Net Force on a Loop in a Nonuniform Field
The exact force is I closed_integral(dl cross B). For a small magnetic dipole in a slowly varying field, F is approximately grad(m dot B). Uniform-field cancellation does not apply when B varies across the loop.
Why this shows up in the exam
Loops near straight wires · Magnetic gradient forces · Comparing forces on near and far sides
Learn the idea
A current loop has zero net force only in a uniform field; field gradients can pull it. Opposite sides of a loop carry opposite currents. Equal fields make their forces cancel, but a nearby wire or varying field makes one side feel a stronger push.
🧠 Memory hook: Uniform fields turn loops; gradients can pull them.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- F = I closed_integral(dl cross B) — net force on a current loop in a general field
- F approximately grad(m dot B) — small-dipole approximation in a slowly varying field
- m = N I A n_hat — loop magnetic moment used in the dipole approximation
How to approach it
- 1Evaluate B at each relevant side
- 2Use I L cross B with direction
- 3Add forces and use the dipole approximation only when the loop is small
Common slip-ups that cost marks
- •Setting net force to zero without checking uniformity
- •Using one field value for a large loop
- •Adding opposite-side force magnitudes with the wrong signs
🌟 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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