Example

Question:

(a) Magnetic field lines show the direction (at every point) along which a small magnetised needle aligns. Do the magnetic field lines also represent the lines of force on a moving charged particle at every point?
(b) If magnetic monopoles existed, how would the Gauss’s law of magnetism be modified?
(c) Does a bar magnet exert a torque on itself due to its own field? Does one element of a current-carrying wire exert a force on another element of the same wire?
(d) Magnetic field arises due to charges in motion. Can a system have magnetic moments even though its net charge is zero?

Solution:

(a) No. The magnetic force is always normal to \(\mathbf{B}\) (\(\mathbf{F} = q\mathbf{v} \times \mathbf{B}\)). It’s misleading to call magnetic field lines as lines of force.
(b) Gauss’s law of magnetism states the flux of \(\mathbf{B}\) through any closed surface is always zero: \[ \oint_S \mathbf{B} \cdot d\mathbf{s} = 0 \] If monopoles existed, right-hand side would equal monopole charge \(q_m\) enclosed by \(S\): \[ \oint_S \mathbf{B} \cdot d\mathbf{s} = \mu_0 q_m \] (c) No. There is no force or torque on an element due to the field produced by itself. But there is a force (or torque) on an element of the same wire (except for a straight wire, where force is zero).
(d) Yes. The average charge may be zero, but mean magnetic moment due to current loops may not be; for instance, in paramagnetic materials with zero net charge but nonzero net dipole moment.