dipole while the field direction can also turn it. If one end of a magnetic dipole sits in a stronger field than the other, the opposite pole forces no longer cancel. The dipole can therefore translate and rotate at the same time.
🧠 Memory hook: A gradient pulls; a misalignment turns.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- F_vector = grad(m_vector dot B_vector) — force on a fixed ideal dipole in a non-uniform static field
- tau_vector = m_vector cross B_vector — simultaneous orienting torque
- F_z = m dB/dz — one-dimensional aligned-dipole form
How to approach it
- 1Check whether field magnitude or direction varies with position
- 2Test for misalignment to decide torque
- 3Use the gradient only after fixing the dipole orientation
Common slip-ups that cost marks
- •Claiming every magnetic field gives net force
- •Ignoring torque in a non-uniform field
- •Using F = mB without a spatial derivative
🌟 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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